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1 Slide 1 / 39 Slide / 39 Graphing Linear Equations th Grade Slide 3 / 39 Table of Contents Vocabulary Review Tables Slope & y-intercept Defining Slope on the Coordinate Plane Tables and Slope Slope Formula Slope Intercept Form Rate of Change Proportional Relationships and Graphing Slope and Similar Triangles Parallel and Perpendicular Lines Solve Systems by Graphing Solve Systems by Substitution Solve Systems by Elimination Choosing Your Strategy Writing Systems to Model Situations Glossary Slide 5 / 39 click on the topic to go to that section Vocabulary words are identified with a dotted underline. Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number. (Click on the dotted underline.) Slide / 39 Links to PRCC sample questions Non-Calculator #3 Calculator # Non-Calculator #7 Non-Calculator #11 Non-Calculator #1 1 Vocab Word Slide / 39 Calculator # Calculator #7 Calculator # Calculator #9 Calculator #1 The charts have parts. Factor whole number that can divide into another number with no remainder. whole number that multiplies with another number to make a third number. Its meaning (s it is used in the lesson.) How many thirds are in 1 whole? How many fifths are in 1 whole? How many ninths are in 1 whole? The underline is linked to the glossary at the end of the Notebook. It can also be printed for a word wall is a factor of 15 3 Examples/ Counterexamples 3 x 5 = 15 3 and 5 are factors of R is not a factor of 1 ack to Instruction Link to return to the instructional page.

2 II III I IV Slide 7 / 39 Vocabulary Review Coordinate Plane: the two dimensional plane or flat surface that is created when the x-axis intersects with the y-axis. lso known as a coordinate graph and the Cartesian plane. Quadrant: any of the four regions created when the x-axis intersects the y-axis. They are usually numbered with Roman numerals. x-axis: horizontal number line that extends indefinitely in both directions from zero. (Right- positive Left- negative) Slide / 39 To graph an ordered pair, such as (, ), you start at the origin (, )and then go left or right on the x-axis depending on the first number and then up or down from there parallel to the y-axis y-axis: vertical number line that extends indefinitely in both directions from zero. (Up- positive Down- negative) Origin: the point where zero on the x-axis intersects zero on the y-axis. The coordinates of the origin are (,). Slide 9 / Slide / 39 So to graph (,), we would go to the right and up from there. (,) Linear Equation: ny equation whose graph is represented by a straight line One way to check this is to create a table of values. Slide 11 / 39 Slide 1 / 39 Tables Geometry Theorem: Through any two points in a plane there can be drawn only one line. Return to Table of Contents

3 Slide 13 / 39 Slide 1 / 39 Given y=3x+, we want to graph our equation to show all of the ordered pairs that make it true. So according to this theorem from Geometry, we need to find points. One way is to create a table of values. Let's consider the equation y= 3x +. We need to find pairs of x and y numbers that make equation true Slide 15 / 39 Slide 1 / 39 Let's find some values for y=3x+. Pick values for x and plug them into the equation,then solve for y. x 3(x)+ y (x,y) -3 3(-3)+ -7 (-3,-7) 3()+ (,) 3()+ (,) Now let's graph those points we just found. x 3(x)+ y (x,y) -3 3(-3)+ -7 (-3,-7) 3()+ (,) 3()+ (,) Notice anything about the points we just graphed? Slide 17 / 39 That's right! The points we graphed form a line. The theorem says we only needed points, so why did we graph 3 points? click for table Slide 1 / 39 Graph y = x+ The third point serves as a check x x x+ y y (x,y) ()+ (,) 3 (3)+ (3,) -1 (-1)+ (-1,) Now graph your points and draw the line. y x Click for graph -

4 Slide 19 / 39 Slide / 39 click for table x -(x)+1 y (x,y) -()+1 1 (,1) 3 -(3)+1-5 (3,-5) -1 -(-1)+1 3 (-1,3) Graph y = -x+1 Now graph your points and draw the line y - x click for table x ¾(x)-3 y (x,y) ¾()-3-3 (,-3) ¾()-3 (,) - ¾(-)-3 - (-,-) Graph y = ¾x-3 Now graph your points and draw the line y x Click for graph - - Click for graph Slide 1 / 39 Slide 1 () / 39 Recall that in the previous example that even though the number in front of x was a fraction, our answers were integers. Why? Discuss at your table. x ¾(x)-3 y (x,y) ¾()-3-3 (,-3) ¾()-3 (,) - ¾(-)-3 - (-,-) Recall that in the previous example that even though the number in front of x was a fraction, our answers were integers. x ¾(x)-3 y (x,y) ¾()-3-3 (,-3) ¾()-3 (,) Why? Discuss at your table. The x-values chosen - ¾(-)-3 are - (-,-) zero, the denominator and the opposite of the denominator. Slide / 39 Slide 3 / 39

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6 Slide 3 / 39 Slide 31 / solution is % bleach. Slide 3 / 39 Create a graph that represents all possible combinations of the number of liters of bleach, contained in the number of liters of the solution. To graph a line, pick two points on the coordinate Students type their answers here plane. line will be drawn through the points. 11 solution is % bleach. Slide 3 () / 39 Create a graph that represents all possible combinations of the number of liters of bleach, contained in the number of liters of the solution. To graph a line, pick two points on the coordinate Students type their answers here plane. line will be drawn through the points. Slide 33 / 39 Slide 3 / 39 The Equation of a Line You only need a few facts about a line to completely describe it: Slope & y-intercept on the Coordinate Plane Its y-intercept (where it crosses the y-axis) "b" Return to Table of Contents Its slope (how much it rises or falls) "m" y = mx + b

7 Slide 35 / 39 Consider this graph of the Cartesian Plane, also called a Coordinate Plane or XY-Plane. The y-intercept ("b")of a line is the point where the line intercepts the y-axis. In this case, the y-intercept of the line is +. Slide 3 / 39 The y-intercept Imagine trying to tell a person how to draw a line on the Cartesian Plane. - This is the ordered pair (,) Slide 37 / 39 1 What is the y-intercept of this line? Slide 37 () / 39 1 What is the y-intercept of this line? b = Slide 3 / What is the y-intercept of this line? Slide 3 () / What is the y-intercept of this line? b =

8 Slide 39 / 39 1 What is the y-intercept of this line? Slide 39 () / 39 1 What is the y-intercept of this line? b = Slide / What is the x-intercept of this line? Slide () / What is the x-intercept of this line? (-,) Slide 1 / 39 1 What is the x-intercept of this line? Slide 1 () / 39 1 What is the x-intercept of this line? (3,)

9 Slide / What is the x-intercept of this line? Slide () / What is the x-intercept of this line? (-,) Slide 3 / 39 1 The graph of the equation x + 3y = intersects the y-axis at the point whose coordinates are Slide 3 () / 39 1 The graph of the equation x + 3y = intersects the y-axis at the point whose coordinates are (,) (,) (,) (,) C (,1) C (,1) D (,) D (,) ) (,) From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11. From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11. Slide / 39 Slide 5 / 39 "Steepness" and "Position" of a Line Defining Slope on the Coordinate Plane Return to Table of Contents

10 Slide / 39 Consider this... Slide 7 / 39 The Slope of a Line n infinite number of lines can pass through the same location on the y-axis...they all have the same y-intercept. The lines all have a different slope. Slope is the steepness of a line. Examples of lines with a y-intercept of are shown on this graph. What's the difference between them (other than their color)? Compare the steepness of the lines on the right. Slope can also be thought of as the rate of change Slide / 39 The Slope of a Line Slide 9 / 39 The Slope of a Line run The red line has a positive slope, since the line rises from left to the right rise The orange line has a negative slope, since the line falls down from left to the right rise run Slide 5 / 39 The Slope of a Line Slide 51 / 39 The Slope of a Line The purple line has a slope of zero, since it doesn't rise at all as you go from left to right on the x-axis. The black line is a vertical line. It has an undefined slope, since it doesn't run at all as you go from the bottom to the top on the y-axis rise = undefined

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12 Slide 5 / 39 Slide 59 / 39 While we can quickly see if the slope of a line is positive, negative or zero...we also need to determine how much slope it has...we have to measure the slope of a line. Slide / 39 Measuring the Slope of a Line rise run The slope of the line is just the ratio of its rise over its run. The symbol for slope is "m". So the formula for slope is: slope = rise run Slide 1 / 39 Measuring the Slope of a Line rise run slope = rise run The slope is the same anywhere on a line, so it can be measured anywhere on the line. Slide / 39 Measuring the Slope of a Line - Keep in mind the direction: Up (+) Down (-) Right (+) Left (-) - - rise run For instance, in this case we measure the slope by using a run from x = to x = +: a run of. During that run, the line rises from y = to y = : a rise of. slope = rise run m = m = 3 Slide 3 / 39 Measuring the Slope of a Line run rise

13 ut we get the same result with a run from x = to x = +3: a run of 3. During that run, the line rises from y = to y = : a rise of. slope = rise run m = 3 Slide / 39 Measuring the Slope of a Line rise run ut we can also start at x = 3 and run to x = : a run of 3. During that run, the line rises from y = 3 to y = 7: a rise of. slope = rise run m = 3 Slide 5 / 39 Measuring the Slope of a Line rise run ut we can also start at x = - and run to x = : a run of. During that run, the line rises from y = - to y = : a rise of. slope = rise run m = m = 3 Slide / 39 Measuring the Slope of a Line - - rise run How is the slope different on this coordinate plane? The line rises, however the run goes left (negative). Therefore, it is said to have a negative slope slope = rise run m = - m = - 3 Slide 7 / 39 Measuring the Slope of a Line run rise *most often the negative sign is placed in the numerator Slide / 39 Slide 9 / 39

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15 Slide 7 / 39 Tables and Slope Return to Table of Contents x Slide 77 / 39 How can slope and the y-intercept be found within the table? Look for the change in the y-values Look for the change in the x-values Write as a ratio (simpified) - this will be the "slope" Determine the corresponding y-value to the x-value of - this will be the "yintercept" y Slide 7 / 39 Slide 79 / 39 Determine the slope and y-intercept from this table. x y x y = is the slope 5 is the y-intercept is the slope click to reveal answer - is the y-intercept Slide / 39 Slide 1 / 39

16 Slide / 39 Slide 3 / 39 Slide / 39 Slide 5 / 39 Slope Formula Return to Table of Contents Slide / 39 Slope is "the rise over the run"of a line. This idea of rise over run of a line on a graph is how we were able to determine the slope of a line. ut slope can be found in other ways than looking at a graph. Slide 7 / 39 Slope is the ratio of change in y (rise) divided by the change in x(run). rise run slope= = change in y change in x line has a constant ratio of change: constant increase constant decrease No change, just constant Or undefined slope

17 Slide / 39 Slide 9 / 39 nother pplication of the Definition of Slope Slope of 3/ Slope of a line is meant to measure how fast it is climbing or descending. 3 feet road might rise 1 foot for every feet of horizontal distance. feet feet 1 foot (The grade of this hill is 3/ =.15= 15%) 3 feet slope of -3/7 The ratio, 1/, which is called slope, is a measure of the steepness of the hill. Engineers call this use of slope grade. What do you think a grade of % means? 7 feet (The grade of this hill is 3/7 =.3= 3%) Slide 9 / 39 so we will define the slope of a line as: (Rise) vertical change between two point on the line slope = horizontal change between two point on the line (Run) Slide 91 / 39 Suppose point P = (x 1, y 1) and Q = (x, y ) are on the line whose slope we want to find. y P(x 1,y 1) Q(x,y ) Horizontal Change (x -x 1) The slope of line PQ= (y-y1) (x -x 1) Vertical Change (y -y 1) (x,y 1) x Slide 9 / 39 Slide 93 / 39 The vertical change between P and Q = y - y 1 The horizontal change = x - x 1 slope = y - y 1 x - x 1

18 Slide 9 / 39 Slide 95 / 39 Slide 9 / 39 Slide 97 / 39 Slide 9 / 39 Slope-Intercept Form y = mx + b Slide 99 / 39 Once you have identified the slope and y-intercept in an equation, it is easy to graph it! To graph y = 3x follow these steps: Plot the y-intercept, in this case (, 5) Use the simplified rise over run to plot the next point - in this case, from (, 5) go UP 3 units and RIGHT 1 unit to plot the next point. Connect the points. Return to Table of Contents

19 Try this...graph y = -x - 3 Slide / 39 Start at the y-intercept - plot it. From the y-intercept, use the slope "m" to plot the next point. How would you use rise over run to plot -? Connect the points. Slide 1 / 39 Try this...graph y = x + 1 (is this in y=mx + b form??) Start at the y-intercept - plot it. From the y-intercept, use the slope "m" to plot the next point. How would you use rise over run to plot it? Connect the points. Did you have different points plotted? Does it make a difference? click to reveal Did you have different points plotted? Does it make a difference? click to reveal Slide / 39 Try this...graph 5x + y = - (is this in y=mx + b form??) Start at the y-intercept - plot it. From the y-intercept, use the slope "m" to plot the next point. How would you use rise over run to plot it? Connect the points. click to reveal Slide 3 / 39 Position of a Line Did you have different points plotted? Does it make a difference? Slide / 39 Slide 5 / 39 What are the similarities and differences between the lines below? The lines were in the form of y = mx+b h(x)=x+ - - q(x)=x+ - - r(x)=x-1 s(x)=x-5

20 Slide / 39 Slide 7 / 39 What determines slope? Examine the following equations: h(x)=x+ - - q(x)=x+ - - r(x)=x-1 s(x)=x-5 y = x + 1 y = 3x + 1 y = -1/ x + 1 y = -x + 1 What do the equations have in common? What is different? So it is the b in y = mx + b that is responsible for the position of the line. Slide / 39 Slide 9 / 39 y=-3x+1 y=x+1 ny equation of the form y = mx + b gives a line where b is the y intercept m is the slope y= y=-1/x y=-7x+1 Slide 1 / 39 Slide 111 / 39 Click for an interactive web site to see how the position of the line changes as you change the slope and the y-intercept.

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22 Slide 11 / 39 Slide 119 / 39 Rate of Change Return to Table of Contents Slide / 39 Slope formula can be used to find the constant of change in a "real world" problem. When traveling on the highway, drivers will set the cruise control and travel at a constant speed this means that the distance traveled is a constant Distance increase. (miles) The graph at the right represents such a trip. The car passed mile-marker at 1 hour and milemarker 1 at 3 hours. Find the slope of the line and what it represents. m= 1 miles- miles = miles = 3 hours-1 hours hours miles hour (1,) Time (hours) (3,1) Slide 11 / 39 If a car passes mile-marker in hours and mile-marker in hours, how many miles per hour is the car traveling? The information above gives us the ordered pairs (,) and (,). Now find the rate of change. So the slope of the line is and the rate of change of the car is miles per hour. Slide 11 () / 39 Slide 1 / 39 If a car passes mile-marker in hours and mile-marker in hours, how many miles per hour is the car traveling? The information above gives us the ordered pairs (,) and (,). Now find the rate of change.

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24 Slide 19 / 39 Slide 13 / 39 1 Two different proportional relationships are represented by the equation and the table. Proportion Proportion y = 9x The rate of change in Proportion is then the rate of change to Proportion. 1.5 E more.5 F less C 5.5 D 3.5 Slide 13 () / 39 1 Two different proportional relationships are represented by the equation and the table. Proportion Proportion y = 9x ().5 (F) less The rate of change in Proportion is then the rate of change to Proportion E more F less Slide 131 / 39 pool cleaning service drained a full pool. The following table shows the number of hours it drained and the amount of water remaining in the pool at that time. Part Students type their answers here Plot the points that show the relationship between the number of hours elapsed and the number of gallons of water left in the pool. C 5.5 D 3.5 Select a place on the grid to plot each point. (Grid on next slide.) Slide 13 / 39 Slide 13 () / 39

25 Slide 133 / 39 3 Part (continued from previous question) The data suggests a linear relationship between the number of hours the pool had been draining and the number of gallons of water remaining in the pool. ssuming the relationship is linear, what does the rate of change represent in the context of this relationship. The number of gallons of water in the pool after 1 hour. The number of hours it took to drain 1 gallon of water. C The number of gallons drained each hour. D The number of gallons of water in the pool when it is full. Slide 133 () / 39 3 Part (continued from previous question) The data suggests a linear relationship between the number of hours the pool had been draining and the number of gallons of water remaining in the pool. ssuming the relationship is linear, what does the rate of change represent in the context of this relationship. The number of gallons of water in the pool C after 1 hour. The number of hours it took to drain 1 gallon of water. C The number of gallons drained each hour. D The number of gallons of water in the pool when it is full. Slide 13 / 39 Part C (continued from previous question) What does the y-intercept of the linear function repressent in the context of this relationship? The number of gallons in the pool after 1 hour. The number of hours it took to drain 1 gallon of water. C The number of gallons drained each hour. D The number of gallons of water in the pool when it is full. Slide 13 () / 39 Part C (continued from previous question) What does the y-intercept of the linear function repressent in the context of this relationship? The number of gallons in the pool after 1 hour. The number of hours it took to drain 1 gallon of water. D C The number of gallons drained each hour. D The number of gallons of water in the pool when it is full. Slide 135 / 39 5 Part D (continued from previous question) Which equation describes this relationship between the time elapsed and the number of gallons of water remaining in the pool? Slide 135 () / 39 5 Part D (continued from previous question) Which equation describes this relationship between the time elapsed and the number of gallons of water remaining in the pool? y = -x + 15, y = -x + 13, C y = -1,x + 13, D y = -1,x + 15, y = -x + 15, y = -x + 13, C y = -1,x + 13, D y = -1,x + 15,

26 Slide 13 / 39 Eric planted a seedling in his garden and recorded its height each week. The equation shown can be used to estimate the height h, in inches, of the seedling after w, weeks since Eric planted the seedling. Part : What does the slope of the graph of the equation represent? The height in inches, of the seedling after w weeks. The height in inches, of the seedling when Eric planted it. C The increases of height in inches, of the seedling each week. D The total increase in the height in inches, of the seedling after w weeks. Slide 13 () / 39 Eric planted a seedling in his garden and recorded its height each week. The equation shown can be used to estimate the height h, in inches, of the seedling after w, weeks since Eric planted the seedling. Part : What does the slope of the graph of the equation represent? C The height in inches, of the seedling after w weeks. The height in inches, of the seedling when Eric planted it. C The increases of height in inches, of the seedling each week. D The total increase in the height in inches, of the seedling after w weeks. Slide 137 / 39 7 Part (continued from previous question) Slide 137 () / 39 7 Part (continued from previous question) The equation estimates the height of the seedlings to be.5 inches after how many weeks? The equation estimates the height of the seedlings to be.5 inches after how many weeks? Slide 13 / 39 Slide 139 / 39 Pavers are being set around a birdbath. The figures below show the first three designs of the pattern. Proportional Relationships Return to Table of Contents Using tiles, build the first five designs that follow the pattern above. Record your results in a table.

27 Slide / 39 Slide 11 / 39 Design number Number of pavers 1 1 Do the coordinate pairs in your table represent a proportional relationship? Graph the data from the table on a coordinate plane. What will you label the x-axis? What will you label the y-axis? Slide 1 / 39 How many tiles will you need for the "n-th" design? Write an equation that would represent the total number of tiles required for any design level. Suppose the birdbath was replaced with two tiles...how would this change the pattern? How would this change the equation? Slide 1 () / 39 How many tiles will you need for the "n-th" design? Write an equation that would represent the total number of tiles required for any design t = n level. t each design level, the number of tiles would increase by two after multiplying by four. Therefore Suppose the the equation birdbath would was replaced be: with two tiles...how would this change the pattern? How would this change t = n the + equation? Slide 13 / 39 Graph both equations on the same coordinate plane. Discuss the similarities and differences in the graphs... Slide 1 / 39 Click for answer Number of of tiles tiles t=n t=n+ Slope & Similar Triangles Return to Table of Contents Design Design Level Level

28 Slide 15 / 39 Slide 1 / 39 Congruent triangles have the same shape and same size. Using the line as the hypotenuse, draw congruent right triangles. How do you know they are congruent? The vertical rise is the same as well as the horizontal run. The simplified ratio is the same as the absolute value of the slope. click to reveal example 1 Slide 17 / 39 Similar triangles have the same shape, however, they are not the same size. The corresponding sides are proportionate. Slide 1 / 39 Sketch two similar right triangles on the line below. Write the ratios to prove they are proportionate. click to reveal example Slide 19 / 39 Slide 15 / 39

29 Slide 151 / 39 Slide 15 / 39 Slide 153 / 39 Slide 15 / 39 Slide 155 / 39 Slide 15 / Line t and ΔEC and ΔFD are shown on the coordinate grid. Which statements are true? Select all that apply. The slope of C is equal to the slope of C. The slope of C is equal to the slope of D. C The slope of C is equal to the slope of line t. D The slope of line t is equal to y t x E The slope of line t is equal to F The slope of line t is equal to

30 75 Line t and ΔEC and ΔFD are shown on the coordinate grid. Which statements are true? Select all that apply. Slide 15 () / 39 The slope of C is equal to the slope of C. The slope of C is equal to the slope of D. C The slope of C is equal to the slope of line t. D The slope of line t is equal to E The slope of line t is equal to F The slope of line t is equal to Slide 15 / 39 y,, C, E Parallel and Perpendicular Lines Return to Table of Contents t x Time (hr.) Family Z Distance (mi.) from home Slope (m) = 7 y-intercept (b) = equation y = 7x Slide 157 / 39 Complete the items below each table. (Click boxes to reveal answers) Time (hr.) Family Distance (mi.) from home Slope (m) = 7 y-intercept (b) = equation y = 7x + If this data from both tables were graphed on the same coordinate plane, what would you notice? The lines at the right are parallel lines. Notice that their slopes are all the same. Slide 159 / 39 Parallel lines all have the slopes because if they change at different rates eventually they would intersect. This also works for vertical and horizontal lines h(x)=x+ q(x)=x+ - r(x)=x-1 s(x)=x Slide 1 / 39 Slide 11 / 39

31 Slide 1 / 39 Slide 13 / 39 Slide 1 / 39 Slide 15 / 39 Slide 1 / 39 Slide 17 / 39 In the diagram the lines form a right angle, when this happens lines are said to perpendicular. Look at their slopes. This time they are not the same instead they are opposite reciprocals h(x)=-3x-11 g(x)= 1 /3x-

32 Slide 1 / 39 Slide 19 / 39 ) y=x- is perpendicular to ) y=- 1 / 5x+1 is perpendicular to C) y-=- 1 / (x-3) is perpendicular to D) 5x-y= is perpendicular to E) y= 1 / x is perpendicular to F) y-9=-5(x-.) is perpendicular to G) y=-(x+) is perpendicular to Perpendicular Equation ank (Drag the equation to complete the statement.) x+y= y= 1 / 5x 1 / 5y=x- y=- 1 / 5x+9 y= 1 / x- y=x+1 y=- 1 / x-3 The rule of using opposite reciprocals will not work for Horizontal and Vertical Lines. Why? Discuss with your table. Slide 19 () / 39 Slide 17 / 39 The rule of using opposite reciprocals will not work for Horizontal and Vertical Lines. Why? Discuss with your Horizontal table. lines have a slope of zero. You can't take the opposite reciprocal of. ut the perpendicular line for a vertical line is a horizontal, and vice-versa. Slide 171 / 39 Slide 17 / 39 Systems Strategy One: Graphing Return to Table of Contents

33 Some vocabulary... Slide 173 / 39 "system" is two or more linear equations. The "solution" to a system is an ordered pair that will work in each equation. One way to find the solution is to graph the equations on the same coordinate plane and find the point of intersection. Slide 17 / 39 Consider this... Suppose you are walking to school. Your friend is 5 blocks ahead of you. You can walk two blocks per minute, your friend can walk one block per minute. How many minutes will it take for you to catch up with your friend? Slide 175 / 39 Slide 17 / 39 First, make a table to represent the problem. Time (min.) Friend's distance from your start (blocks) Your distance from your start (blocks) locks Next, plot the points on a graph Time (min.) Time (min. ) Friend's distance from your start (blocks) Your distance from your start(blocks) locks Slide 177 / 39 The point where they intersect is the solution to the system (5,) is the solution. In the context of the problem this means after 5 minutes, you will meet your friend at block. Slide 17 / 39 Solve the system of equations graphically. y = x -3 y = x Time (min.)

34 Slide 179 / 39 Slide 1 / 39 Solve the system of equations graphically. Solve the system of equations graphically. x + y = 3 x - y = 3x + y = 11 x - y = Write the equation for the green dashed line Slide 11 / 39 Solve using graphing move y = -3x-1 What is this point of intersection? (move the hand!) (-1, ) y = move x+ Write the equation for the blue solid line Slide 1 / 39 Now take the ordered pair we just found and substitute it into the equation to prove that it is a solution for both lines. y = -3x-1 (, ) -1 y = x+ Slide 13 / 39 Slide 1 / 39 Solve by Graphing Solve by Graphing y = x + 3 y = -x - 3 y= -3x + y= x -

35 Parallel lines do not intersect! click to reveal Slide 15 / 39 What's the problem here? y= x + y= x - Therefore there is no solution. No ordered pair that will work in OTH equations click to reveal ( ) x + y = 5 -x -x y = -x + 5 Slide 1 / 39 Solve by Graphing First - transform the equations into y = mx + b form (slope-intercept form) y = -x + y = -x + 5 Now graph the two transformed lines. x + y = 5 becomes y = -x + 5 Slide 17 / 39 What's the problem? y = -x becomes y = -x + 5 Slide 1 / 39 5 Solve the system by graphing. y = -x + y = x +1 The equations transform to the same line. So we have infinitely many solutions. (3,1) Click for multiple choice answers. (1,3) C (-1,3) click to reveal click to reveal D no solution Slide 19 / 39 Slide 19 / 39 Solve the system by graphing. y =.5x - 1 y = -.5x -1 (,-1) Click for multiple choice answers. 7 Solve the system by graphing. x + y = 3 x - y = (,) Click for multiple choice answers. (,) (.,.) C infinitely many C (, -1) D no solution D no solution

36 Slide 191 / 39 Solve the system by graphing. y = 3x + 3 y = 3x - 3 (,) Click for multiple choice answers. (3,3) Slide 19 / 39 9 Solve the system by graphing. y = 3x + y = 1x + 1 (3,) Click for multiple choice answers. (-3,-) C infinitely many C infinitely many D no solution D no solution Slide 193 / 39 9 On the accompanying set of axes, graph and label the following lines: y=5 x = - y = x+5 Calculate the area, in square units, of the triangle formed by the three points of intersection. 91 The equation of the line s is The equation of the line t is Slide 19 / 39 The equations of the lines s and t form a system of equations. The solution Students type of their equations answers here is located at Point P. From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11. Slide 19 () / 39 Slide 195 / The equation of the line s is The equation of *Note: the line This t question is should be practiced on the computer in the PRCC sample test The equations of so the that lines students s and see how t form to graph a system the two of equations. The solution Students type of their lines equations answers the herecomputer. is located at Point P. 9 The table shows two systems of linear equations. Indicate whether each system of the equations has no solution, one solution or infinitely many solutions by selecting the correct cell in the table. Select one cell per column. Students type their answers here

37 Slide 195 () / 39 9 The table shows two systems of linear equations. Indicate whether each system of the equations has no solution, one solution or infinitely many solutions by selecting the correct cell in the table. Select one cell per column. Students type their answers here Slide 19 / 39 Systems Strategy Two: Substitution Return to Table of Contents Slide 197 / 39 Solve the system of equations graphically. y = x +.1 y = -x - 1. NOTE Slide 19 / 39 Substitution Explanation Graphing can be inefficient or approximate. nother way to solve a system is to use substitution. Substitution allows you to create a one variable equation. Slide 199 / 39 Solve the system using substitution. Why was it difficult to solve this system by graphing? y = x +.1 y = -x - 1. y = -x start with one equation x +.1 = -x substitute x +.1 for y in equation +x -.1 +x -.1 3x = solve for x x = -.5 Substitute -.5 for x in either equation and solve for y. y = x +.1 y = ( -.5) +.1 y = 3. Since x = -.5 and y = 3., the solution is (-.5, 3.) CHECK: See if (-.5, 3.) satisfies the other equation. y = -x = -(-.5) - 1.? 3. = 5-1.? 3. = 3. Slide / 39 Solve the system using substitution. (*Note: Equations can be moved on the page to show substitution into the y of the second equation.) y = -x +1 ( ) -3 y + 3x = 1

38 Slide () / 39 Solve the system using substitution. (*Note: Equations can be moved on the page to show substitution into the y of the second equation.) -3 (-x + 1) + 3x = 1 y = -x +1 x - + 3x ( = 1 ) 9x - = 1-3 y + 3x = 1 9x = 3 x = 7 y = -(7) + 1 y = y = Slide 1 / 39 Solve the system using substitution. x = -5y - 39 ( ) x = -y - 3 (7, ) Slide 1 () / 39 Slide / 39 Solve the system using substitution. -y - 3 = -5y - 39 x = -y -3 x = -5y - 39 y - 3 = -39 x = -(-9) - 3( ) y = -3 x = 9-3 x = -y - 3 y = -9 x = (, -9) Examine each system of equations. Which variable would you choose to substitute? Why? y = x - 9. y = -x + 9 y = -3x 7x - y = y = x + 1 x = y + 1 Slide 3 / 39 Slide / Examine the system of equations. Which variable would you substitute? 9 Examine the system of equations. Which variable would you substitute? x + y = 5 y = - x y - = x y + x = x y x y

39 Slide 5 / Examine the system of equations. Which variable would you substitute? x - y = x + 3y = x y Slide 7 / 39 Your class of is going on a trip. There are four drivers and two types of vehicles, vans and cars. The vans seat six people, and the cars seat four people, including drivers. How many vans and cars does the class need for the trip? Let v = the number of vans and c = the number of cars Slide 9 / 39 Now solve this system using substitution. What happens? x + y = 5x + 5y = x + y = -solve the first equation for x x = - y 5( - y) + 5y = -substitute - y for x in nd equation 3-5y + 5y = -solve for y 3 = -FLSE! Since 3 = is a false statement, the system has no solution. Slide / 39 Sometimes you need to rewrite one of the equations so that you can use the substitution method. For example: The system: Is equivalent to: 3x -y = 5 y = 3x -5 x + 5y = - x + 5y = - Using substitution you now have: x + 5(3x-5) = - -solve for x x + 15x - 5 = - -distribute the 5 17x - 5 = - -combine x's 17x = 17 -at 5 to both sides x = 1 - divide by 17 Substitute x = 1 into one of the equations. (1) + 5y = - + 5y = - 5y = - y = - The ordered pair (1,-) satisfies both equations in the original system. 3x -y = 5 x + 5y = - 3(1) - (-) = 5 (1) + 5(-) = = 5 - = - - = - Slide / 39 Set up the system: Drivers: v + c = People: v + c = Solve the system by substitution. v + c = -solve the first equation for v. v = -c + -substitute -c + for v in the (-c + ) + c = second equation -c + + c = -solve for c -c + = -c = - c = 1 v + c = v + 1 = -substitute for c in the 1st equation v = 3 -solve for v Since c = 1 and v = 3, they should use 1 car and 3 vans. Check the solution in the equations: v + c = v + c = = (3) + (1) = = 1 + = = Slide / 39 Now solve this system using substitution. What happens? x + y = -3 x + y = - x + y = -3 - solve the first equation for x x = -3 - y (-3 - y) + y = - - sub y for x in nd equation - - y + y = - - solve for y - = - - TRUE! - there are infinitely many solutions

40 Slide 11 / 39 Slide 1 / 39 How can you quickly decide the number of solutions a system has? 9 3x - y = - y = 3x + 1 Different slopes No Infinitely Many Same slope; different y- intercept (Parallel Lines) Same slope; same y-intercept (Same Line) C 1 solution no solution infinitely many solutions Slide 13 / 39 Slide 1 / x + 3y = y = 1 x 3 9 y = x x -.5y = 1 solution 1 solution no solution no solution C infinitely many solutions C infinitely many solutions Slide 15 / 39 Slide 1 / x + y = 5 x + y = 1 y = x - 7 y = 3x + 1 solution 1 solution no solution no solution C infinitely many solutions C infinitely many solutions

41 Slide 17 / 39 Slide 1 / 39 1 Solve each system by substitution. y = x - 3 y = -x + 5 Click for multiple choice answers. (,9) (-,-9) Solve each system by substitution. y = x - y = - Click for multiple choice answers. (-,-) (-,) C (,1) D (1,) C (,-) D (,) Slide 19 / 39 Slide / 39 3 Solve each system by substitution. y + x = -1 y = x + 1 Click for multiple choice answers. (1,) Solve each system by substitution. x = -5y + 5 x = y - 7 Click for multiple choice answers. (,.5) (1,1) (5,) C (,-) C (,5) D (-,) D (,5) Slide 1 / 39 Slide / 39 5 Solve each system by substitution. y = -3x + 3 -y + x = 19 Click for multiple choice answers. (,5) (-7,5) C (,-3) Systems Strategy Three: Elimination D (,-5) Return to Table of Contents

42 Slide 3 / 39 Slide / 39 When both linear equations of a system are in Standard Form, x + y = C, you can solve the system using elimination. You can add or subtract the equations to eliminate a variable. How do you decide which variable to eliminate? First, look to see if one variable has the same or opposite coefficients. If so, eliminate that variable. Second, look for which coefficients have a simple least common multiple. Eliminate that variable. Slide 5 / 39 If the variables have the same coefficient, you can subtract the two equations to eliminate the variable. If the variables have opposite coefficients, you add the two equations to eliminate the variable. Sometimes, you need to multiply one, or both, equations by a number in order to create a common coefficient. Slide / 39 Solve by Elimination - Click on the terms to eliminate and they will disappear, then add the two equations together. ( 5x + y = -x - y = -3 ) Slide 7 / 39 Solve by Elimination - Click on the terms and they will disappear then add the two equations together. Slide / 39 Solve by Elimination - There are ways to complete this problem. See both examples. 3x + y = 15-3x( -3y = -1) Multiplication by -1 5x + y = 17 -x + y = - 5x + y = 17 -x + y = - Subtraction

43 Solve the system by elimination. x + 3y = 1 x - 3y = Slide 9 / 39 Slide 3 / 39 Solve each system by elimination. x + y = x - y = Pull Click for multiple choice answers. (5,1) (-5,-1) C (1,5) D no solution Slide 31 / 39 Slide 3 / 39 7 Solve each system by elimination. x + y = -5 x - y = -3 Click for multiple choice answers. (-,1) Solve each system by elimination. x + y = - 3x + y = - Click for (,) multiple choice answers. (-1,-) (3,5) C (-,-1) C (,) D infinitely many D (-,) Slide 33 / 39 Slide 3 / 39 9 Solve each system by elimination. x - y = 5 x - y = -7 Click for multiple choice answers. no solution 1 Solve each system by elimination. 3x + y = -5x + y = 3 Click for multiple choice answers. (,-7) (,11) (7,) C (-,-11) C (,7) D (11,-) D infinitely many

44 Slide 35 / 39 Slide 3 / 39 Examine each system of equations. Which variable would you choose to eliminate? What do you need to multiply each equation by? Sometimes, it is not possible to eliminate a variable by adding or subtracting the equations. When this is the case, you need to multiply one or both equations by a nonzero number in order to create a common coefficient. Then add or subtract the equations. x + 5y = -1 x + y = 3x + y = 1 5x - y = -39 3x + y = x - 3y = Slide 37 / 39 In order to eliminate the y, you need to multiply first. 3x + y = - 5x - y = 1 Multiply the second equation by so the coefficients are opposites. (5x - y = 1) x - y = 3 Now solve by adding the equations together. 3x + y = - + x - y = 3 13x = x = Solve for y, by substituting x = into one of the equations. 3x + y = - 3() + y = - + y = - y = -1 y = - So (,-) is the solution. Check: 3x + y = - 5x - y = 1 3() + (-) = - 5() - (-) = = - + = 1 - = - 1 = 1 Slide 39 / 39 Slide 3 / 39 Now solve the same system by eliminating x. What do you multiply the two equations by? 3x + y = - 5x - y = 1 Multiply the first equation by 5 and the second equation by 3 so the coefficients will be the same 5(3x + y = -) 3(5x - y = 1) 15x + y = -5 15x - y = 5 Now solve by subtracting the equations. 15x + y = -5-15x - y = 5 y = - y = - Solve for x, by substituting y = - into one of the equations. 3x + y = - 3x + (-) = - 3x + -1 = - 3x = x = So (,-) is the solution. Check: 3x + y = - 5x - y = 1 3() + (-) = - 5() - (-) = = - + = 1 - = - 1 = 1 Slide / Which variable can you eliminate with the least amount of work? 11 Which variable can you eliminate with the least amount of work? x y 9x + y = 15 -x + y = 3 x y 3x - 7y = - -x + 15y = 9

45 Slide 1 / Which variable can you eliminate with the least amount of work? Slide / What will you multiply the first equation by in order to solve this system using elimination? x y x - 3y = -7 x + y = 3 x + 5y = 3x - y = 37 Now solve it... Slide () / What will you multiply the first equation by in order to solve this system using elimination? Slide 3 / What will you multiply the first equation by in order to solve this system using elimination? x + 5y = 3x - y = 37 You'd multiply the first equation by. 5 (11, - ) Now solve it... 3x + y = -19 x - 1y = 19 Now solve it... Slide 3 () / What will you multiply the first equation by in order to solve this system using elimination? 3x + y = You'd -19 multiply x - 1y = 19 the first equation by. (-5,-) Now solve it... x + 3y = 3x + y = Slide / What will you multiply the first equation by in order to solve this system using elimination? Now solve it...

46 Slide () / What will you multiply the first equation by in order to solve this system using elimination? x + 3y = 3x + y You'd = multiply the first equation by -3. (-,) Now solve it... Slide 5 / 39 Systems Choose Your Strategy Return to Table of Contents Slide / 39 Set up the system: Slide 7 / 39 number of tickets sold: a + s = 9 money collected: 3a + s = 7 ltogether 9 tickets were sold for a basketball game. n adult ticket costs $3. student ticket costs $1. Ticket sales were $7. Let a = adults s = students First eliminate one variable. a + s = 9 - in both equations s has the same - (3a + s = 7) coefficient so you subtract the -a+ = -17 equations in order to eliminate it. a = 9 -solve for a Then, find the value of the eliminated variable. a + s = s = 9 -substitute 9 for a in 1st equation s = 3 -solve for s There were 9 adult tickets and 3 student tickets sold. (9, 3) Slide / 39 Check: a + s = 9 3a + s = = 9 3(9) + 3 = 7 9 = = 7 7 = 7 Slide 9 / piece of glass with an initial temperature of 99 º F is cooled at a rate of 3.5 º F/min. t the same time, a piece of copper with an initial temperature of º F is heated at a rate of.5º F/min. Let m = the number of minutes and t = the temperature in F. Which system models the given information? 11 Which method would you use to solve the system? C graphing substitution elimination t = m t = +.5m click for equations C t = m t = +.5m t = m t = +.5m t = m t = -.5m Now solve it... m = 1.5 t = 1.5 This means that in 1.5 minutes, the temperatures will both be 1.5º C. click for answer

47 Slide 9 () / Which method would you use to solve the system? C graphing substitution elimination Now solve it... t = m t = +.5m click for equations ) Substitution m = 1.5 t = 1.5 This means that in 1.5 minutes, the temperatures will both be 1.5º C. click for answer Slide 5 / What method would you choose to solve the system? s - 3t = t = -s -1 C graphing substitution elimination Slide 5 () / What method would you choose to solve the system? s - 3t = t = -s -1 C graphing substitution elimination ) Substitution Now solve the system! Click for multiple choice answers. 1 Slide 51 / 39 (, -) s - 3t = 1 (, ) C (, -) t = -s -1 1 D (-, ) Slide 51 () / 39 Slide 5 / 39 Now solve the system! Click for multiple choice answers. 1 (, -) s - 3t = 1 (, ) C (, -) t = -s What method would you choose to solve the system? C graphing substitution elimination y = 3x - 1 y = x 1 D (-, )

48 Slide 5 () / 39 Slide 53 / What method would you choose to solve the system? 1 Now solve it! C graphing substitution elimination y = 3x - 1 y = x ) substitution Click for multiple choice answers. (1, ) (-, -1) C (-1, ) y = 3x - 1 y = x D (-1, -) Slide 53 () / 39 Slide 5 / 39 1 Now solve it! 13 What method would you choose to solve the system? Click for multiple choice answers. (1, ) (-, -1) C (-1, ) y = 3x - 1 y = x D C graphing substitution elimination 3m - n = 1 3m - n = -1 D (-1, -) Slide 5 () / 39 Slide 55 / What method would you choose to solve the system? 1 Now solve it! C graphing substitution elimination 3m - n = 1 3m - n = -1 C) elimination Click for multiple choice answers. (-, -1) (-1, -1) C (-1, 1) 3m - n = 1 3m - n = -1 D (1, 1)

49 1 Now solve it! Slide 55 () / 39 Slide 5 / What method would you choose to solve the system? Click for multiple choice answers. (-, -1) (-1, -1) C (-1, 1) 3m - n = 1 3m - n = -1 C graphing substitution elimination y = -x y = -.5x + 3 D (1, 1) Slide 5 () / 39 Slide 57 / What method would you choose to solve the system? 1 Now solve it! Click for multiple choice answers. C graphing substitution elimination y = -x y = -.5x + 3 (-, 1) (, -) C (-, ) y = -x y = -.5x + 3 D (1, -) 1 Now solve it! Click for multiple choice answers. Slide 57 () / 39 Slide 5 / What method would you choose to solve the system? (-, 1) (, -) C (-, ) y = -x y = -.5x + 3 C C graphing substitution elimination x - y = x + 3y = 1 D (1, -)

50 Slide 5 () / 39 Slide 59 / What method would you choose to solve the system? 1 Now solve it! Click for multiple choice answers. C graphing substitution elimination x - y = x + 3y = 1 C (, 5) (-, 7) C (-, ) x - y = x + 3y = 1 D (, ) 1 Now solve it! Slide 59 () / 39 Slide / What method would you choose to solve the system? Click for multiple choice answers. (, 5) (-, 7) C (-, ) x - y = x + 3y = 1 D C graphing substitution elimination u = v 3u - 3v = 7 D (, ) Slide () / 39 Slide 1 / What method would you choose to solve the system? 13 Now solve it! C graphing substitution elimination u = v 3u - 3v = 7 Click for multiple choice answers. 7 (, ) (, ) 9 C (, 7) 9 u = v 3u - 3v = 7 D (7, 7 )

51 Slide 1 () / 39 Slide / Now solve it! Click for multiple choice answers. 7 (, ) (, ) 9 C (, 7) 9 u = v 3u - 3v = Choose a strategy and then answer the question. What is the value of the y-coordinate of the solution to the system of equations x y = 1 and x + y = 7? 1-1 C 3 D D (7, 7 ) From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11. Slide () / Choose a strategy and then answer the question. What is the value of the y-coordinate of the solution to the system of equations x y = 1 and x + y = 7? 1-1 C 3 D Slide 3 / system of equations is shown. Students type their answers here What is the solution (x,y) of the system of equations? x = y = From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11. Slide 3 () / system of equations is shown. Students type their answers here What is the solution (x,y) of the system of equations? x = y = (, -) Slide / Two lines are graphed on the same coordinate plane. The lines only intersect at the point (3,). Which of these systems of linear equations could represent the two lines? Select all that apply. D E C

52 Slide () / Two lines are graphed on the same coordinate plane. The lines only intersect at the point (3,). Which of these systems of linear equations could represent the two lines? Select all that apply. D, E Slide 5 / 39 Systems Modeling Situations C E Return to Table of Contents Slide / 39 Slide 7 / 39 group of 1 people is spending five days at a summer camp. The cook ordered 1 pounds of food for each adult and 9 pounds of food for each child. total of 1, pounds of food was ordered. Part : Using your work from part, find: (1) the total number of adults in the group Part : Write an equation or a system of equations that describes the above situation and define your variables. Pull Pull () the total number of children in the group From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11. Slide / 39 Slide 9 / 39 Tanisha and Rachel had lunch at the mall. Tanisha ordered three slices of pizza and two colas. Rachel ordered two slices of pizza and three colas. Tanisha s bill was $., and Rachel s bill was $5.5. What was the price of one slice of pizza? What was the price of one cola? Sharu has $.35 in nickels and dimes. If he has a total of thirty-two coins, how many of each coin does he have? Pull Pull From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11. From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11.

53 Slide 7 / 39 Slide 71 / 39 en had twice as many nickels as dimes. ltogether, en had $.. How many nickels and how many dimes did en have? Pull 13 Your class receives $15 for selling 5 packages of greeting cards and gift wrap. pack of cards costs $ and a pack of gift wrap costs $9 Set up a system and solve. How many packages of cards were sold? From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11. You will answer how many packages of gift wrap in the next question. Slide 7 / 39 Slide 73 / Your class receives $15 for selling 5 packages of greeting cards and gift wrap. pack of cards costs $ and a pack of gift wrap costs $9 Set up a system and solve. How many packages of gift wrap were sold? 13 The sum of two numbers is 7, and their difference is 15. What is the larger number? 1 31 C 3 D 3 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11. Slide 7 / 39 Slide 75 / Ramon rented a sprayer and a generator. On his first job, he used each piece of equipment for hours at a total cost of $9. On his second job, he used the sprayer for hours and the generator for hours at a total cost of $. What was the hourly cost for the sprayer? 13 What is true of the graphs of the two lines 3y - = -5x and 3x = y -1? no intersection intersect at (,-) C intersect at (-,) D are identical From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11.

54 Slide 7 / 39 Slide 77 / You have 15 coins in your pocket that are either quarters or nickels. They total $.75. Set up a system to solve. Which method will you use? (Solving it comes later...) You have 15 coins in your pocket that are either quarters or nickels. They total $.75. How many quarters do you have? graphing C substitution elimination Slide 7 / 39 Slide 79 / You have 15 coins in your pocket that are either quarters or nickels. They total $.75. How many nickels do you have? 1 Julia went to the movies and bought one jumbo popcorn and two chocolate chip cookies for $5.. Marvin went to the same movie and bought one jumbo popcorn and four chocolate chip cookies for $.. How much does one chocolate chip cookie cost? $.5 $.75 C $1. D $. From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11. Slide / 39 Slide 1 / Mary and my had a total of yards of material from which to make costumes. Mary used three times more material to make her costume than my used, and yards of material was not used. How many yards of material did my use for her costume? 1 The tickets for a dance recital cost $5. for adults and $. for children. If the total number of tickets sold was 95 and the total amount collected was $, how many adult tickets were sold? From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11. From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 11.

55 Slide / school is selling t-shirts and sweatshirts for a fundraiser. The table shows the number of t-shirts and the number of sweatshirts in each of the three recent orders. The total cost of and are given. Each t-shirt has the same cost, and each sweatshirt has the same cost. Slide () / school is selling t-shirts and sweatshirts for a fundraiser. The table shows the number of t-shirts and the number of sweatshirts in each of the three recent orders. The total cost of and are given. Each t-shirt has the same cost, and each sweatshirt has the same cost. 19 The system of equations shown can be used to represent the situation. x + y = 3 Part : What is the total cost of 1 t-shirt and 1 sweatshirt? { 3x + y = 35 Slide 3 / 39 1 Part (continued from previous question) Select the choices to correctly complete the following statement. In the system of equations, x represents and y represents. (Type in for x first, then for y.) the number of t-shirts in the order the number of sweatshirts in the order C the cost, in dollars, of each t-shirt D the cost, in dollars, of each sweatshirt { x + y = 3 3x + y = 35 Slide / Part C (continued from previous question) { Students type their answers here If the system of equations is graphed in a coordinate plane, what are the coordinates (x, y) of the intersection of two lines? (, ) x + y = 3 3x + y = 35 The system of equations shown can be used to represent the situation. x + y = 3 Part : What is the total cost of 1 t-shirt and 1 sweatshirt? { 3x + y = 35 Slide 3 () / 39 1 Part (continued from previous question) Select the choices to correctly complete the following statement. In the system of equations, x represents and y represents. (Type in for x first, then for y.) { x + y = 3 3x + y = 35 the number of t-shirts in the order the number of sweatshirts in the order C the cost, in dollars, of each t-shirt D the cost, in dollars, of each sweatshirt C = x value D = y value Slide () / Part C (continued from previous question) { Students type their answers here If the system of equations is graphed in a coordinate plane, what are the coordinates (x, y) of the intersection of two lines? (, 11) (, ) x + y = 3 3x + y = 35

56 Slide 5 / 39 1 Part D (continued from previous question) { Students type their answers here x + y = 3 3x + y = 35 What is the total cost in dollars, of order C? $ Slide 5 () / 39 1 Part D (continued from previous question) { Students type their answers here What is the total cost in dollars, of order C? $ x + y = 3 3x + y = 35 3 Slide / 39 Slide 7 / 39 Coordinate Plane Glossary Slide / 39 Elimination Return to Table of Contents The two dimensional plane or flat surface that is created when the x-axis intersects with the y-axis. a.k.a. Cartesian Plane or Coordinate Graph Slide 9 / 39 Geometry Theorem Plot lines and points! ack to Instruction The process of eliminating one of the variables in a system of equations. Through any two points in a plane there can be drawn only one line. System: x - 3y = - x + y = Eliminate the y variable x - 3y = - (3)( x + y ) = (3) x - 3y = - +( 1x + 3y = 7) 1x = 7 x = 5 (5) - 3y = - - 3y = - -3y = -1 y = : (5,) ack to Instruction ack to Instruction

57 Slide 9 / 39 Grade Slide 91 / 39 Linear Equation unit engineers use to measure the steepness of a hill. ny equation whose graph is a line. feet 5 feet = 5 grade of hill is. 3 meters 5 meters grade of hill is 3 5. The sign warns cars the hill has a grade of 7. ack to Instruction slope intercept form: y = mx + b where "b" is the line's y-intercept and "m" is its slope. point slope form: y - y 1 = m(x - x 1) where "(x 1,y 1)" is a point on the line and "m" is its slope. standard form: ax + by = c where a is nonnegative and a and b cannot both be. ack to Instruction Slide 9 / 39 Origin Slide 93 / 39 Parallel The point where zero on the x-axis intersects zero on the y-axis. The point (,). Two lines that have the same slope and never interesent. Used to graph coordinates! right from origin (,) down 3 from origin (,-3) ack to Instruction ack to Instruction Slide 9 / 39 Perpendicular Slide 95 / 39 Proportional Relationship Two lines that interset and form a right angle. When two quantities have the same relative size. Right ngle 1 if weight is proportional to age, then a weight of 3kg on the 1st day means it will weigh kg on the nd day, 9kg on the 3rd day, 3kg on the th day, etc. ack to Instruction ack to Instruction

58 Slide 9 / 39 Quadrant Slide 97 / 39 Slope ny of the four regions created when the x-axis intersects the y-axis that are usually numbered with Roman numerals. How much a line rises or falls Steepness of a line The ratio of a line's rise over its run y = mx + b II I "m" = slope III IV formula for slope: m = y - y 1 x - x 1 "Steepness" and "Position" of a Line "First Quadrant" ack to ack to Instruction Instruction Slide 9 / 39 Slide 99 / 39 Substitution System The process of putting in a value in place of another. Two or more linear equations working together. System: x - 3y = - x + y = x - 3y = - y = - x x - 3( -x) = - x x = - 1x = 7 x = 5 y = - (5) y = Check: : (5,) (5) - 3() = = - - = - (5) + () = + = = ack to Instruction y = x -3 y = x - 1 3x + y = 11 x - y = x + y = 3 x - y = ack to Instruction Slide 3 / 39 System Slide 31 / 39 Table of Values n ordered pair that will work in each equation. Numbers or quantities arranged in rows and columns. system 3x + y = 11 x - y = x = + y 3( + y) = y = 11 y = -7 y = -7 x = + (-7/) x = -3 7/ (-7/, -3 7/) solution ack to Instruction columns rows x 1 3 y x 1 3 3(x)+7 3()+7 3(1)+7 3()+7 3(3)+7 y (x,y) (,7) (1,) (,13) (3,1) ack to Instruction