Graphing Linear Equations. 8th Grade

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2 Slide 2 / 309 Graphing Linear Equations 8th Grade

3 Slide 3 / 309 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 click on the topic to go to that section

4 Slide 4 / 309 Links to PARCC sample questions Non-Calculator #3 Calculator #4 Non-Calculator #7 Non-Calculator #11 Non-Calculator #16 Calculator #6 Calculator #7 Calculator #8 Calculator #9 Calculator #12

5 Slide 5 / 309 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. How many thirds are in 1 whole? (Click on the dotted underline.) 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.

6 1 Vocab Word Slide 6 / 309 The charts have 4 parts. Factor A whole number that can divide into another number with no remainder. A whole number that multiplies with another number to make a third number. 2 Its meaning (As it is used in the lesson.) R is a factor of 15 3 Examples/ Counterexamples 3 x 5 = 15 3 and 5 are factors of 15 3 is not a factor of 16 Back to Instruction 4 Link to return to the instructional page.

7 Slide 7 / 309 Vocabulary Review Coordinate Plane: the two dimensional plane or flat surface that is created when the x-axis intersects with the y-axis. Also known as a coordinate graph and the Cartesian plane. II I Quadrant: any of the four regions created when the x-axis intersects the y-axis. They are usually numbered with Roman numerals. III IV x-axis: horizontal number line that extends indefinitely in both directions from zero. (Right- positive Left- negative) 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 (0,0).

8 Slide 8 / 309 To graph an ordered pair, such as ( 4, 8), you start at the origin (0, 0)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

9 Slide 9 / 309 So to graph (4,8), we would go 4 to the right and up 8 from there (4,8)

10 Slide 10 / 309 Linear Equation: Any equation whose graph is represented by a straight line. One way to check this is to create a table of values.

11 Slide 11 / 309 Tables Return to Table of Contents

12 Slide 12 / 309 Geometry Theorem: Through any two points in a plane there can be drawn only one line.

13 Slide 13 / 309 Given y=3x+2, 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 2 points

14 Slide 14 / 309 One way is to create a table of values. Let's consider the equation y= 3x + 2. We need to find pairs of x and y numbers that make equation true

15 Let's find some values for y=3x+2. Pick values for x and plug them into the equation,then solve for y. Slide 15 / 309 x 3(x)+2 y (x,y) -3 3(-3)+2-7 (-3,-7) 0 3(0)+2 2 (0,2) 2 3(2)+2 8 (2,8)

16 Slide 16 / 309 Now let's graph those points we just found x 3(x)+2 y (x,y) (-3)+2-7 (-3,-7) 0 3(0)+2 2 (0,2) 2 3(2)+2 8 (2,8) Notice anything about the points we just graphed? -8-10

17 Slide 17 / 309 That's right! The points we graphed form a line. The theorem says we only needed 2 points, so why did we graph 3 points? The third point serves as a check

18 Slide 18 / 309 Graph y = 2x+4 click for table x x 2x+4 y y (x,y) 0 2(0)+4 4 (0,4) 3 2(3)+4 10 (3,10) -1 2(-1)+4 2 (-1,2) y 10 8 Now graph your points and draw the line x Click for graph -10

19 Slide 19 / 309 click for table Graph y = -2x+1 x -2(x)+1 y (x,y) 0-2(0)+1 1 (0,1) 3-2(3)+1-5 (3,-5) -1-2(-1)+1 3 (-1,3) Now graph your points and draw the line y x Click for graph

20 Slide 20 / 309 click for table x ¾(x)-3 y (x,y) 0 ¾(0)-3-3 (0,-3) 4 ¾(4)-3 0 (4,0) -4 ¾(-4)-3-6 (-4,-6) Graph y = ¾x-3 Now graph your points and draw the line y x Click for graph -10

21 Slide 21 / 309 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) 0 ¾(0)-3-3 (0,-3) 4 ¾(4)-3 0 (4,0) -4 ¾(-4)-3-6 (-4,-6)

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32 11 A solution is 20% bleach. Slide 32 / 309 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. A line will be drawn through the points. From PARCC sample test

33 Slide 33 / 309 Slope & y-intercept on the Coordinate Plane Return to Table of Contents

34 Slide 34 / 309 The Equation of a Line You only need a few facts about a line to completely describe it: Its y-intercept (where it crosses the y-axis) "b" Its slope (how much it rises or falls) "m" y = mx + b

35 Slide 35 / 309 Consider this graph of the Cartesian Plane, also called a Coordinate Plane or XY-Plane. Imagine trying to tell a person how to draw a line on the Cartesian Plane.

36 Slide 36 / 309 The y-intercept 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 This is the ordered pair (0,4)

37 Slide 37 / What is the y-intercept of this line?

38 Slide 38 / What is the y-intercept of this line?

39 Slide 39 / What is the y-intercept of this line?

40 Slide 40 / What is the x-intercept of this line?

41 Slide 41 / What is the x-intercept of this line?

42 Slide 42 / What is the x-intercept of this line?

43 Slide 43 / The graph of the equation x + 3y = 6 intersects the y-axis at the point whose coordinates are A (0,2) B (0,6) C (0,18) D (6,0) From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

44 Slide 44 / 309 Defining Slope on the Coordinate Plane Return to Table of Contents

45 Slide 45 / 309 "Steepness" and "Position" of a Line

46 Slide 46 / 309 Consider this... An infinite number of lines can pass through the same location on the y-axis...they all have the same y-intercept Examples of lines with a y-intercept of are shown on this graph. What's the difference between them (other than their color)?

47 Slide 47 / 309 The Slope of a Line The lines all have a different slope. Slope is the steepness of a line Compare the steepness of the lines on the right. Slope can also be thought of as the rate of change

48 Slide 48 / 309 The Slope of a Line 10 run The red line has a positive slope, since the line rises from left to the right. rise

49 Slide 49 / 309 The Slope of a Line 10 The orange line has a negative slope, since the line falls down from left to the right rise run

50 Slide 50 / 309 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

51 Slide 51 / 309 The Slope of a Line 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 0 = undefined

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60 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 60 / 309 Measuring the Slope of a Line rise run

61 Slide 61 / 309 Measuring the Slope of a Line 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 rise run

62 Slide 62 / 309 Measuring the Slope of a Line slope = rise run The slope is the same anywhere on a line, so it can be measured anywhere on the line. rise run Keep in mind the direction: Up (+) Down (-) Right (+) Left (-)

63 For instance, in this case we measure the slope by using a run from x = 0 to x = +6: a run of 6. During that run, the line rises from y = 0 to y = 8: a rise of 8. slope = rise run Slide 63 / 309 Measuring the Slope of a Line run rise m = 8 6 m =

64 But we get the same result with a run from x = 0 to x = +3: a run of 3. Slide 64 / 309 Measuring the Slope of a Line 10 8 During that run, the line rises from y = 0 to y = 4: a rise of 4. slope = rise run run rise m =

65 But we can also start at x = 3 and run to x = 6 : a run of 3. During that run, the line rises from y = 3 to y = 7: a rise of 4. Slide 65 / 309 Measuring the Slope of a Line rise run slope = rise run m =

66 But we can also start at x = -6 and run to x = 0: a run of 6. During that run, the line rises from y = -8 to y = 0: a rise of 8. slope = rise run Slide 66 / 309 Measuring the Slope of a Line rise -6 run m = 8 6 m =

67 How is the slope different on this coordinate plane? Slide 67 / 309 Measuring the Slope of a Line 10 The line rises 8, however the run goes left 6(negative). Therefore, it is said to have a negative slope run slope = rise run rise m = 8-6 m = *most often the negative sign is placed in the numerator

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76 Slide 76 / 309 Tables and Slope Return to Table of Contents

77 Slide 77 / 309 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 0 - this will be the "yintercept" x y

78 Slide 78 / 309 x y = 2 is the slope 5 is the y-intercept

79 Slide 79 / 309 Determine the slope and y-intercept from this table. x y is the slope click to reveal answer -4 is the y-intercept

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85 Slide 85 / 309 Slope Formula Return to Table of Contents

86 Slide 86 / 309 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. But slope can be found in other ways than looking at a graph.

87 Slide 87 / 309 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 A line has a constant ratio of change: A constant increase A constant decrease No change, just constant Or undefined slope

88 Slide 88 / 309 Another Application of the Definition of Slope Slope of a line is meant to measure how fast it is climbing or descending. A road might rise 1 foot for every 10 feet of horizontal distance. 10 feet 1 foot The ratio, 1/10, 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 4% means?

89 Slide 89 / 309 Slope of 3/20 3 feet 20 feet (The grade of this hill is 3/20 =.15= 15%) 3 feet slope of -3/7 7 feet (The grade of this hill is 3/7 =.43= 43%)

90 Slide 90 / 309 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)

91 Slide 91 / 309 Suppose point P = (x 1, y 1 ) and Q = (x 2, y 2 ) are on the line whose slope we want to find. y Q(x 2,y 2 ) Vertical Change (y 2 -y 1 ) P(x 1,y 1 ) Horizontal Change (x 2 -x 1 ) The slope of line PQ= (y 2-y 1 ) (x 2 -x 1 ) (x 2,y 1 ) x

92 Slide 92 / 309 The vertical change between P and Q = y 2 - y 1 The horizontal change = x 2 - x 1 slope = y 2 - y 1 x 2 - x 1

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98 Slide 98 / 309 Slope-Intercept Form y = mx + b Return to Table of Contents

99 Slide 99 / 309 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 (0, 5) Use the simplified rise over run to plot the next point - in this case, from (0, 5) go UP 3 units and RIGHT 1 unit to plot the next point. Connect the points.

100 Slide 100 / 309 Try this...graph y = -2x - 3 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 -2? Connect the points. Did you have different points plotted? Does it make a difference? click to reveal

101 Slide 101 / 309 Try this...graph 4y = x + 12 (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

102 Slide 102 / 309 Try this...graph 5x + y = -4 (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

103 Slide 103 / 309 Position of a Line

104 Slide 104 / 309 What are the similarities and differences between the lines below? h(x)=x+6 q(x)=x+2 r(x)=x s(x)=x

105 Slide 105 / 309 The lines were in the form of y = mx+b.

106 Slide 106 / h(x)=x+6 q(x)=x+2 r(x)=x s(x)=x So it is the b in y = mx + b that is responsible for the position of the line.

107 Slide 107 / 309 What determines slope? Examine the following equations: y = 2x + 1 y = 3x + 1 y = -1/2 x + 1 y = -x + 1 What do the equations have in common? What is different?

108 Slide 108 / y=-3x+1 y=x+1 2 y= y=-1/2x y=-7x+1

109 Slide 109 / 309 Any equation of the form y = mx + b gives a line where b is the y intercept m is the slope

110 Slide 110 / 309 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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119 Slide 119 / 309 Rate of Change Return to Table of Contents

120 Slide 120 / 309 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 increase. Distance (miles) The graph at the right represents such a trip. The car passed mile-marker 60 at 1 hour and milemarker 180 at 3 hours. Find the slope of the line and what it represents. m= 180 miles-60 miles = 120 miles = 3 hours-1 hours 2 hours 60 miles hour (1,60) Time (hours) (3,180) So the slope of the line is 60 and the rate of change of the car is 60 miles per hour.

121 Slide 121 / 309 If a car passes mile-marker 100 in 2 hours and mile-marker 200 in 4 hours, how many miles per hour is the car traveling? The information above gives us the ordered pairs (2,100) and (4,200). Now find the rate of change.

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130 Slide 130 / Two different proportional relationships are represented by the equation and the table. Proportion A Proportion B y = 9x The rate of change in Proportion A is then the rate of change to Proportion B. A 1.5 E more B 2.5 F less C 25.5 From PARCC sample test D 43.5

131 Slide 131 / A 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. Students type their answers here Part A Plot the points that show the relationship between the number of hours elapsed and the number of gallons of water left in the pool. Select a place on the grid to plot each point. (Grid on next slide.) From PARCC sample test

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133 Slide 133 / Part B (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. Assuming the relationship is linear, what does the rate of change represent in the context of this relationship. A The number of gallons of water in the pool after 1 hour. B 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. From PARCC sample test

134 Slide 134 / Part C (continued from previous question) What does the y-intercept of the linear function repressent in the context of this relationship? A The number of gallons in the pool after 1 hour. B 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. From PARCC sample test

135 Slide 135 / 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? A y = -600x + 15,000 B y = -600x + 13,2000 C y = -1,200x + 13,200 D y = -1,200x + 15,000 From PARCC sample test

136 Slide 136 / 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 A: What does the slope of the graph of the equation represent? A The height in inches, of the seedling after w weeks. B 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. From PARCC sample test

137 Slide 137 / Part B (continued from previous question) The equation estimates the height of the seedlings to be 8.25 inches after how many weeks? From PARCC sample test

138 Slide 138 / 309 Proportional Relationships Return to Table of Contents

139 Slide 139 / 309 Pavers are being set around a birdbath. The figures below show the first three designs of the pattern. Using tiles, build the first five designs that follow the pattern above. Record your results in a table.

140 Slide 140 / 309 Design number Number of pavers 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?

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142 Slide 142 / 309 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?

143 Slide 143 / 309 Graph both equations on the same coordinate plane. Discuss the similarities and differences in the graphs... Click for answer Number of of tiles tiles t=4n t=4n Design Design Level Level

144 Slide 144 / 309 Slope & Similar Triangles Return to Table of Contents

145 Slide 145 / 309 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? click to reveal example

146 Slide 146 / 309 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

147 Slide 147 / 309 Similar triangles have the same shape, however, they are not the same size. The corresponding sides are proportionate

148 Slide 148 / 309 Sketch two similar right triangles on the line below. Write the ratios to prove they are proportionate. click to reveal example

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156 75 Line t and ΔECA and ΔFDB are shown on the coordinate grid. Which statements are true? Select all that apply. A The slope of AC is equal to the slope of BC. B The slope of AC is equal to the slope of BD. C The slope of AC is equal to the slope of line t. D The slope of line t is equal to Slide 156 / 309 y t x E The slope of line t is equal to F The slope of line t is equal to From PARCC sample test

157 Slide 157 / 309 Complete the items below each table. (Click boxes to reveal answers) Family Z Family A Time (hr.) Distance (mi.) from home Time (hr.) Distance (mi.) from home Slope (m) = 70 y-intercept (b) = 0 equation y = 70x Slope (m) = 70 y-intercept (b) = 10 equation y = 70x + 10 If this data from both tables were graphed on the same coordinate plane, what would you notice?

158 Slide 158 / 309 Parallel and Perpendicular Lines Return to Table of Contents

159 Slide 159 / 309 The lines at the right are parallel lines. Notice that their slopes are all the same. 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+6 q(x)=x r(x)=x-1 s(x)=x

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167 Slide 167 / 309 In the diagram the 2 lines form a right angle, when this happens lines are said to perpendicular. h(x)=-3x-11 Look at their slopes. This time they are not the same instead they are opposite reciprocals g(x)= 1 / 3 x-2

168 Slide 168 / 309 A) y=4x-2 is perpendicular to B) y=- 1 / 5 x+1 is perpendicular to C) y-2=- 1 / 4 (x-3) is perpendicular to D) 5x-y=8 is perpendicular to E) y= 1 / 6 x is perpendicular to F) y-9=-5(x-.4) is perpendicular to G) y=-6(x+2) is perpendicular to Perpendicular Equation Bank (Drag the equation to complete the statement.) 6x+y=10 y= 1 / 5 x 1 / 5 y=x-2 y=- 1 / 5 x+9 y= 1 / 6 x-6 y=4x+1 y=- 1 / 4 x-3

169 Slide 169 / 309 The rule of using opposite reciprocals will not work for Horizontal and Vertical Lines. Why? Discuss with your table.

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172 Slide 172 / 309 Systems Strategy One: Graphing Return to Table of Contents

173 Some vocabulary... Slide 173 / 309 A "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.

174 Slide 174 / 309 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?

175 Slide 175 / 309 First, make a table to represent the problem. Time (min.) Friend's distance from your start (blocks) Your distance from your start (blocks)

176 Next, plot the points on a graph. Slide 176 / Time (min. ) Friend's distance from your start (blocks) Your distance from your start(blocks) Blocks Time (min.)

177 Slide 177 / 309 The point where they intersect is the solution to the system. 20 Blocks (5,10) is the solution. In the context of the problem this means after 5 minutes, you will meet your friend at block Time (min.) 15

178 Slide 178 / 309 Solve the system of equations graphically. y = 2x -3 y = x - 1 Solution

179 Slide 179 / 309 Solve the system of equations graphically. 2x + y = 3 x - 2y = 4 Solution

180 Slide 180 / 309 Solve the system of equations graphically. 3x + y = 11 x - 2y = 6 Solution

181 Write the equation for the green dashed line Slide 181 / 309 Solve using graphing move y = -3x-1 y = move 4x+6 Write the equation for the blue solid line What is this point of intersection? (move the hand!) (-1, 2)

182 Slide 182 / 309 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 2 y = 4x+6

183 Slide 183 / 309 Solve by Graphing y = 2x + 3 y = -4x - 3 Solution

184 Slide 184 / 309 Solve by Graphing y= -3x + 4 y= x - 4 Solution

185 Slide 185 / 309 What's the problem here? y= 2x + 4 y= 2x - 4 Parallel lines do not intersect! click to reveal Therefore there is no solution. No ordered pair that will work in BOTH equations click to reveal ( )

186 Slide 186 / 309 Solve by Graphing First - transform the equations into y = mx + b form (slope-intercept form) 2x + y = 5-2x -2x y = -2x + 5 2y = -4x y = -2x + 5 Now graph the two transformed lines.

187 Slide 187 / 309 What's the problem? 2x + y = 5 becomes y = -2x + 5 2y = 10-4x becomes y = -2x + 5 The equations transform to the same line. So we have infinitely many solutions. click to reveal click to reveal

188 Slide 188 / Solve the system by graphing. y = -x + 4 y = 2x +1 Solution A (3,1) Click for multiple choice answers. B (1,3) C (-1,3) D no solution

189 Slide 189 / Solve the system by graphing. y = 0.5x - 1 y = -0.5x -1 A (0,-1) Click for multiple choice answers. Solution B (0,0) C infinitely many D no solution

190 Slide 190 / Solve the system by graphing. 2x + y = 3 x - 2y = 4 A (2,4) Click for multiple choice answers. Solution B (0.4, 2.2) C (2, -1) D no solution

191 Slide 191 / Solve the system by graphing. y = 3x + 3 y = 3x - 3 Solution A (0,0) Click for multiple choice answers. B (3,3) C D infinitely many no solution

192 Slide 192 / Solve the system by graphing. y = 3x + 4 4y = 12x + 16 A (3,4) Click for multiple choice answers. B (-3,-4) C D infinitely many no solution

193 Slide 193 / On the accompanying set of axes, graph and label the following lines: y=5 x = - 4 y = x+5 Solution Calculate the area, in square units, of the triangle formed by the three points of intersection. From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

194 Slide 194 / The equation of the line s is The equation of the line t is 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 PARCC sample test

195 Slide 195 / 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 From PARCC sample test

196 Slide 196 / 309 Systems Strategy Two: Substitution Return to Table of Contents

197 Slide 197 / 309 Solve the system of equations graphically. y = x y = -2x NOTE

198 Slide 198 / 309 Substitution Explanation Graphing can be inefficient or approximate. Another way to solve a system is to use substitution. Substitution allows you to create a one variable equation.

199 Slide 199 / 309 Solve the system using substitution. Why was it difficult to solve this system by graphing? y = x y = -2x y = -2x x = -2x x x x = -7.5 x = start with one equation -substitute x for y in equation -solve for x Substitute -2.5 for x in either equation and solve for y. y = x y = ( -2.5) y = 3.6 Since x = -2.5 and y = 3.6, the solution is (-2.5, 3.6) CHECK: See if (-2.5, 3.6) satisfies the other equation. y = -2x = -2(-2.5) - 1.4? 3.6 = 5-1.4? 3.6 = 3.6

200 Slide 200 / 309 Solve the system using substitution. (*Note: Equations can be moved on the page to show substitution into the y of the second equation.) y = -2x +14 ( ) -3 y + 3x = 21

201 Slide 201 / 309 Solve the system using substitution. x = -5y - 39 ( ) x = -y - 3

202 Slide 202 / 309 Examine each system of equations. Which variable would you choose to substitute? Why? y = 4x y = -2x + 9 y = -3x 7x - y = 42 y = 4x + 1 x = 4y + 1

203 Slide 203 / Examine the system of equations. Which variable would you substitute? 2x + y = 5 2y = 10-4x Solution A B x y

204 Slide 204 / Examine the system of equations. Which variable would you substitute? 2y - 8 = x y + 2x = 4 Solution A B x y

205 Slide 205 / Examine the system of equations. Which variable would you substitute? x - y = 20 2x + 3y = 0 Solution A B x y

206 Slide 206 / 309 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 2x + 5y = -8 2x + 5y = -8 Using substitution you now have: 2x + 5(3x-5) = -8 -solve for x 2x + 15x - 25 = -8 -distribute the 5 17x - 25 = -8 -combine x's 17x = 17 -at 25 to both sides x = 1 - divide by 17 Substitute x = 1 into one of the equations. 2(1) + 5y = y = -8 5y = -10 y = -2 The ordered pair (1,-2) satisfies both equations in the original system. 3x -y = 5 2x + 5y = -8 3(1) - (-2) = 5 2(1) + 5(-2) = = = -8-8 = -8

207 Slide 207 / 309 Your class of 22 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

208 Slide 208 / 309 Set up the system: Drivers: v + c = 4 People: 6v + 4c = 22 Solve the system by substitution. v + c = 4 -solve the first equation for v. v = -c + 4 -substitute -c + 4 for v in the 6(-c + 4) + 4c = 22 second equation -6c c = 22 -solve for c -2c + 24 = 22-2c = -2 c = 1 v + c = 4 v + 1 = 4 -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 = 4 6v + 4c = = 4 6(3) + 4(1) = 22 4 = = = 22

209 Slide 209 / 309 Now solve this system using substitution. What happens? x + y = 6 5x + 5y = 10 x + y = 6 -solve the first equation for x x = 6 - y 5(6 - y) + 5y = 10 -substitute 6 - y for x in 2nd equation 30-5y + 5y = 10 -solve for y 30 = 10 -FALSE! Since 30 = 10 is a false statement, the system has no solution.

210 Slide 210 / 309 Now solve this system using substitution. What happens? x + 4y = -3 2x + 8y = -6 x + 4y = -3 - solve the first equation for x x = -3-4y 2(-3-4y) + 8y = -6 - sub. -3-4y for x in 2nd equation -6-8y + 8y = -6 - solve for y -6 = -6 - TRUE! - there are infinitely many solutions

211 Slide 211 / 309 How can you quickly decide the number of solutions a system has? 1 Solution Different slopes No Solution Infinitely Many Same slope; different y- intercept (Parallel Lines) Same slope; same y-intercept (Same Line)

212 Slide 212 / x - y = -2 y = 3x + 2 A B C 1 solution no solution infinitely many solutions Solution

213 Slide 213 / x + 3y = 8 y = 1 x 3 A B C 1 solution no solution infinitely many solutions Solution

214 Slide 214 / y = 4x 2x - 0.5y = 0 A B C 1 solution no solution infinitely many solutions Solution

215 Slide 215 / x + y = 5 6x + 2y = 1 Solution A B C 1 solution no solution infinitely many solutions

216 Slide 216 / y = 2x - 7 y = 3x + 8 Solution A B C 1 solution no solution infinitely many solutions

217 Slide 217 / Solve each system by substitution. y = x - 3 y = -x + 5 Click for multiple choice answers. A (4,9) Solution B (-4,-9) C (4,1) D (1,4)

218 Slide 218 / Solve each system by substitution. y = x - 6 y = -4 Solution Click for multiple choice answers. A (-10,-4) B (-4,2) C (2,-4) D (10,4)

219 Slide 219 / Solve each system by substitution. y + 2x = -14 y = 2x + 18 Solution Click for multiple choice answers. A (1,20) B (1,18) C (8,-2) D (-8,2)

220 Slide 220 / Solve each system by substitution. 4x = -5y + 50 x = 2y - 7 Click for multiple choice answers. A (6,6.5) Solution B (5,6) C (4,5) D (6,5)

221 Slide 221 / Solve each system by substitution. y = -3x y + 4x = 19 Click for multiple choice answers. A (6,5) Solution B (-7,5) C (42,-103) D (6,-5)

222 Slide 222 / 309 Systems Strategy Three: Elimination Return to Table of Contents

223 Slide 223 / 309 When both linear equations of a system are in Standard Form, Ax + By = C, you can solve the system using elimination. You can add or subtract the equations to eliminate a variable.

224 Slide 224 / 309 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.

225 Slide 225 / 309 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.

226 Slide 226 / 309 Solve by Elimination - Click on the terms to eliminate and they will disappear, then add the two equations together. ( 5x + y = 44-4x - y = -34 )

227 Slide 227 / 309 Solve by Elimination - Click on the terms and they will disappear then add the two equations together. 3x + y = 15-3x( -3y = -21)

228 Slide 228 / 309 Solve by Elimination - There are 2 ways to complete this problem. See both examples. Multiplication by -1 5x + y = 17-2x + y = -4 5x + y = 17-2x + y = -4 Subtraction

229 Slide 229 / 309 Solve the system by elimination. 4x + 3y = 16 2x - 3y = 8 Pull

230 Slide 230 / Solve each system by elimination. x + y = 6 x - y = 4 Click for multiple choice answers. A (5,1) Solution B (-5,-1) C (1,5) D no solution

231 Slide 231 / Solve each system by elimination. 2x + y = -5 2x - y = -3 Click for multiple choice answers. A (-2,1) Solution B (-1,-2) C (-2,-1) D infinitely many

232 Slide 232 / Solve each system by elimination. 2x + y = -6 3x + y = -10 Solution A (4,2) Click for multiple choice answers. B (3,5) C (2,4) D (-4,2)

233 Slide 233 / Solve each system by elimination. 4x - y = 5 x - y = -7 Click for multiple choice answers. A no solution Solution B (4,11) C (-4,-11) D (11,-4)

234 Slide 234 / Solve each system by elimination. 3x + 6y = 48-5x + 6y = 32 Click for multiple choice answers. A (2,-7) Solution B (7,2) C (2,7) D infinitely many

235 Slide 235 / 309 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.

236 Slide 236 / 309 Examine each system of equations. Which variable would you choose to eliminate? What do you need to multiply each equation by? 2x + 5y = -1 x + 2y = 0 3x + 8y = 81 5x - 6y = -39 3x + 6y = 6 2x - 3y = 4

237 Slide 237 / 309 In order to eliminate the y, you need to multiply first. 3x + 4y = -10 5x - 2y = 18 Multiply the second equation by 2 so the coefficients are opposites. 2(5x - 2y = 18) 10x - 4y = 36 Now solve by adding the equations together. 3x + 4y = x - 4y = 36 13x = 26 x = 2 Solve for y, by substituting x = 2 into one of the equations. 3x + 4y = -10 3(2) + 4y = y = -10 4y = -16 y = -4 So (2,-4) is the solution. Check: 3x + 4y = -10 5x - 2y = 18 3(2) + 4(-4) = -10 5(2) - 2(-4) = = = = = 18

238 Slide 238 / 309 Now solve the same system by eliminating x. What do you multiply the two equations by? 3x + 4y = -10 5x - 2y = 18 Multiply the first equation by 5 and the second equation by 3 so the coefficients will be the same 5(3x + 4y = -10) 3(5x - 2y = 18) 15x + 20y = x - 6y = 54 Now solve by subtracting the equations. - 15x + 20y = x - 6y = 54 26y = -104 y = -4 Solve for x, by substituting y = -4 into one of the equations. 3x + 4y = -10 3x + 4(-4) = -10 3x = -10 3x = 6 x = 2 So (2,-4) is the solution. Check: 3x + 4y = -10 5x - 2y = 18 3(2) + 4(-4) = -10 5(2) - 2(-4) = = = = = 18

239 Slide 239 / Which variable can you eliminate with the least amount of work? A B x y 9x + 6y = 15-4x + y = 3 Solution

240 Slide 240 / Which variable can you eliminate with the least amount of work? A B x y 3x - 7y = -2-6x + 15y = 9 Solution

241 Slide 241 / Which variable can you eliminate with the least amount of work? A B x y x - 3y = -7 2x + 6y = 34 Solution

242 Slide 242 / What will you multiply the first equation by in order to solve this system using elimination? 2x + 5y = 20 3x - 10y = 37 Now solve it...

243 Slide 243 / What will you multiply the first equation by in order to solve this system using elimination? 3x + 2y = -19 x - 12y = 19 Now solve it...

244 Slide 244 / What will you multiply the first equation by in order to solve this system using elimination? x + 3y = 4 3x + 4y = 2 Now solve it...

245 Slide 245 / 309 Systems Choose Your Strategy Return to Table of Contents

246 Slide 246 / 309 Altogether 292 tickets were sold for a basketball game. An adult ticket costs $3. A student ticket costs $1. Ticket sales were $470. Let a = adults s = students

247 Slide 247 / 309 Set up the system: number of tickets sold: a + s = 292 money collected: 3a + s = 470 First eliminate one variable. a + s = in both equations s has the same - (3a + s = 470) coefficient so you subtract the 2-2a+ 0 = -178 equations in order to eliminate it. a = 89 -solve for a Then, find the value of the eliminated variable. a + s = s = 292 -substitute 89 for a in 1st equation s = 203 -solve for s There were 89 adult tickets and 203 student tickets sold. (89, 203) Check: a + s = 292 3a + s = = 292 3(89) = = = = 470

248 Slide 248 / A piece of glass with an initial temperature of 99 º F is cooled at a rate of 3.5 º F/min. At the same time, a piece of copper with an initial temperature of 0 º F is heated at a rate of 2.5º F/min. Let m = the number of minutes and t = the temperature in F. Which system models the given information? Solution A B C t = m t = m t = m t = m t = m t = 0-2.5m

249 Slide 249 / Which method would you use to solve the system? A B C graphing substitution elimination t = m t = m click for equations Now solve it... m = 16.5 t = This means that in 16.5 minutes, the temperatures will both be 41.25º C. click for answer

250 Slide 250 / What method would you choose to solve the system? 4s - 3t = 8 t = -2s -1 A B C graphing substitution elimination

251 Slide 251 / Now solve the system! Click for multiple choice answers. 1 2 A (, -2) 4s - 3t = B (, 2) C (2, -2) 1 2 D (-2, ) t = -2s -1

252 Slide 252 / What method would you choose to solve the system? A B C graphing substitution elimination y = 3x - 1 y = 4x

253 Slide 253 / Now solve it! Click for multiple choice answers. A (1, 4) B (-4, -1) y = 3x - 1 y = 4x C (-1, 4) D (-1, -4)

254 Slide 254 / What method would you choose to solve the system? A B C graphing substitution elimination 3m - 4n = 1 3m - 2n = -1

255 Slide 255 / Now solve it! Click for multiple choice answers. A (-2, -1) B (-1, -1) 3m - 4n = 1 3m - 2n = -1 C (-1, 1) D (1, 1)

256 Slide 256 / What method would you choose to solve the system? A B C graphing substitution elimination y = -2x y = -0.5x + 3

257 Slide 257 / Now solve it! Click for multiple choice answers. A (-6, 12) B (2, -4) y = -2x y = -0.5x + 3 C (-2, 4) D (1, -2)

258 Slide 258 / What method would you choose to solve the system? A B C graphing substitution elimination 2x - y = 4 x + 3y = 16

259 Slide 259 / Now solve it! Click for multiple choice answers. A (6, 5) B (-4, 7) 2x - y = 4 x + 3y = 16 C (-4, 4) D (4, 4)

260 Slide 260 / What method would you choose to solve the system? A B C graphing substitution elimination u = 4v 3u - 3v = 7

261 Slide 261 / Now solve it! Click for multiple choice answers A (, ) 9 7 B (, 28 ) u = 4v 3u - 3v = 7 C (28, 7) D (7, 7 ) 4

262 Slide 262 / 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 2y = 1 and x + 4y = 7? A 1 B -1 C 3 D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

263 Slide 263 / A system of equations is shown. Students type their answers here What is the solution (x,y) of the system of equations? x = y = From PARCC sample test

264 Slide 264 / Two lines are graphed on the same coordinate plane. The lines only intersect at the point (3,6). Which of these systems of linear equations could represent the two lines? Select all that apply. A D B E C From PARCC sample test

265 Slide 265 / 309 Systems Modeling Situations Return to Table of Contents

266 Slide 266 / 309 A group of 148 people is spending five days at a summer camp. The cook ordered 12 pounds of food for each adult and 9 pounds of food for each child. A total of 1,410 pounds of food was ordered. Part A: Write an equation or a system of equations that describes the above situation and define your variables. Pull From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

267 Slide 267 / 309 Part B: Using your work from part A, find: (1) the total number of adults in the group Pull (2) the total number of children in the group

268 Slide 268 / 309 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 $6.00, and Rachel s bill was $5.25. What was the price of one slice of pizza? What was the price of one cola? Pull From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

269 Slide 269 / 309 Sharu has $2.35 in nickels and dimes. If he has a total of thirty-two coins, how many of each coin does he have? Pull From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

270 Slide 270 / 309 Ben had twice as many nickels as dimes. Altogether, Ben had $4.20. How many nickels and how many dimes did Ben have? Pull From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

271 Slide 271 / Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9 Set up a system and solve. How many packages of cards were sold? Solution You will answer how many packages of gift wrap in the next question.

272 Slide 272 / Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9 Set up a system and solve. How many packages of gift wrap were sold? Solution

273 Slide 273 / The sum of two numbers is 47, and their difference is 15. What is the larger number? A 16 B 31 C 32 Solution D 36 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

274 Slide 274 / Ramon rented a sprayer and a generator. On his first job, he used each piece of equipment for 6 hours at a total cost of $90. On his second job, he used the sprayer for 4 hours and the generator for 8 hours at a total cost of $100. What was the hourly cost for the sprayer? Solution From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

275 Slide 275 / What is true of the graphs of the two lines 3y - 8 = -5x and 3x = 2y -18? A no intersection Solution B intersect at (2,-6) C intersect at (-2,6) D are identical

276 Slide 276 / You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. Set up a system to solve. Which method will you use? (Solving it comes later...) A B C graphing substitution elimination Solution

277 Slide 277 / You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many quarters do you have? Solution

278 Slide 278 / You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many nickels do you have? Solution

279 Slide 279 / Julia went to the movies and bought one jumbo popcorn and two chocolate chip cookies for $5.00. Marvin went to the same movie and bought one jumbo popcorn and four chocolate chip cookies for $6.00. How much does one chocolate chip cookie cost? A $0.50 B $0.75 C $1.00 D $2.00 Solution From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

280 Slide 280 / Mary and Amy had a total of 20 yards of material from which to make costumes. Mary used three times more material to make her costume than Amy used, and 2 yards of material was not used. How many yards of material did Amy use for her costume? Solution From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

281 Slide 281 / The tickets for a dance recital cost $5.00 for adults and $2.00 for children. If the total number of tickets sold was 295 and the total amount collected was $1220, how many adult tickets were sold? Solution From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

282 Slide 282 / A 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 A and B are given. Each t-shirt has the same cost, and each sweatshirt has the same cost. The system of equations shown can be used to represent the situation. 2x + 2y = 38 Part A: What is the total cost of 1 t-shirt and 1 sweatshirt? From PARCC sample test { 3x + y = 35

283 Slide 283 / Part B (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.) { 2x + 2y = 38 3x + y = 35 A the number of t-shirts in the order B the number of sweatshirts in the order C the cost, in dollars, of each t-shirt D the cost, in dollars, of each sweatshirt From PARCC sample test

284 Slide 284 / Part C (continued from previous question) { 2x + 2y = 38 3x + y = 35 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? (, ) From PARCC sample test

285 Slide 285 / Part D (continued from previous question) { 2x + 2y = 38 3x + y = 35 Students type their answers here What is the total cost in dollars, of order C? $ From PARCC sample test

286 Slide 286 / 309 Glossary Return to Table of Contents

287 Slide 287 / 309 Coordinate Plane 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 Plot lines and points! Back to Instruction

288 Slide 288 / 309 Elimination The process of eliminating one of the variables in a system of equations. System: 2x - 3y = -2 4x + y = 24 Eliminate the y variable 2x - 3y = -2 (3)( 4x + y ) = 24(3) 2x - 3y = -2 +( 12x + 3y = 72) 14x = 70 x = 5 2(5) - 3y = y = -2-3y = -12 y = 4 Solution: (5,4) Back to Instruction

289 Slide 289 / 309 Geometry Theorem Through any two points in a plane there can be drawn only one line. Back to Instruction

290 Slide 290 / 309 Grade A unit engineers use to measure the steepness of a hill. 10 feet 5 feet 10 = 2 5 grade of hill is 2. 3 meters 25 meters The sign warns cars the hill has a grade of 7. grade of hill is Back to Instruction

291 Slide 291 / 309 Linear Equation Any equation whose graph is a line. 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 0. Back to Instruction

292 Slide 292 / 309 Origin The point where zero on the x-axis intersects zero on the y-axis. The point (0,0). (0,0) Used to graph coordinates! down 3 from origin right 4 from origin (4,-3) Back to Instruction

293 Slide 293 / 309 Parallel Two lines that have the same slope and never interesent. Back to Instruction

294 Slide 294 / 309 Perpendicular Two lines that interset and form a right angle. Right Angle Back to Instruction

295 Slide 295 / 309 Proportional Relationship When two quantities have the same relative size if weight is proportional to age, then a weight of 3kg on the 1st day means it will weigh 6kg on the 2nd day, 9kg on the 3rd day, 30kg on the 10th day, etc. Back to Instruction

296 Slide 296 / 309 Quadrant Any of the four regions created when the x-axis intersects the y-axis that are usually numbered with Roman numerals. II I III IV "First Quadrant" Back to Instruction

297 Slide 297 / 309 Slope How much a line rises or falls Steepness of a line The ratio of a line's rise over its run y = mx + b "m" = slope formula for slope: m = y 2 - y 1 x 2 - x 1 "Steepness" and "Position" of a Line Back to Instruction