FLC Ch 3. Ex 1 Plot the points Ex 2 Give the coordinates of each point shown. Sec 3.2: Solutions and Graphs of Linear Equations

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1 Math 100 Elementary Algebra Sec 3.1: The Rectangular Coordinate System x-axis and y-axis origin ordered pair x-coordinate y-coordinate quadrants (I, II, III, and IV) Rectangular/Cartesian Coordinate System Ex 1 Plot the points Ex 2 Give the coordinates of each point shown. Identify quadrants these points lie in. A(, ) B(, ) C(, ) D(, ) E(, ) F(, ) Ex 3 Plot the points (4,3) and ( 4, 1) and graph a straight line that passes through them. a) Does the point ( 2, 0) lie on the line? b) ( 6, 2)? Sec 3.2: Solutions and Graphs of Linear Equations Defn A linear equation in two variables is an equation that can be written in the form Ax + By = C, where A, B, and C are real numbers and A and B are not both zero. Ax + By = C is called form. Ex 4 Find the missing coordinate to complete the ordered-pair solution to the given equation. a) PP y = 2x + 3 b) 3x + 1 y = 11 4 ( 6, ) (3, ) (,8) ( 13, ) 4 Ans ( 6, 15), (3, 3) Page 1 of 13

2 Ex 5 For each equation, tell which of the given ordered pairs are solutions. a) y = 8x 3 (0,3), (5, 16), (1,5) b) y = 4 (3, 4), ( 4, 4), (0, 4) Ex 6 PP Complete the given ordered pairs and use the results to graph the solution set for the equation. x + 3y = 6 ( 3, ), (0, ), (3, ) Ex 7 Graph using a table. y = 1 x + 2 y = 2 x = 3 3 Independent/Dependent Variables x is the variable and y is the variable. Note: You may print graph paper from my website. (PR sec 3.3&3.4 hw) Page 2 of 13

3 Recall: What is a linear equation? Sec 3.3: Graphing Linear Equations Using Intercepts Very Important: 1) The graph of a linear equation is a straight line. (Converse is true as well.) 2) Be able to recognize a linear equation. Defn The x-intercept of a line is the point where the line crosses the x-axis; it has the form (a, 0). The y-intercept of a line is the point where the line crosses the y-axis; it has the form (0, b). Note: These definitions can be extended to all graphs. (Does not apply to only lines.) Example Ex 8 Identify the x- and y-intercepts. Ex 9 Find the x- and y-intercepts then graph the equation. a) b) What number should we choose? 3x + 2y = 6 4x + 5y =?? Page 3 of 13

4 Ex 9 Find the x- and y-intercepts then graph the equation. c) d) y = 2 3 x 2 y = x 2 Sec 3.4: Graphing Linear Equations Using Slope We use the word slope to describe the steepness of a line or incline of objects, such as a hill. Slope The slope of the line passing through the points (x 1, y 1 ) and (x 2, y 2 ) is given by m = slope = rise change in y = = y 2 y 1, where x run change in x x 2 x 2 x 1 1 Slope of a Straight Line Positive Slope Negative Slope Zero Slope Undefined Slope (No Slope) Ex 10 Find the slope of a straight line that passes through the given pair of points. a) b) (5,6) and ( 5, 2) and 3 ( 3,1) (3, 6) Page 4 of 13

5 Ex 10 Find the slope of a straight line that passes through the given pair of points. c) d) Prac Prob 5 7 ( 1, 1) and ( 3, 2) and 2 ( 1, 1) (4, 7) 2 Ex 11 Graph the line with the given slope and y-intercept. Fill the entire graph grid. (*tests/quizzes*) a) b) m = 3 ; (0, 2) m = 3; (0,1) 4 How will I be graded on these problems? Ex 12 Find the slope and y-intercept of the line. Ex 13 Graph the line that has an x-intercept of ( 3,0) and y-intercept of (0,6). What is the slope of this line? Page 5 of 13

6 Ex 14 Find the value of y if the line through (1, y) and (7, 3) has slope 2. Slope-Intercept Form of a Line y = mx + b, where m is the slope and (0, b) is the y-intercept Ex 15 Find the slope and y-intercept of each line. a) b) c) d) e) y = x + 3 y = x + 3 y = 1 2 x 10 4x 3y = y = 1 5 x 7 Different Methods of Graphing a Line Constructing a table of values/xy-table (Sec 3.2) Finding x- and y-intercepts (Sec 3.3) Using the slope and y-intercept (Sec 3.4) Ex 16 Graph each linear equation using any method. Fill the entire graph grid. (*tests/quizzes*) a) b) c) d) y = 3x + 1 y = x y = 2 5 x y = 2 5 x 2 Page 6 of 13

7 Ex 17 Graph the lines. a) 5y = 15 b) 3x 7 = 5 Find the slope and intercepts. a) m x int: y int: b) m x int: y int: What form does the equation of a horizontal line have? A vertical line? Every horizontal line has the form. Every vertical line has the form. Ex 18 Graph using any method. Fill the entire graph grid. (*tests/quizzes*) a) b) c) y = 3 x + 1 y + 3x = 1 2x + 5y = 15 4 PP PP d) e) f) 1 2 y 2x + 4 = 0 y = 5 x 2x + 4y 1 = ) Plotting x- vs y-intercept for (0,b). 2) boxes = slope (marked down for inaccurate graphs) Page 7 of 13

8 g) h) i) y = 2 3 x y = x + 1 3x 4y = 12 Do 3.6 examples 26, 27, and 28a. Sec 3.5: Finding the Equation of a Line Slope Intercept Form of a Line Point Slope Form of a Line y = mx + b y y 1 = m(x x 1 ) To Find the Equation of a Line I. If we are NOT given the y-intercept (this is true for most cases) use y 1 = m(x x 1 ). (point-slope form) 1. Substitute the given values of x 1, y 1, and m. 2. Solve for y (if we are asked to put it in slope-intercept form.) II. III. IV. If we are given the y-intercept (0, b), use y = mx + b. (slope-intercept form) 1. Substitute the given values of m and b. If the slope is zero, the equation is a horizontal line and has the form y = a number. If the slope is undefined (zero in the denominator), the equation is a vertical line and has the form x = a number. Ex 19 Find an equation of a line that has the given slope and passes through the given point. Write all answers in slope-intercept form, if possible. a) b) c) m = 1 2 ; (0, 1) m = 3; (6, 1) m = 2; (3, 1 3 ) Page 8 of 13

9 Ex 19 Find an equation of a line that has the given slope and passes through the given point. Write all answers in slope-intercept form, if possible. d) e) f) m = 4; (0,2) m is undefined; (2,4) m = 0; (2,4) Ex 20 Find an equation of a line passing through the given points. a) ( 1, 19), (2,2) b) ( 3, 5), (3, 1) c) (2, 3), (2, 1) d) ( 3, 6), (3, 2) e) PP (π, 6), (3, 6) f) PP ( 1 3, 1 5 ), ( 1 3, 2 7 ) Ex 21 Write an equation for each line. Write answer in slope-intercept form. Write answer in point-slope AND slope-intercept form. Defn Parallel lines are two straight lines that have the same slope but different y-intercepts. Parallel lines never touch. Perpendicular lines have slopes whose product is 1. Slope of line l If l m, then slope of line m is: If l m, then slope of line m is: Page 9 of 13

10 Ex 22 Find an equation of a line with the given conditions. Write answer in slope-intercept form when possible. (Note: This does NOT mean we must use the slope intercept formula.) a) Passes through (5, 6) and has undefined slope b) Passes through (5, 6) and is parallel to 4y = 3x + 8 c) Passes through (5, 6) and is perpendicular to 4y = 3x + 8 (Note) d) Parallel to the line y = 2 and passing through the point ( π, 6). e) Perpendicular to the x-axis and passing through the point ( π, 6). f) Has x-intercept ( 3,0) and y-intercept (0, 5, ). Ex 23 (0, 2). Find the slope of a line parallel to the line that crosses the point (3, 1) and has a y-intercept of Ex 24 One line passes through the points (3, 4) and ( 3, 1). Find the slope of a second line that is perpendicular to the first. Page 10 of 13

11 Ex 25 Prac Prob Do the points (1,3), (3,7), and (5,11) lie on the same line? Sec 3.6: Graphing Linear Inequalities in Two Variables Ex 26 (Review-section 2.7) Ex 27 Graph the solution to the inequality. Graph the region described by the inequality. x > 3 x > 3 Ex 28 Graph the region described by the inequality. a) b) y 3x 1 2y < 3x c) d) 3x + 5y > 15 3x 4y 8 0 Page 11 of 13

12 Ex 29 Review (For Exam 4) Write an equation of a line that meets the specified conditions. Write answers in slope-intercept form. a) has slope 2 and passes through (0, 3 ) b) has slope 3 and passes through (2, 1) 3 2 c) has 0 slope and d) has undefined slope e) is perpendicular to passes through ( 2,5) and passes through ( 2,5) the x-axis and passes through (π, π) f) passes through (1, 7) and ( 3, 5) Ex 30 a) Find the slope of 4x + 5y = 20. b) Find an equation of the line parallel c) Find an equation of the line perpendicular to 4x + 5y = 20 passing through (1,2). to 4x + 5y = 20 passing through (1,2). Write answer in slope-intercept form. Write answer in slope-intercept form. Ex 31 Find the equation of the line. Ex 32 Find the slope of the line passing through ( 2 3, 1 3 ) and ( 5 2, 7 5 ) Ex 33 Graph 3y = 2x 9 and fill the entire graph grid. Page 12 of 13

13 Ex 34 A pool company learned that by pricing a new pool toy at $10, local sales will reach 200 a week. Lowering the price to $9 will cause sales to rise to 250 a week. a) Assume that the relationship is linear. Write the equation that describes this situation in slopeintercept form. b) Predict the weekly sales of the toy if the price is $7. Write answer in a complete sentence. Ex 35 Identify the slope and a point on the line. Use this information to graph the line. y 2 = 3 (x + 1) 4 Ex 36 Write in point-slope form the equation of a line with slope 13 passing through ( 10,11). Ex 37 Find the slope of each line. 2x 4 5 y = 7 y = 3 x = 1 3 Ex 38 Graph 2x + 3y = 6 and 3x 2y > 6. Page 13 of 13