Name: Date: Instructor: Section: 81 Chapter 3 LINEAR EQUATIONS IN TWO VARIABLES 3.1 Reading Graphs; Linear Equations in Two Variables Learning Objectives 1 Interpret graphs. 2 Write a solution as an ordered pair. 3 Decide whether a given ordered pair is a solution of a given equation. 4 Complete ordered pairs for a given equation. 5 Complete a table of values. 6 Plot ordered pairs. Key Terms Use the vocabulary terms listed below to complete each statement in exercises 1-10. bar graph line graph linear equation in two variables ordered pair x-axis y-axis rectangular coordinate system quadrant origin coordinates 1. A(n) ___________________ is a pair of numbers written with parentheses in which the order of the numbers is important. 2. A(n) ___________________ is one of the four regions in the plane determined by a rectangular coordinate system. 3. The vertical line in a rectangular coordinate system is called the ___________________. 4. A(n) ___________________ is a series of line segments that connect points representing data. 5. The numbers in an ordered pair are called the ___________________ of the corresponding point. 6. The horizontal line in a rectangular coordinate system is called the ___________________. 7. A point at which the x-axis and y-axis of a rectangular coordinate system intersect is called the ___________________. 8. A(n) ___________________ is a series of bars arranged either vertically or horizontally to show comparisons of data. Name: Date: Instructor: Section: 82 9. The x-axis and y-axis placed at a right angle at their zero points form a ___________________. 10. A(n) ___________________ 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. Objective 1 Interpret graphs. The graphs below show the total number of degrees awarded by Jefferson University for the years 1990 – 1995 and the distribution of degrees awarded over this period. Use these graphs to answer the questions in Exercises 1 – 3. 1. Between which two years did the total number of degrees awarded show the greatest decline? 1.________________ 2. About how many more students received M.B.A. degrees in 1995 than 1994? 2.________________ 3. Between which two years did the total number of degrees awarded show the smallest change? 3.________________ Name: Date: Instructor: Section: 83 The graphs below show the usage of a mathematics help center by subject and by day of the week. Use these graphs to answer the questions in Exercises 4 – 6. 4. On which day was the center used one-and-a-half times as much as Monday? 4.________________ 5. If 4500 students used the math center during the school year, how many of these students were not enrolled in geometry or pre-calculus courses? Assume each student is enrolled in only one math class. 5.________________ 6. Which day had the greatest decrease in usage of the center as compared to the previous day? 6.________________ Objective 2 Write a solution as an ordered pair. Write each solution as an ordered pair. 7. x = 4 and y = 7 7.________________ Name: Date: Instructor: Section: 84 8. x = −2 and y = −3 8.________________ 9. 13 y = and x = 0 9.________________ Objective 3 Decide whether a given ordered pair is a solution of a given equation. Decide whether the given ordered pair is a solution of the given equation. 10. 5x − 2y = 6; (2,−2) 10.________________ 11. ( 1 ) 3 2x −3y =1; 0, 11.________________ 12. 2x = 3y; (3,2) 12.________________ 13. ( 1 ) 2 x =1− 2y; 0,− 13.________________ Objective 4 Complete ordered pairs for a given equation. For each of the given equations, complete the ordered pairs beneath it. 14. y = 2x −5 (a) (2, ) (b) (0, ) (c) ( ,3) (d) ( ,−7) (e) ( ,9) 14.________________ 15. y = 3+ 2x (a) (−4, ) (b) (2, ) (c) ( ,0) (d) (−2, ) 15.________________ Name: Date: Instructor: Section: 85 (e) ( ,−7) 16. 5x + 4y =10 (a) (2, ) (b) (4, ) (c) ( ,3) (d) (0, ) (e) ( ,0) 16.________________ 17. x = −2 (a) ( ,−2) (b) ( ,0) (c) ( ,19) (d) ( ,3) (e) ( 2 ) 3 ,− 17.________________ Objective 5 Complete a table of values. Complete the table of ordered pairs for each equation. 18. 4 6 2 1 4 x y xy + = 18.________________ 19. 3 2 4 0 4 0 x y xy + = 19.________________ 20. 4 0 6 0 6 yxy − = − 20.________________ 21. 3 4 6 0 0 2x y x y − = − 21.________________ Name: Date: Instructor: Section: 86 22. 4 3 12 0 01 x y x y + = − 22.________________ 23. 4 0 4 06 yx y − = − 23.________________ Objective 6 Plot ordered pairs. Plot the following ordered pairs on a coordinate system. 24. (7,1) 24.________________ 25. (−2,4) 25.________________ 26. (−2,−7) 26.________________ 27. (4,−2) 27.________________ 28. (0,−4) 28.________________ 29. (−5,0) 29.________________ 30. (0,0) 30.________________ Name: Date: Instructor: Section: 87 Chapter 3 LINEAR EQUATIONS IN TWO VARIABLES 3.2 Graphing Linear Equations in Two Variables Learning Objectives 1 Graph linear equations by plotting ordered pairs. 2 Find intercepts. 3 Graph linear equations of the form Ax + By = 0 . 4 Graph linear equations of the form y = k or x = k. 5 Use a linear equation to model data. Key Terms Use the vocabulary terms listed below to complete each statement in exercises 1-6. graph of a linear equation x-intercept y-intercept line through the origin horizontal line vertical line 1. The graph of a ___________________ has an equation of the form y = k. 2. The ___________________ in two variables is a straight line. 3. A point where a graph intersects the x-axis is called a(n) ___________________. 4. The graph of a ___________________ has an equation of the form x = k. 5. A point where a graph intersects the y-axis is called a(n) ___________________. 6. A ___________________ passes through the point (0, 0). Name: Date: Instructor: Section: 88 Objective 1 Graph linear equations by plotting ordered pairs. Complete the ordered pairs for each equation. Then graph the equation by plotting the points and drawing a line through them. 1. ( ) ( ) ( ) 3 0, ,0 2, x + y = 1.________________ 2. ( ) ( ) ( ) 3 0 0, 4,3, y + = − 2.________________ 3. ( ) ( ) ( ) 2 4 0, ,0 2, y − = x − 3.________________ Name: Date: Instructor: Section: 89 4. ( ) ( ) ( ) 4 0 ,0 , 2 ,3 x − = − 4.________________ 5. ( ) ( ) ( )4 0, ,0 2, x − y = − 5.________________ 6. ( ) ( ) ( ) 3 2 0, ,0 2, y = x − 6.________________ Name: Date: Instructor: Section: 90 7. ( ) ( ) ( ) 1 0, ,0 4, x − y = − 7.________________ 8. ( ) ( ) ( ) 2 3 6 0, ,0 3, x + y = − 8.________________ Objective 2 Find intercepts. Find the intercepts for the graph of each equation. 9. −5x + 2y =10 9.________________ 10. 3x + 2y =12 10.________________ 11. 2x + 4y = 0 11.________________ 12. 4x + 5y = 8 12.________________ 13. 5x − 2y =10 13.________________ Name: Date: Instructor: Section: 91 14. 4x + 3y = 9 14.________________ 15. 3x + 2y = −2 15.________________ 16. 5x − 3y =12 16.________________ 17. 2x + 9y = −9 17.________________ 18. 3x + 4y = 9 18.________________ Objective 3 Graph linear equations of the form Ax + By = 0 . Graph each equation. 19. 3x + y = 6 19.________________ 20. 6x + 5y =15 20.________________ Name: Date: Instructor: Section: 92 21. 4x − y = 4 21.________________ 22. 5x − 2y = −10 22.________________ 23. x + 5y = 0 23.________________ Name: Date: Instructor: Section: 93 Objective 4 Graph linear equations of the form y = k or x = k. Graph each equation. 24. x = 3 24.________________ 25. y = 0 25.________________ 26. y = −2 26.________________ Name: Date: Instructor: Section: 94 27. x −1 = 0 27.________________ 28. y + 3 = 0 28.________________ Objective 5 Use a linear equation to model data. Simplify each expression. 29. The enrollment at Lincolnwood High School decreased during the years 1990 to 1995. If x = 0 represents 1990, x =1represents 1991, and so on, the number of students enrolled in the school can be approximated by the equation y = −85x + 2435. Use this equation to approximate the number of students in each year from 1990 through 1995. 29.________________ Name: Date: Instructor: Section: 95 30. Suppose that the demand and price for a certain model of calculator are related by the equation 3 y = 45 − 5 x, where y is the price (in dollars) and x is the demand (in thousands of calculators). Assuming that this model is valid for a demand up to 50,000 calculators, find the price at each of the following levels of demand. (a) 0 calculators (b) 5000 calculators (c) 20,000 calculators (d) 45,000 calculators 30.________________ Name: Date: Instructor: Section: 97 Chapter 3 LINEAR EQUATIONS IN TWO VARIABLES 3.3 The Slope of a Line Learning Objectives 1 Find the slope of a line, given two points. 2 Find the slope from the equation of a line. 3 Use slopes to determine whether two lines are parallel, perpendicular, or neither. Key Terms Use the vocabulary terms listed below to complete each statement in exercises 1-10. rise run slope subscript notation positive slope negative slope horizontal line vertical line parallel perpendicular 1. A line with ___________________ falls from left to right. 2. The ratio of the change in y to the change in x along a line is called the ___________________ of the line. 3. The ___________________ is the horizontal change between two points on a line – that is, the change in x-values. 4. A ___________________ has undefined slope. 5. ___________________ is a way of indicating nonspecific values, such as 1 x and 2 x . 6. A line with ___________________ rises from left to right. 7. Two lines that are ___________________ have the same slope. 8. The ___________________ is the vertical change between two points on a line – that is, the change in y-values. 9. A ___________________ has slope equal to zero. 10. Two lines that are ___________________ have slopes whose product is −1. Name: Date: Instructor: Section: 98 Objective 1 Find the slope of a line, given two points. Find the slope of each line. 1. Through (4,3) and (3,5) 1.________________ 2. Through (2,3) and (6,7) 2.________________ 3. Through (−3,2) and (7, 4) 3.________________ 4. Through (5,−2) and (2,7) 4.________________ 5. Through (2,−4) and (−3,−1) 5.________________ 6. Through (7, 2) and (−7,3) 6.________________ 7. Through (−7,−7) and (2,−7) 7.________________ 8. Through (−4,−4) and (−2,−2) 8.________________ 9. Through (−4,6) and (−4,−1) 9.________________ 10. Through (2,−7) and (−2,1) 10.________________ Objective 2 Find the slope from the equation of a line. Find the slope of each line. 11. y = −5x 11.________________ Name: Date: Instructor: Section: 99 12. 1 y = 2 x + 5 12.________________ 13. 25 y = − x − 4 13.________________ 14. 47 y = − x + 9 14.________________ 15. 4y = 3x + 7 15.________________ 16. 2x + 7y = 7 16.________________ 17. 4x − 3y = 0 17.________________ 18. y = −4 18.________________ 19. x = 0 19.________________ 20. 3x = 4y 20.________________ Objective 3 Use slopes to determine whether two lines are parallel, perpendicular, or neither. In each pair of equations, give the slope of each line, and then determine whether the two lines are parallel, perpendicular, or neither. 21. 5 2 5 11 y x y x = − − = + 21.________________ 22. 14 4 4 3 y x y x = + = − 22.________________ Name: Date: Instructor: Section: 100 23. 7 3 x y x y − + = − − = − 23.________________ 24. 2 2 7 2 2 5 x y x y + = − = 24.________________ 25. 4 2 8 4 3 x y x y + = + = − 25.________________ 26. 9 3 2 3 5 x y x y + = − = 26.________________ 27. 4 2 7 5 3 11 x y x y + = + = 27.________________ 28. 8 2 7 3 x y x y + = = − 28.________________ 29. 4 0 7 0 yy + = − = 29.________________ 30. 90 yx == 30.________________ Name: Date: Instructor: Section: 101 Chapter 3 LINEAR EQUATIONS IN TWO VARIABLES 3.4 Equations of a Line Learning Objectives 1 Write an equation of a line by using its slope and y-intercepts. 2 Graph a line by using its slope and a point on the line. 3 Write an equation of a line by using its slope and any point on the line. 4 Write an equation of a line by using two points on the line. 5 Find an equation of a line that fits a data set. Key Terms Use the vocabulary terms listed below to complete each statement in exercises 1-3. slope-intercept form point-slope form standard form 1. A linear equation is written in ___________________ if it is in the form y = mx + b , where m is the slope and (0,b) is the y-intercept. 2. A linear equation is written in ___________________ if it is in the form ( ) 1 1 y − y = m x − x , where m is the slope and ( ) 1 1 x , y is a point on the line. 3. A linear equation in two variables written in the form Ax + By = C , with A and B both not 0, is in ___________________. Objective 1 Write an equation of a line by using its slope and y-intercepts. Write an equation in slope-intercept form for each of the following lines. 1. 23 m = ; b = −4 1.________________ 2. m = −2; b = 0 2.________________ 3. m = −7; b = −2 3.________________ 4. Slope 1 2 ; − y-intercept (0,−3) 4.________________ Name: Date: Instructor: Section: 102 5. Slope –4; y-intercept (0,0) 5.________________ 6. Slope 0; y-intercept (0,−4) 6.________________ Objective 2 Graph a line by using its slope and a point on the line. Graph the line passing through the given point and having the given slope. 7. (4,−2); m = −1 7.________________ 8. ( ) 23 −3,−2 ; m = 8.________________ Name: Date: Instructor: Section: 103 9. (2, 4) ; undefined slope 9.________________ 10. ( ) 52 1,−3 ; m = − 10.________________ 11. (−2,−2); m = 0 11.________________ Name: Date: Instructor: Section: 104 12. (3,−1); m = 2 12.________________ Objective 3 Write an equation of a line by using its slope and any point on the line. Write an equation for the line passing through the given point and having the given slope. Write the equations in the standard form Ax + By = C . 13. ( ) 13 5, 4 ; m = 13.________________ 14. (−2, 4); m = 2 14.________________ 15. ( ) 23 −3,−1 ; m = − 15.________________ 16. ( ) 34 −4,−7 ; m = 16.________________ 17. (0,0); m = 0 17.________________ Name: Date: Instructor: Section: 105 18. (−4,−2); undefined slope 18.________________ 19. (2,−2); m = −1 19.________________ Objective 4 Write an equation of a line by using two points on the line. Write an equation for the line passing through each pair of points. Write the equations in standard form Ax + By = C . 20. (2,3) and (7,5) 20.________________ 21. (3,−4) and (2,7) 21.________________ 22. (1,−2) and (−2,8) 22.________________ 23. (−3,4) and (−3,−7) 23.________________ 24. (7, 2) and (−2,−4) 24.________________ Name: Date: Instructor: Section: 106 25. (2,6) and (−4,6) 25.________________ 26. ( 1 2 ) 2 3 , and ( 3 ) 2 − ,2 26.________________ Objective 5 Find an equation of a line that fits a data set. The total expenditures (in millions of dollars) for the purchase of memorabilia collectibles is given below. Use the information in the chart to answer questions 27 – 30. Year x Millions of dollars (y) 1993 0 84 1994 1 101 1995 2 123 1996 3 136 1997 4 160 1998 5 181 1999 6 196 27. Use the data from 1994 and 1999 to find the slope of the line that approximates this information. Then use the slope to find the equation of the line in slope-intercept form. 27.________________ 28. To see how well the equation in exercise 27 approximates the ordered pairs (x, y) in the table of data, let x = 4 (for 1997) and find y. 28.________________ Name: Date: Instructor: Section: 107 29. Use the data from 1995 and 1998 to find the slope of the line that approximates this information. Then use the slope to find the equation of the line in slope-intercept form. 29.________________ 30. To see how well the equation in exercise 29 approximates the ordered pairs (x, y) in the table of data, let x = 3 (for 1996) and find y. 30.________________