State the x- and y-intercepts of each equation. Then use the intercepts to graph the equation. 1. y = 2x + 7 To find the x-intercept, substitute 0 for y and solve for x. y = 2x + 7 0 = 2x + 7 7 = 2x 3.5 = x The x-intercept is 3.5. The equation is written in slope-intercept form. Find b. y = 2x + 7 b = 7 The y-intercept is 7. Use the points (3.5, 0) and (0, 7) to graph the equation. esolutions Manual - Powered by Cognero Page 1
2. To find the x-intercept, substitute 0 for y and solve for x. The x-intercept is 4. The equation is written in slope-intercept form. Find b. y = x + 3 b = 3 The y-intercept is 3. Use the points ( 4, 0) and (0, 3) to graph the equation. esolutions Manual - Powered by Cognero Page 2
3. 12x + 9y = 15 To find the x-intercept, substitute 0 for y and solve for x. 12x + 9y = 15 12x + 9(0) = 15 12x = 15 x = x = The x-intercept is. To find the y-intercept, substitute 0 for x and solve for y. 12x + 9y = 15 12(0) + 9y = 15 9y = 15 y = y = The y-intercept is. esolutions Manual - Powered by Cognero Page 3
4. The table shows the cost for a clothing store to buy jeans and khakis. The total cost for Saturday s shipment, $1,800, is represented by the equation 15x + 20y = 1,800. Use the x- and y-intercepts to graph the equation. Then interpret the x- and y-intercepts. Use the x- and y-intercepts to graph the function. To find the x-intercept, let y = 0. 15x + 20y = 1,800 15x + 20(0) = 1,800 15x = 1,800 x = 120 The x-intercept is 120. To find the y-intercept, let x = 0. 15x + 20y = 1,800 15(0) + 20y = 1,800 20y = 1,800 y = 90 The y-intercept is 90. Graph the points (120, 0) and (0, 90) on a coordinate plane. Then connect the points. esolutions Manual - Powered by Cognero Page 4
The x-intercept is at the point (120, 0). This means that if the store purchased only jeans, they would have 120 pairs of jeans for a total cost of $1,800. The y-intercept is at the point (0, 90). This means that if the store purchased only khakis, they would have 90 pairs of khakis for a total cost of $1,800. 5. The total number of legs, 1,500, on four-legged and two-legged animals in a zoo can be represented by the equation 4x + 2y = 1,500. Use the x- and y-intercepts to graph the equation. Then interpret the x- and y-intercepts. Use the x- and y-intercepts to graph the function. To find the x-intercept, let y = 0. 4x + 2y = 1,500 4x + 2(0) = 1,500 4x = 1,500 x = 375 The x-intercept is 375. To find the y-intercept, let x = 0. 4x + 2y = 1,500 4(0) + 2y = 1,500 2y = 1,500 y = 750 The y-intercept is 750. esolutions Manual - Powered by Cognero Page 5
Graph the points (375, 0) and (0, 750) on a coordinate plane. Then connect the points. The x-intercept is at the point (375, 0). This means that if the zoo had only four-legged animals, there would be 375 of them for a total of 1,500 legs. The y-intercept is at the point (0, 750). This means that if the zoo had only twolegged animals, there would be 750 of them for a total of 1,500 legs. 6. Multiple Representations The table shows the group rate for admission tickets for adults and children to an amusement park. a. Symbols The total cost of a group s tickets is $1,350. Write an equation to represent the number of adults and children s tickets purchased. b. Words What are the x- and y-intercepts and what do they represent? c. Graphs Use the x- and y-intercepts to graph the equation. Use the graph to find the number of children s tickets purchased if 20 adult tickets were purchased. esolutions Manual - Powered by Cognero Page 6
a. The total cost, $1,350, of a group s tickets is the sum of the cost of the adults tickets purchased, 45x, and of the children s tickets purchased, 30y. So, the equation is 45x + 30y = 1,350. b. To find the x-intercept, let y = 0. 45x + 30y = 1,350 45x + 30(0) = 1,350 45x = 1,350 x = 30 The x-intercept is 30. This means that if only adults bought tickets, 30 tickets would be sold for a total of $1,350. To find the y-intercept, let x = 0. 45x + 30y = 1,350 45(0) + 30y = 1,350 30y = 1,350 y = 45 The y-intercept is 45. This means that if only children s tickets were purchased, 45 tickets would be purchased for a total of $1,350. c. Graph the points (30, 0) and (0, 45) on a coordinate plane. Then connect the points. Locate the point with an x-coordinate of 20. The y-coordinate is 15. So, 15 children s tickets would be purchased if 20 adult tickets were purchased for a total of $1,350. esolutions Manual - Powered by Cognero Page 7
7. Find the Error Carmen is finding the x-intercept of the equation 3x 4y =12. Find her mistake and correct it. After 3x = 12, Carmen didn t divide both sides by 3 to get the x-intercept of 4. 3x 4y = 12 3x 4(0) = 12 3x = 12 x = 4 8. Persevere with Problems The perimeter of a rectangle that is x units wide and y units long is 24 centimeters. a. Write an equation in standard form for the perimeter. b. Find the x- and y-intercepts. Does either intercept make sense as a solution for this situation? Explain. a. Rewrite the perimeter equation 2w + 2l = P specifically for this exercise. 2x + 2y = 24. b. To find the x-intercept, let y = 0. 2x + 2y = 24 2x + 2(0) = 24 2x = 24 x = 12 The x-intercept is 12. To find the y-intercept, let x = 0. 2x + 2y = 24 2(0) + 2y = 24 2y = 24 y = 12 The y-intercept is 12. Sample answer: The x-intercept is at the point (12, 0) and the y-intercept is at the point (0, 12). These points are not solutions in this situation because the length or width of the rectangle cannot be 0. esolutions Manual - Powered by Cognero Page 8
9. Model with Mathematics Write two equations, one with an x-intercept but no y-intercept, and one with a y- intercept but no x-intercept. x-intercept equation: y-intercept equation: Sample answers are given. x-intercept equation: x = 2 A vertical line will have no y-intercept because it is parallel to the y-axis. y-intercept equation: y = 2 A horizontal line will have no x-intercept because it is parallel to the x-axis. 10. State the x- and y-intercepts of the equation. Then use the intercepts to graph the equation. To find the x-intercept, substitute 0 for y and solve for x. The x-intercept is. The equation is written in slope-intercept form. Find b. esolutions Manual - Powered by Cognero Page 9
The y-intercept is. Use the points (, 0) and (0, ) to graph the equation. esolutions Manual - Powered by Cognero Page 10
Copy and Solve State the x- and y-intercepts of each equation. Then use the intercepts to graph the equation on a separate sheet of grid paper. 11. 2x + 3y = 24 To find the x-intercept, substitute 0 for y and solve for x. 2x + 3y = 24 2x + 3(0) = 24 2x = 24 x = 12 The x-intercept is 12. To find the y-intercept, substitute 0 for x and solve for y. 2x + 3y = 24 2(0) + 3y = 24 3y = 24 y = 8 The y-intercept is 8. Use the points (12, 0) and (0, 8) to graph the line. esolutions Manual - Powered by Cognero Page 11
12. To find the x-intercept, substitute 0 for y and solve for x. The x-intercept is 18. The equation is written in slope-intercept form. Find b. The y-intercept is 16. Use the points ( 18, 0) and (0, 16) to graph the equation. esolutions Manual - Powered by Cognero Page 12
13. 5x + 3y = 30 To find the x-intercept, substitute 0 for y and solve for x. 5x + 3y = 30 5x + 3(0) = 30 5x = 30 x = 6 The x-intercept is 6. To find the y-intercept, substitute 0 for x and solve for y. 5x + 3y = 30 5(0) + 3y = 30 3y = 30 y = 10 The y-intercept is 10. Use the points (6, 0) and (0, 10) to graph the line. 14. Tiffany has 15 teaspoons of chocolate chips. She uses teaspoons for each muffin. The total amount of chocolate chips that she has left y after making x muffins can be given by. Graph the equation. Then interpret the x- and y-intercepts. Find the intercepts. Then graph the equation. esolutions Manual - Powered by Cognero Page 13
To find the x-intercept, let y = 0 and solve for x. The x-intercept is 10. The equation is written in slope-intercept form. Find b. The y-intercept is 15. The x-intercept represents the number of muffins baked by using all of the chocolate chips. The y-intercept represent the amount of chips before she baked any muffins. 15. Use Math Tools Miriam has $440 to pay a painter to paint her basement. The painter charges $55 per hour. The equation y = 440 55x represents the amount of money y she pays after x number of hours worked by the painter. Graph the equation. Then interpret the x- and y-intercepts. esolutions Manual - Powered by Cognero Page 14
Find the intercepts. Then graph the equation. To find the x-intercept, let y = 0 and solve for x. The x-intercept is 8. The equation is written in slope-intercept form. Find b. The y-intercept is 440. The x-intercept represents the number of hours the painter worked to finish the basement. The y-intercept represents the total amount of money she has to pay the painter. 16. Match each equation to the appropriate graph below. esolutions Manual - Powered by Cognero Page 15
All the graphs have the same y-intercept. Look for the graph with the correct x-intercepts. To find the x-intercept, let y = 0. 2x - 3y = 6 2x 3(0) = -6 2x = -6 x = 3 The x-intercept is 3. This matches the fourth graph. 3x 2y = -6 3x 2(0) = -6 3x = -6 x = 2 The x-intercept is -2. This matches the first graph. esolutions Manual - Powered by Cognero Page 16
3x + 2y = 6 3x + 2(0) = 6 3x = 6 x = 2 The x-intercept is 2. This matches the second graph. 2x + 3y = 6 2x + 3(0) = 6 2x = 6 x = 3 The x-intercept is 3. This matches the third graph. 17. The equation 12x 10y = 600 represents the total amount Student Council spent on supplies for a school fundraiser. Fill in the boxes to make a true statement. The x-intercept of the line is and the y-intercept is. To find the x-intercept, let y = 0. 12x 10y = 600 12x 10(0) = 600 12x = 600 x = 50 The x-intercept of the function is 50. To find the y-intercept, let x = 0. 12x 10y = 600 12(0) 10y = 600 10y = 600 y = -60 The y-intercept of the function is -60. Simplify each expression. 18. 3(x + 6) Use the Distributive Property to simplify. 3(x + 6) = 3x 18 19. Use the Distributive Property first, then simplify the subtraction. = 2x + 1 esolutions Manual - Powered by Cognero Page 17
20. Combine like terms. 21. = t + 5 Combine like terms. = 4x + 6 22. Use the Distributive Property first, then simplify the addition. 23. = x + 20 Combine like terms. = 6a 6 esolutions Manual - Powered by Cognero Page 18