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320 Without graphing, determine whether each of the following systems of equations has one solution, no solution, or many solutions: 33. 3x − 2y + 13 = 0 3x + y + 7 = 0 34. 4x + 6y − 14 = 0 2x + 3y − 7 = 0 35. x − 3y + 2 = 0 3x − 9y + 11 = 0 36. 15x + 3y = 10 5x + y = −3 38. 3x − y + 2 = 0 37. 2x − 4y = 6 x − 2y = 3 9x − 3y + 6 = 0 Solve the following systems of equations by using the Graphical method: 39. y = −2x − 1 y = 3x − 11 40. y = 2x + 3 y = −2x − 1 41. 2x − 3y − 6 = 0 x + 2y − 10 = 0 42. 3x + 4y − 5 = 0 2x − y + 4 = 0 43. 2y = x y = −x + 3 44. 3y = 2x y = −3x + 11 45. x + 4y = 8 2x + 5y = 13 46. x + y = 3 2x − y = 12 47. x + 4y + 12 = 0 9x − 2y − 32 = 0 48. x − y − 1 = 0 2x + 3y − 12 = 0 49. 3x + 2y = 5 y = 2x − 1 50. 4x + 3y = 12 9 − 3x = y Solve the following systems of equations by using either the Substitution method or the Elimination method: 57. 0.4x − 0.5y = –0.8 0.3x − 0.2y = 0.1 58. 0.2x − 0.3y = –0.6 0.5x + 0.2y = 2.3 59. 5x – 5y ==-–55 3 2 y y xx – -- === 222 33 44 y 60. x + = 2 4 2 2y x + = 4 3 6 3 61. (2x + 1) – 2(y + 7) = –1 4(x + 5) + 3(y – 1) = 28 62. 2(3x + 2) + 5(2y + 7) = 13 3(x + 1) – 4(y – 1) = –15 63. Find the value of two numbers if their sum is 95 and the difference between the larger number and the smaller number is 35. 64. Find the value of two numbers if their sum is 84 and the difference between the larger number and the smaller number is 48. Solve the following systems of equations by using the Elimination method: 51. 8x + 7y = 23 7x + 8y = 22 52. 2x + y = 8 3x + 2y = 7 53. 9x − 2y = −32 x + 4y = −12 54. 5x − 2y + 3 = 0 3x − 2y − 1 = 0 55. 4x + 3y = 12 18 − 6x = 2y 56. 2x + y + 2 = 0 6x = 2y 65. 300 tickets were sold for a theatrical performance. The tickets cost $28 for adults and $15 for kids. If $7,230 was collected, how many adults and how many children attended this play? 66. 640 tickets were sold for a soccer game between Toronto FC and Liverpool FC. The tickets cost $35 for adults and $20 for students. If $16,250 was collected from sales, how many adults and students attended the game? 9 Self-Test Exercises Answers to all problems are available at the end of the textbook. 1. Write the following equations in the form Ax + By = C: 5. Graph the equation 2x − 3y = 9 using a table of values with 4 points. a. y = 2 4 x -– 2 3 2x + b.6y6-– 2x + 1 = 00 = 4 2. Three vertices of a rectangle ABCD have the points A (–3, 4), B (5, 4), and C (5, –1). Find the coordinates of the 4th vertex and the area of the rectangle. 3. Find the slope and y-intercept of the following lines: a. 2x – 3y + 6 = 0 b. 3x + 4y – 5 = 0 4. Find the equation of the line, in standard form, that passes through the points P (–4, 5) and Q (1, 1). 6. Use the slope of the lines to determine whether the pairs of lines are parallel: a. 3y = 6x – 9 and 4x − 2y = –6 b. 3y + 4x = 0 and 3x + 4y = 2 7. Write the equation of a line, in standard form, that is parallel to 3x – 2y + 9 = 0 and that passes through the point (–6, –3). (Hint: Parallel lines will have the same slope.) 8. Graph the equation 3y + 4x = 0 using the x-intercept, y-intercept, and another point on the line. 9. Given the following slopes (m) and y-intercepts (b), write the equations in standard form: b. m = 2 , b = –2 a. m =-– 1 , b = –4 2 3 Chapter 9 | Graphs and Systems of Linear Equations 321 10. Graph the line that contains the point (–3, 5) and that has a slope of –3 . 4 11. Write the equation of a line parallel to 2x + 3y = 6 and that passes through the point P(–6, –1). 12. Write the equation of the line perpendicular to 5x – y = 4 and that passes through the point P(1, 2). 13. Write the equation of the line passing through the points P(–3, 5) and Q(5, –1). 14. Write the equation of the line that passes through the origin and that is perpendicular to the line passing through the points P(–3, 5) and Q(5, –1). 15. Write the equation of the line having an x-intercept equal to 5 and y-intercept equal to –3. 16. Write the equation of the line passing through the origin and that is parallel to y = 5x − 1. Without graphing, determine whether each of the following systems of equations has one solution, no solution, or many solutions: 17. 4x + 3y − 16 = 0 and 2x − y + 2 = 0 18. x − 3y + 11 = 0 and 2x − 6y + 4 = 0 19. y = 4x + 8 and 8x − 2y + 8 = 0 Solve the following systems of equations by using the Graphical method: 20. y = 3x + 6 6x – 2y + 12 = 0 21. 2x + 3y + 4 = 0 3x – y + 7 = 0 Solve the following systems of equations by using the Substitution method: 22. 2x + 4y + 6 = 0 y – 3x + 9 = 0 23. 3x + y = –8 2x + 3y = 4 Solve the following systems of equations by using the Elimination method: 24. 6x − 4y + 3 = 0 4x − 6y − 3 = 0 25. 3x + 5y − 19 = 0 5x − 2y + 11 = 0 26. Find the value of two numbers if their sum is 65 and the difference between the larger number and the smaller number is 5. 27. In a coin box, there are 3 times as many quarters (25¢) as dimes (10¢). If the total value of all these coins is $21.25, how many quarters are there? 28. The cost of admission to a concert was $75 for adults and $50 for students. If 400 tickets were sold and $28,125 was collected, how many adult tickets were sold? 29. Henry works in a computer store and earns $500 a month plus a commission of 10% on the sales he makes. The relationship between his earnings (y) in a month and the number of computers he sells (x) is given by the equation y = 500 + 0.10x. a. What would his commission be if his earnings are $14,000 in a month? b. What would his earnings be if his sales are $250,000 in a month? 30. The fixed costs (FC) of a factory for the month are $5,000 and the variable costs (VC) to manufacture each product are $5. The total costs (TC) for the month are the sum of the fixed costs and the variable costs per unit, multiplied by the number of products produced and sold (x). The relationship between TC, FC, VC, and x is expressed by the equation TC = (VC)x + FC. a. What would be the total cost if 90 products were sold this month? b. How many products were sold this month if the total cost is $11,375? Self-Test Exercises