6.7 Solving Systems of Linear Equations 365 b) If the two lines in a system of equations have the same slope but different y-intercepts, how many solutions will the system have? Explain. No solution; the lines do not intersect. c) If the two lines in a system of equations have the same slope and the same y-intercept, how many solutions will the system have? Explain. Infinitely many solutions; the lines coincide. 66. Indicate whether the graph shown represents a consistent, inconsistent, or dependent system. a) y x Line 2 Line 1 Consistent b) y x Line 2 Line 1 Inconsistent c) y x Line 1 and Line 2 Dependent Challenge Problems/Group Activities 67. Solve the following system of equations for u and v by first substituting x for u 1 and y for . v 1 u v 1 2 8 + = u v 3 1 3 − = 1 2 , 1 3 ⎛ ⎝⎜ ⎞ ⎠⎟ 68. Develop a system of linear equations that has (6, 5) as its solution. Explain how you developed your system of equations. Answers will vary. 69. Construct a system of two linear equations that has an infinite number of solutions. Explain how you know the system has an infinite number of solutions. Answers will vary. 70. Construct a system of two linear equations that has no solution. Explain how you know the system has no solution. Answers will vary. 63. MODELING—Repair Costs Car repair costs at Steve’s Garage can be modeled by the equation C x 200 50 , = + and the repair costs at Greg’s Garage can be modeled by the equation C x 375 25 , = + where x represents the number of hours of labor. The graph below shows the repair costs at Steve’s Garage and at Greg’s Garage for up to 5 hours of labor. Assuming this trend continues, use the substitution method to determine the number of hours of labor it would take for the total cost at each garage to be the same. 7 hours Number of Hours of Labor Cost ($) 1 0 2 3 4 5 100 0 200 300 400 500 600 C x Greg’s Garage Steve’s Garage Repair Costs 64. Salaries Jose’s salary can be modeled by the equation y t 39,000 1200 , = + and Charlie’s salary can be modeled by the equation y t 45,000 700 , = + where t represents the number of years since 2020. The graph below shows Jose’s salary and Charlie’s salary for up to 10 years after 2020. Assuming this trend continues, use the substitution method to determine when Jose’s salary will equal Charlie’s salary. 12 years after 2020 or in 2032 Years After 2020 Salary ($) 0 1 2 3 4 5 6 7 8 9 10 30,000 40,000 50,000 60,000 y t Charlie’s salary Jose’s salary Salaries Concept/Writing Exercises 65. a) If the two lines in a system of equations have different slopes, how many solutions will the system have? Explain. One unique solution; the lines intersect at one and only one point. $ $
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