Systems of Linear Equations and Inequalities Review Questions

UNIT #5 – SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES REVIEW QUESTIONS Part I Questions 1. Which of the following points is a solution to the system shown below? (1)  5, 12  (3) 1, 10  y  x7 (2)  2, 9  (4)  4, 7  y  11  x To solve a system, the point must make both equations true. The point  2, 9  : (2) 9  2  7 and 9  11  2 2. The system shown below has a solution when x  1 . What must be the value of b in the second equation? (1) 10 (3) 5 (2) 2 (4) 6 y  3  1  5  3  5  8 y  3x  5 y  2 x  b 1, 8  is a solution 8   2  1  b 8  2  b  b  10 (1) 3. Which of the following is the x-coordinate of the solution to the system shown below? (1) x  5 (3) x  3 y  4x  7 (2) x  7 (4) x  2 x  y  3 x   4 x  7   3 5 x  7  3 5 x  10 (4) x  2 4. The linear equation y  2 x  6 is shown graphed below. If it creates a system with the linear equation y  x  3 , which of the following would be the y-coordinate of the solution? y (1) y  3 (3) y  8 (2) y  1 (4) y  4 1, 4  x (4) 5. The point  2, 5 is a solution to the system of equations x  2 y  12 . Which of the following equations 3x  y  1 would it not be a solution to? (1) 4 x  y  13 (3) 6 x  2 y  2 (2) 2 x  3 y  11 (4) 2 x  4 y  12 All choices from (1) to (3) can be obtained by adding, subtracting the two equations or multiplying both sides of one of the equations. But, in (4), only the left-side was multiplied by 2. COMMON CORE ALGEBRA I, UNIT REVIEWS – UNIT #5 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 (4) 6. Which of the following values of x solves the system shown below? (1) x  5 (2) x  7 3x  2 y  19 (3) x  3 2 x  2 y  6 (4) x  25 After adding the equations, the y  terms cancel each other and result in: 5 x  25  x  5 (1) 7. Which value of y below is the solution to the system shown below? (1) y  6 (3) y  4 x  2 y  27 (2) y  1 (4) y  8 2 x  3 y  46 2 x  4 y  54 2 x  3 y  46  y  8 y8 (4) 8. The sum of two integers is 23 and the positive difference of the same two integers is 13. What is the product of these two integers? (1) 90 (3) 46 (2) 75 (4) 299 x  y  23 x  y  23 18  y  23 y5 x  y  13 2 x  36  2 x 36  2 2 x  y  18  5  90 x  18 (1) 9. The line y   x  3 and parabola y  x 2  6 x  7 are graphed below. Which of the following represents the y solution set to the equation: Since we have graphed the two sides of the equation, we simply take the x-coordinates of their intersections: (1, 2) and (4, -1). Hence, x = 1 and 4. x2  6x  7   x  3 (1) 1, 5 (3) 1, 4 x (2) 1, 2 (4) 0, 4 (3) 10. Which of the following points lies in the solution set of the system of inequalities shown graphed below? y (1) 1, 0  (3)  2, 2  (2)  4, 0  (4)  2,  4  A point will lie in the solution set of a system of inequalities if it lies in the overlapping regions that have been shaded. See the four points plotted. x (4) COMMON CORE ALGEBRA I, UNIT REVIEWS – UNIT #5 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 Free Response Questions 11. Explain how you can tell that the point  5, 13 is a solution to the system shown below, without formally solving the system? y  4x  7 2 x  y  23 A point will lie in the solution set of a system of equations if it makes both equations true. So, we can tell it is a solution by checking the truth value of each equation. y  4x  7 2 x  y  23 13  4  5   7 2  5   13  23 13  20  7 13  13 10  13  23 23  23 TRUE TRUE 12. If the point  2, 5 is a solution to the system of equations shown below, then determine the missing values of b and m. Show how you arrive at your answer. y  3x  b y  mx  9 5  3  2  b Because (2, 5) is a solution, it must make both equations TRUE. Substitute the point into both to find the missing parameters. 5  6b b  1 5  m  2  9 5  2m  9 4  2m m  2 13. Solve the following system of equation using the method of substitution. Show the work that leads to your answer. y  2 x  10 2 x  5  2 x  10   6 2 x  10 x  50  6 8 x  50  6 50  50 2x  5y  6 8 x  56 8 x 56  8 8 x  7 Final Solution: y  2  7   10 y  14  10 y4 x  7 and y  4 or  7, 4  14. Would the point  5, 10  lie in the solution set of the system of inequalities shown below? Justify your answer. x2 y  3x  7 To be a solution, it must make both inequalities true. It certainly makes x > 2 TRUE because x = 5 and 5 is greater than 2. Now we need to check the second inequality. y  3x  7 10  3  5   7 10  15  7 10  8 TRUE COMMON CORE ALGEBRA I, UNIT REVIEWS – UNIT #5 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 YES! Because it makes both inequalities true, the point lies in the solution set of the system. 15. Danny used the method of elimination to solve the system below: 4 x  3 y  12 2x  y  5 (a) Danny first rewrote the second equation as 6 x  3 y  15 . What property justifies writing this equation? (b) Is the x-coordinate that solves the system greater or less than one? Justify your answer. 4 x  3 y  12 6 x  3 y  15 The Multiplication Property of Equality. 2 x  3 2 x 3  2 2 x  1.5 It is greater than 1. 16. Solve the following system of equations graphically. Label your equations. 2 y  5 x  8 1 y   x2 2 1 2 y  intercept  b  2 slope  m   2 y  5 x  8 5 x  5 x 2 y  5x  8 2 y 5x  8  2 2 5 y  x4 2 5 2 y  intercept  b  4 slope  m  Final Solution:  2, 1 17. Solve the following system by the method of elimination. 2x  5y  5 3 x  2 y  9 2x  5 y  5 6 x  15 y  15 3  2 x  5 y   3  5  2  3 x  2 y   2  9  2 x  5  3  5 6 x  4 y  18 11 y  33 2 x  15  5 15  15 11 y 33  11 11 y3 2 x  10 2 x 10  2 2 x  5 COMMON CORE ALGEBRA I, UNIT REVIEWS – UNIT #5 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 Final Solution: x  5 and y  3 or  5, 3  18. At a local movie theatre, sodas cost $4.50 and bags of popcorn cost $1.50. If Kirk buys three times as many bags of popcorn as sodas and pays a total of $36, answer the following questions. (a) If s is the number of sodas bought and p is the number of bags of popcorn, write a system of equations that models this scenario and explain how your system models the problem. 4.50 s  1.50 p  36 Money spent on sodas + money spent on popcorn  total amount spent p  3s The number of bags of popcorn, p, is three times the numbers of soda, s . (b) Determine how many of each item Kirk bought. Do this by using an algebraic solution to the system you wrote in (a). 4.50 s  1.50 p  36 4.50 s  1.50  3 s   36 4.50 s  4.50 s  36 9 s  36 9 s 36  9 9 s4 p  3s Kirk bought 4 sodas! Now, we find the number of bags of popcorn. Kirk bought 12 bags of popcorn!! Kirk often takes many kids to the movies. p  3  4 p  12 19. Jody is supplying his food truck for the week. He buys 32 pounds of pork and 50 pounds of chicken for a total of $185. He realizes he did not purchase enough of either and returns to buy 8 more pounds of pork and 20 more pounds of chicken for a total of $65.60. Determine the unit price, in dollars per pound, for the pork and the chicken. 32 p  50c  185 8 p  20c  65.60 Let: 8 p  51.60  65.60 32 p  80c  262.4 p = the price per pound for pork c = the price per pound for chicken 8 p  20  2.58   65.60 32 p  50c  185 51.60  51.60 8 p  14 30c  77.4 30c 77.4  30 30 c  2.58 32 p  50c  185 4  8 p  20c   4  65.60  8 p 14  8 8 p  1.75 20. The graphs of two functions, f  x  and g  x  , intersect only once. Selected values of the functions are shown in the table below. Based on the table, state the solution to the equation: f  x  g  x Explain how you arrived at your choice. x 3 1 0 2 5 8 f  x 7 4 2 8 0 5 g  x 8 2 5 8 4 3 The solution to this equation is x = 2. This equation is saying that we should find the value of values of x that make the outputs of these two functions the same. That only happens when x = 2. COMMON CORE ALGEBRA I, UNIT REVIEWS – UNIT #5 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 21. Given the system of inequalities shown below, answer the following questions. y  2x  3 yx0 (a) Is the point  1, 5 in the solution set? Justify without first graphing. y  2x  3 5  2  1   3 5  2  3 51 y x  0 5  1  0 40 TRUE TRUE (b) Graph the solution set on the axes provided. 22. Marcus is working at a local pizzeria where he makes $12.50 per hour and is also working at the university bookstore where he makes $9.50 per hour. He must make at least $300 per week to cover his expenses but cannot work more than 30 hours per week in order to attend classes. If p represents the hours he works at the pizzeria and b represents the hours he works at the bookstore, do the following. (a) Write a system of inequalities that models this situation. 12.50 p  9.50b  300 p  b  30 Money made at the pizzeria + money made at the bookstore must be at least $300. The hours worked at the pizzeria + the hours worked at the bookstore cannot be more than 30. (b) The bookstore would like Marcus to pick up a weekly workload of 18 hours. Does this value for b lie in the solution set of the system of inequalities? Explain. 12.50 p  9.50b  300 If Marcus works 18 hours at the bookstore, i.e. b = 18, then the most he can work at the pizzeria, where he gets paid more, is 12 hours, so p = 12. 12.50  12   9.50  18   300 150  171  300 321  300 TRUE COMMON CORE ALGEBRA I, UNIT REVIEWS – UNIT #5 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 Yes, this value of b does lie in the solution set of the system of inequalities because there is a value of p that will go with this value and make both inequalities TRUE.