Mid-Chapter Quiz: Lessons 3-1 through 3-4. Solve each system of equations. SOLUTION: Add both the equations and solve for x.
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1 1. Solve each system of equations. Add both the equations and solve for x. 6x = 18 Divide both sides by 6. x = 3 Substitute 3 for x in the second equation and solve for y. The solution is (3, 1). 2. Substitute 5x 2 for y in the first equation and solve for x. Substitute 1 for x in the second equation and solve for y. The solution is (1, 3). esolutions Manual - Powered by Cognero Page 1
2 3. Multiply the first equation by 3 and add with the second equation. Substitute 3 for x in the first equation and solve for y. The solution is (3, 5). 4. Multiply the first equation by 2 and add with the second equation. Substitute 10 for x in the first equation and solve for y. The solution is (10, 6). esolutions Manual - Powered by Cognero Page 2
3 5. Solve each system of inequalities by graphing esolutions Manual - Powered by Cognero Page 3
4 8. 9. MULTIPLE CHOICE Which statement best describes the graphs of the two equations? x + 4y = 8 3x + 12y = 2 A The lines are parallel. B The lines are the same. C The lines intersect in only one point. D The lines intersect in more than one point, but are not the same. Rewrite the equations in the slope intercept form. Since the slopes are equal and the y-intercepts are not equal, they are parallel line. Therefore, option A is the correct answer. esolutions Manual - Powered by Cognero Page 4
5 Solve each system of equations. 10. Multiply the second equation by 2 and add with the third equation. Substitute 4 for z in the second equation and solve for y. Substitute 7 and 4 for y and z in the first equation respectively and solve for x. The solution is (3, 7, 4). esolutions Manual - Powered by Cognero Page 5
6 11. Add the first and the third equations. Multiply the third equation by 3 and add with the second equation. Multiply the fourth equation by 3 and add with the fifth equation. Substitute 1 for x in the fourth equation and solve for y. Substitute 1 and 2 for x and y in the first equation respectively and solve for z. The solution is (1, 2, 1). esolutions Manual - Powered by Cognero Page 6
7 12. Subtract the third equation from the first equation. Multiply the fourth equation by 4 and add with the second equation. Substitute 2 for x in the third equation and solve for y. Substitute 2 for x in the second equation and solve for z. The solution is (2, 5, 3). esolutions Manual - Powered by Cognero Page 7
8 13. Subtract the third equation from the first equation. Multiply the fourth equation by 3 and add with the second equation. Substitute 2 for y in the fourth equation and solve for x. Substitute 1 and 2 for x and y in the third equation and solve for z. The solution is (1, 2, 4). esolutions Manual - Powered by Cognero Page 8
9 14. MULTIPLE CHOICE Seela rented a raft from River Rafter s Inc. She paid $100 to rent the raft and $25 for each hour. Martin rented a raft from Oscar s Outdoor Shop. He paid $50 to rent the raft and $35 per hour. For what number of hours will both rafting companies charge the same amount? F 0 G 4 H 5 J 10 Let x be number of rafting hours. The equation representing the situation is. Solve for x. Therefore, option H is the correct answer. esolutions Manual - Powered by Cognero Page 9
10 15. CARPENTRY Cal s Carpentry makes tables and chairs. The process involves some carpentry time and some finishing time. The carpentry times and finishing times are listed in the table below. Cal s Carpentry can work for a maximum of 108 carpentry hours and 20 finishing hours per day. The profit is $35 for a table and $25 for a chair. How many tables and chairs should be made each day to maximize profit? a. Using c for the number of chairs and t for the number of tables, write a system of inequalities to represent this situation. b. Draw the graph showing the feasible region. c. Determine the number of tables and chairs that need to be made to maximize profit. What is the maximum profit? a. Let c and t be the number of chair and table. Optimization function: Constraints: b. c. The vertices of the feasible region are (0, 0), (0, 20), (34, 3) and (36, 0). Substitute the points (0, 0), (0, 20), (34, 3) and (36, 0) in the function f (x, y) = 25c + 35t. The maximum value is 955 at (34, 3). Therefore, 34 chairs and 3 tables will give a maximum profit of $955. esolutions Manual - Powered by Cognero Page 10
11 16. DRAMA On opening night of the drama club s play, they made $1366. They sold a total of 199 tickets. They charged $8.50 for each adult ticket and $5.00 for each child s ticket. Write a system of equations that can be used to find the number of adult tickets and the number of children s tickets sold. Let a and c are represent number of adult tickets and number of children s ticket respectively. Total number of sold tickets is 199. That is,. Total collection cost is $1366. That is,. Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and the minimum values of the given function. 17. The vertices of the feasible region are ( 4, 3), (0,5) and (2, 3). Substitute the points ( 4, 3), (0,5) and (2, 3) in the function f (x, y) = 4x 3y. The maximum value is 17 at (2, 3) and the minimum value is 15 at (0, 5). esolutions Manual - Powered by Cognero Page 11
12 18. The vertices of the feasible region are ( 10, 1), ( 4, 1), (2.4, 3.8), ( 2, 6) and ( 10, 6). Substitute the points ( 10, 1), ( 4, 1), (2.4, 3.8), ( 2, 6) and ( 10, 6) in the function f (x, y) = 2x + y. The maximum value is 1 at (2.4, 3.8) and the minimum value is 26 at ( 10, 6). 19. GEOMETRY An isosceles trapezoid has shorter base of measure a, longer base of measure c, and congruent legs of measure b. The perimeter of the trapezoid is 58 inches. The average of the bases is 19 inches and the longer base is twice the leg plus 7. a. Find the lengths of the sides of the trapezoid. b. Find the area of the trapezoid. a. The equation representing the situation is Simplify the second equation. Substitute 38 for a + c in the first equation and solve for b. esolutions Manual - Powered by Cognero Page 12
13 Substitute 10 for b in the third equation and solve for c. Substitute 27 for c in the second equation and solve for a. Therefore, the measures of the sides a, b and c are 11 in., 10 in. and 27 in. respectively. b. The area of a trapezoid is. The bases and the height of the trapezoid is 11 in., 27 in., and 6 in. Substitute the values in the area formula and simplify. Therefore, area of the trapezoid is 114 in 2. esolutions Manual - Powered by Cognero Page 13
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