University:
University of Michigan
Course:
MATH 115 | Calculus I
Academic year:

2024
Views:

253

Pages:

3
Author:

Luciano Kirby

Question 1 What are the solutions to the following system of equations? y = x² + 3x - 7 3x - y = -2 Answer y = x² + 3x - 7 3x - y = -2 I'll plug the top into the bottom since it is already y= 3x - (x² + 3x - 7) = -2 distribute the negative and get rid of parenthesis 3x - x² - 3x + 7 = -2 condense -x² + 7 = -2 re-write as quadratic with positive x² x² - 9 = 0 solve for x x = -3, x = 3 plug in each x to get its y y = -7, y = 11 solutions: (-3, -7) and (3, 11) Question 2 What is the value of x - y, if xy = 144, x + y = 30, and x>y? Answer (1) xy = 144 (2) x + y = 30 ⇒ y = 30 - x Substitute (2) for y back in (1): x(30 - x) = 144 → x² - 30x + 144 = 0 Factoring (x - 6)(x - 24) = 0 Either x = 6 or x = 24 Since we know that x > y ⇒ x = 24, y = 6 ⇒ x - y = 24 - 6 = 18 Question 3 What is the solution of the system of equations? y = 2x - 3 5x + y = 11 A(2,1) B(1,2) C(3,-4) D(1,-1) Answer Let: y = 2x - 3 (1) 5x + y = 11 (2) Solve by substitution. Substitute (1) to (2) and solve for x: 5x + (2x - 3) = 11 7x - 3 = 11 7x = 14 x = 2 Solve for y using (1): y = 2(2) - 3 y = 1 So, the solution of the system is: (2,1) which is choice A. Question 4 Solve the system: 3x + y = 9 2x - y = 1 Answer 3x + y = 9 2x - y = 1 Add equation (1) and (2): 5x = 10, x = 2 Substitute x = 2 into equation (1) and solve for y: 3 * 2 + y = 9 6 + y = 9 y = 9 - 6 y = 3 Thus, the solution is (2,3) 3x + y = 9 (1) 2x - y = 1 (2) Add equation (1) and (2) 5x = 10 x = 2 Substitute x = 2 into equation (1) and solve for y: 3 * 2 + y = 9 6 + y = 9 y = 9 - 6 y = 3 Thus, the solution is (2, 3)