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

2024
Views:

16

Pages:

2
Author:

Luciano Kirby

Question 1 In a nonlinear system of equations or inequalities, _____ equation or inequality must be nonlinear. Answer In a nonlinear system of equations or inequalities, at least one equation or inequality must be nonlinear. Question 2 Write and solve a system of equations for each situation. Check your answers A shop has one-pound bags of peanuts for $2 and three-pound bags of peanuts for $5.50. If you buy 5 bags and spend $17, how many of each size bag did you buy? Answer 1: x + y = 5, and 2x + 5.5y = 17 2: x = 5 - y and x = 8.5 - 2.75y 3: y | x(1) | x(2) | 0 | 5 | 8.5 | 1 | 4 | 5.75 | 2 | 3 | 3 | 3 | 2 | 0.25 | 4 | 1 | -2.5 | You bought 3 one-pound bags and 2 three-pound bags. Question 3 What is the solution to the system of equations below? 2x + 3y = 17 3x + 6y = 30 Answer You have the following system of equations: 2x + 3y = 17 3x + 6y = 30 In order to solve the previous system, proceed a follow: Multiply the first equation by 2: (2x + 3y = 17)(2) 4x + 6y = 34 -3x - 6y = -30 x + 0 = 4 x = 4 Then, replace the previous value of x into any of the systems of equations, for instance, into the first equation, and solve for y: 2x + 3y = 17 2(4) + 3y = 17 8 + 3y = 17 3y = 17 - 8 3y = 9 y = 9/3 y = 3 Then, the solution to the given system of equations is: (4,3) Question 4 A roadside vegetable stand sells pumpkins for $5 each and squashes for $3 each. One day they sold 6 more squash than pumpkins, and their sales totaled $98. Write and solve a system of equations to find how many pumpkins and quash they sold? Answer Let p be the number of pumpkins and s be the number of squash they sold. Since they sell each pumpkin for $5, they will earn 5p dollars selling p pumpkins. Since they sell each squash for $3, they will earn 3s dollars selling s squash. The total amount they will earn is then 5p + 3s. If their sales totaled $98, then 5p + 3s = 98. If they sold 6 more squash than pumpkins, then the number of squash they sold, s, is 6 more than the number of pumpkins they sold, p. This gives the equation s = 6 + p. You then have the system {5p + 3s = 98, s = 6 + p}. Since the second equation is already solved for p you should use the substitution method to solve. Substitute s = 6 + p into 5p + 3s = 98 and solve for p: 5p + 3s = 98 5p + 3(6 + p) = 98 5p + 18 + 3p = 98 8p + 18 = 98 8p = 80 p = 10 The vegetable stand then sold p = 10 pumpkins and s = 6 + p = 6 + 10 = 16 squash.