Problem 171 - Elementary Algebra

Question

Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for \(\$ 114\) or she is able to buy 2 shirts and 4 sweaters for \(\$ 164 .\) How much does a shirt cost? How much does a sweater cost?

Step-by-Step Solution

Answer

The cost of a shirt is \$16\ and the cost of a sweater is \$33\.

Step 1: Define Variables

Let the cost of a shirt be denoted by \(s\) and the cost of a sweater be denoted by \(w\).

Step 2: Write Down Equations from the Problem

From the information given, create the following two equations: \(3s + 2w = 114\) and \(2s + 4w = 164\).

Step 3: Simplify the Second Equation

Divide the second equation \(2s + 4w = 164\) by 2 to simplify, resulting in \(s + 2w = 82\).

Step 4: Solve for \(s\)

Subtract the simplified second equation from the first equation: \[3s + 2w - (s + 2w) = 114 - 82\] Simplify to get: \[2s = 32\] Therefore, \s = 16\. The cost of a shirt is \$16\.

Step 5: Solve for \(w\)

Substitute \(s = 16\) back into the simplified second equation \((s + 2w = 82)\): \[16 + 2w = 82\] Subtract 16 from both sides: \[2w = 66\] Therefore, \w = 33\. The cost of a sweater is \$33\.

Key Concepts

system of equations

In this exercise, we encounter a system of equations to determine the cost of shirts and sweaters. A system of equations is a set of two or more equations with the same variables. Our goal is to find values for these variables that solve all the equations in the system. We created two equations based on the problem statement:
1. \(3s + 2w = 114\)
2. \(2s + 4w = 164\)
These equations represent relationships between the cost of shirts (\