Chapter 11, Problem 28CT

To determine

The price of each clothing item where Megan went clothes shopping and bought $2$ pairs of jeans, $2$ camisoles, and $4$ T-shirts for $\$90.00$. At the same store, Paige bought one pair of jeans and $3$ T-shirts for $\$42.50$, while Kara bought $1$ pair of jeans, $3$ camisoles, and $2$ T-shirts for $\$62.00$.

Answer to Problem 28CT

Solution:

The price of a pair of jeans is $\$24.50$, the price of a camisole is $\$8.50$ and the price of one T-shirt is $\$6.00$.

Explanation of Solution

Given information:

Megan went clothes shopping and bought $2$ pairs of jeans, $2$ camisoles, and $4$ T-shirts for $\$90.00$.

Paige bought one pair of jeans and $3$ T-shirts for $\$42.50$.

Kara bought $1$ pair of jeans, $3$ camisoles, and $2$ T-shirts for $\$62.00$.

Explanation:

Let $x$ be the price of a pair of jeans, $y$ be the price of a camisoles and $z$ be the price of one T-shirt.

Megan went clothes shopping and bought $2$ pairs of jeans, $2$ camisoles, and $4$ T-shirts for $\$90.00$.

Therefore, $2x+2y+4z=90$.… (1)

Paige bought one pair of jeans and $3$ T-shirts for $\$42.50$,

Therefore, $x+3z=42.50$.… (2)

Kara bought $1$ pair of jeans, $3$ camisoles, and $2$ T-shirts for $\$62.00$.

Therefore, $x+3y+2z=62$… (3)

Thus, the system of equation is $\{\begin{array}{l}2x+2y+4z=90\\ x+3z=42.50\\ x+3y+2z=62\end{array}$.

To solve the above system of equation using elimination method, Multiply equation (3) by $2$ and subtract equation (1) from it.

$\begin{array}{l}2x+6y+4z=124\\ \underset{\_}{-2x+2y+4z=90}\\ 4y=34\end{array}$

$\Rightarrow y=\frac{34}{4}=8.5$

Substitute $y=8.5$ in (1),

$\Rightarrow 2x+2\left(8.5\right)+4z=90$

$\Rightarrow 2x+4z=73$

$\Rightarrow x+2z=36.5$

Subtract $x+2z=36.5$ from equation (2),

$\begin{array}{l}x+3z=42.50\\ \underset{\_}{-x+2z=36.5}\\ z=6\end{array}$

Now substitute $z=6$ in equation (2)

$\Rightarrow x+3\left(6\right)=42.50$

$\Rightarrow x=42.50-18$

$\Rightarrow x=24.5$

Thus, $x=24.5,y=8.5,z=6$.

Therefore, the price of a pair of jeans is $\$24.50$, the price of a camisole is $\$8.50$ and the price of one T-shirt is $\$6.00$.