Q 192 - Intermediate Algebra

Question

In the following exercises, solve the given problem.  

After watching a major musical production at the theater, the patrons can purchase souvenirs. If a family purchases $4$ t-shirts, the video and $1$ stuffed animal their total is \($135$. A couple buys $2$ t-shirts, the video and $3$ stuffed animal for their nieces and spends \)$115$. Another couple buys $2$ t-shirts, the video and $1$ stuffed animal and their total is $$85$. What is the cost of each item? 

Step-by-Step Solution

Answer

The cost of one t-shirt is $$25$.

The cost of the video is $$20$.

The cost of one stuffed animal is $$15$.

Step 1. Identify and name what we are to find

We need to find the cost of the t-shirt, video, and stuffed animal.

Let $x$ represent the cost of one t-shirt, $y$ represent the cost of the video, and $z$ represent the cost of one stuffed animal.

Step 2. Form the equations

The price $4$ t-shirts, the video and $1$ stuffed animal is $$135$. So the first equation is

$4x+y+z=135 ...\left(1\right)$

The price of $2$ t-shirts, the video and $3$ stuffed animals is $$115$. So the second equation is

$2x+y+3z=115 ...\left(2\right)$

The price of $2$ t-shirts, the video and $1$ stuffed animal is $$85$. So the third equation is

$2x+y+z=85 ...\left(3\right)$

Step 3. Find x by elimination

Subtract the third equation from the first equation

$\begin{array}{rcl}4x+y+z-(2x+y+z)& =& 135-85\\ 4x+y+z-2x-y-z& =& 50\\ 2x& =& 50\\ x& =& 25\end{array}$

Step 4. Substitute the value of x in the second and third equation

Substitute $25$ for $x$ in the second equation

$\begin{array}{rcl}2x+y+3z& =& 115\\ 2\times 25+y+3z& =& 115\\ 50+y+3z-50& =& 115-50\\ y+3z& =& 65 ...\left(4\right)\end{array}$

Substitute $25$ for $x$ in the third equation

$\begin{array}{rcl}2x+y+z& =& 85\\ 2\times 25+y+z& =& 85\\ 50+y+z-50& =& 85-50\\ y+z& =& 35 ...\left(5\right)\end{array}$

Step 5. Find the value of z by elimination

Subtract the fifth equation from the fourth equation

$\begin{array}{rcl}y+3z-(y+z)& =& 65-35\\ y+3z-y-z& =& 30\\ 2z& =& 30\\ z& =& 15\end{array}$

Substitute $15$ for $z$ in the fifth equation

$\begin{array}{rcl}y+z& =& 35\\ y+15& =& 35\\ y& =& 35-15\\ y& =& 20\end{array}$

So the price of one t-shirt is $$25$, the video is $$20$, and one stuffed animal is $$15$