Q. 375 - Intermediate Algebra

Question

After attending a major league baseball game, the patrons often purchase souvenirs. If a family purchases $4$ t-shirts, a cap, and $1$ stuffed animal their total is \($135$. A couple buys $2$ t-shirts, a cap and $3$ stuffed animals for their nieces and spends \)$115$. Another couple buys $2$ t-shirts, a cap and $1$ stuffed animal and their total is $$85$. What is the cost of each item?

Step-by-Step Solution

Answer

The price of a t-shirt is $$25$, the price of a cap is $$20$, and the price of a stuffed animal is $$15$.

Step 1. Assuming the values.

Let the price of a t-shirt be $$x$,

the price of the cap be $$y$,

and the price of a stuffed animal is $$z$.

Step 2. Forming the equations.

The number of equations formed,

$$\begin{gathered} 4x+y+z=135...........\left(1\right) \\ 2x+y+3z=115.........\left(2\right) \\ 2x+y+z=85............\left(3\right) \end{gathered}$$

Eliminating z from equations (1) and (2).

Eliminating $z$ from equations (1) and (2).

$$\begin{gathered} -12x-3y-3z=-405........\left(1\right)\times -3 \\ 2x+y+3z=115.................\left(2\right) \end{gathered}$$

Solving the equations, we get,

$$\begin{gathered} -10x-2y=-290 \\ 10x+2y=290 \\ 5x+y=145...........\left(4\right) \end{gathered}$$

Step 4. Eliminating z from equations (2) and (3).

Eliminating $z$ from equations (2) and (3).

$$\begin{gathered} 2x+y+3z=115.........\left(2\right) \\ -6x-3y-3z=-255....\left(3\right)\times -3 \end{gathered}$$

Solving the equations, we get,

$$\begin{gathered} -4x-2y=-140 \\ 2x+y=70...........\left(5\right) \end{gathered}$$

Step 5. Finding the value of x .

Solving equation (4) and (5).

$$\begin{gathered} 5x+y=145..........\left(4\right) \\ 2x+y=70............\left(5\right) \end{gathered}$$

Subtracting the equations, we get,

$$\begin{gathered} 3x=75 \\ x=25 \end{gathered}$$

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

Step 6. Finding the value of y .

Substituting $x=25$ in the equation

$2x+y=70$

$$\begin{gathered} 2\left(25\right)+y=70 \\ 50+y=70 \\ y=70-50 \\ y=20 \end{gathered}$$

The cost of a cap is $$20$

Step 7. Finding the value of z .

Substituting $$\begin{gathered} x=25 \\ y=20 \end{gathered}$$ in the equation

$4x+y+z=135$,

$$\begin{gathered} 4\left(25\right)+20+z=135 \\ 100+20+z=135 \\ 120+z=135 \\ z=135-120 \\ z=15 \end{gathered}$$

The price of a stuffed animal is $$15$