Chapter 4.3, Problem 52HP

To determine

Find the price of each item.

Answer to Problem 52HP

Option A is correct.

  $shirt,\$12.95;pants,\$15.99;shoes,\$23.49$

Explanation of Solution

Given information:

Table shows the amount spent on each item.

  

Options:-

  $\begin{array}{l}Ashirt,\$12.95;pants,\$15.99;shoes,\$23.49\\ Bshirt,\$15.99;pants,\$12.95;shoes,\$23.49\\ Cshirt,\$15.99;pants,\$23.49;shoes,\$12.95\\ Dshirt,\$23.49;pants,\$15.99;shoes,\$12.95\end{array}$

Calculation:

Let consider price of each item as

  $\begin{array}{l}x=shirtprice\\ y=pantsprice\\ z=shoesprice\end{array}$

Now make a system of equations,

  $\begin{array}{l}3x+4y+2z=\mathrm{149.79....}\left(1\right)\\ 5x+3y+3z=\mathrm{183.19......}\left(2\right)\\ 6x+5y+z=\mathrm{181.14......}\left(3\right)\end{array}$

Solve equation $\left(1\right)$ and $\left(3\right)$ to eliminate one variable, by multiplying equation $\left(3\right)$ to $-2$

  $\begin{array}{l}addtheequations\\ -2\left(6x+5y+z=181.14\right)\\ \underset{\_}{\begin{array}{l}-12x-10y-2z=-362.28\\ 3x+4y+2z=149.79\end{array}}\\ -9x-6y=-212.49\end{array}$

Now consider equation $\left(2\right)$ and $\left(3\right)$ by multiplying equation $\left(3\right)$ to $-3$ ,

  $\begin{array}{l}addtheequations\\ -3\left(6x+5y+z=181.14\right)\\ \underset{\_}{\begin{array}{l}-18x-15y-3z=-362.28\\ 5x+3y+3z=183.19\end{array}}\\ -13x-12y=-360.23\end{array}$

Now multiply equation $-9x-6y=-212.49$ with $-2$ and add to $-13x-12y=-360.23$ we get,

  $\begin{array}{l}\underset{\_}{\begin{array}{l}-13x-12y=-360.23\\ 18x+12y=424.98\end{array}}\\ 5x=64.75\\ x=64.75/5=12.95\end{array}$

Now substitute $x=12.95$ in equation, $-13x-12y=-360.23$ we get,

  $\begin{array}{l}-13\left(12.95\right)-12y=-360.23\\ -168.35-12y=-360.23\\ -12y=-191.88\\ y=15.99\end{array}$

Now substitute $x=12.95,y=15.99$ in equation $3x+4y+2z=149.79$ we get,

  $\begin{array}{l}3\left(12.95\right)+4\left(15.99\right)+2z=149.79\\ 38.85+63.96+2z=149.79\\ 2z=149.79-63.96-38.85\\ z=23.98\end{array}$

Hence,

  $shirt,\$12.95;pants,\$15.99;shoes,\$23.49$