Chapter 9.7, Problem 20IP

To determine

To represent : the given situation by a system of equations and solve it.

To interpret the solution of the system of equations.

Answer to Problem 20IP

The system of equations representing the given situation:

  $\begin{array}{l}x+y=5\\ 5x+10y=35\end{array}$

The solution of the system of equations is $\text{(3},2)$ .

From the solution of the system of equations, it can be concluded thatWil bought $3$

  $\$5$ T-shirts (represented by $x$ in the equations) and $2$

  $\$10$ T-shirts (represented by $y$ in the equations) at $\$35$ .

Explanation of Solution

Given information :

Wil paid $\$35$ for $5$ T-shirts. Some of the T-shirts cost $\$5$ each and the rest cost $\$10$ each.

Calculation :

As per problem,

Let the number of $\$5$ T-shirts bought by Wil be $x$ and the number of $\$10$ T-shirts be $y$ .

Total T-shirts bought by Wil is $5$ , and the amount paid for the $5$ T-shirts is $\$35$ .

The given situationcan be represented by the followingsystem of equations:

  $\begin{array}{l}x+y=5\\ 5x+10y=35\end{array}$

Solve the first equation for $x$ since its coefficient is $1$ .

  $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}\text{}x+y=5\text{ [Write the first equation]}\\ x+y-y=5-y\text{ [Subtract }y\text{ from each side]}\\ x=5-y\text{ [Simplify}]\end{array}$

Find the value of $y$ by substituting $5-y$ for $x$ in the second equation.

  $\begin{array}{l}\text{ 5}x+10y=35\text{ [Write the second equation]}\\ \text{5(5}-y)+10y=35\text{ [Substitute }x=5-y\text{]}\\ 25-5y+10y=35\text{ [Distributive property}]\\ 25+5y=35\text{ [Combine like terms}]\\ 25+5y-25=35-25\text{ [Subtract 25 from each side}]\\ \text{ 5}y=10\text{ [Simplify}]\\ y=2\text{ [Divide both sides by 5}]\end{array}$

Substitute $2$ for $y$ in either equation to find the value of $x$ .

  $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}x+y=5\text{ [Write the first equation]}\\ x+2=5\text{ [}y=2\text{]}\\ x=3\text{ [Subtract 2 from each side}]\end{array}$

The solution of this system of equations is $\text{(3},2)$ .

Thus, Wil bought $3$

  $\$5$ T-shirts (represented by $x$ in the equations) and $2$

  $\$10$ T-shirts (represented by $y$ in the equations) at $\$35$ .