Chapter 9.7, Problem 11IP

To determine

To solve : the given system of equations.

Answer to Problem 11IP

The solution of the given system of equations is $\text{(}5,16)$ .

Explanation of Solution

Given information :

The given system of equations:

  $\begin{array}{l}x+\frac{1}{4}y=9\\ x+y=21\end{array}$

Calculation :

As per problem,

Consider the system of equations:

  $\begin{array}{l}x+\frac{1}{4}y=9\\ x+y=21\end{array}$

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

  $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}\text{}x+\frac{1}{4}y=9\text{ [Write the first equation]}\\ x+\frac{1}{4}y-\frac{1}{4}y=9-\frac{1}{4}y\text{ [Subtract }\frac{1}{4}y\text{ from each side]}\\ x=9-\frac{1}{4}y\text{ [Simplify}]\end{array}$

Find the value of $y$ by substituting $9-\frac{1}{4}y$ for $x$ in the second equation.

  $\begin{array}{l}x+y=21\text{ [Write the second equation]}\\ 9-\frac{1}{4}y+y=21\text{ [Substitute }x=9-\frac{1}{4}y\text{]}\\ 9+\frac{3}{4}y=21\text{ [Combine like terms}]\\ 9+\frac{3}{4}y-9=21-9\text{ [Subtract 9 from each side}]\\ \frac{3}{4}y=12\text{ [Simplify}]\\ y=12\cdot \frac{4}{3}\text{ [Multiply both sides by }\frac{4}{3}]\\ y=16\text{ [Simplify}]\end{array}$

Substitute $16$ for $y$ in either equation to find the value of $x$ .

  $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}\text{}x+\frac{1}{4}y=9\text{ [Write the first equation]}\\ x+\frac{1}{4}(16)=9\text{ [}y=16\text{]}\\ x+4=9\text{ [Simplify}]\\ x=5\text{ [Subtract 4 from each side}]\end{array}$

The solution of this system of equations is $\text{(}5,16)$ .