Question

Solve each system of equations by using either substitution or elimination

$\begin{array}{c}4x+5y=20\\ 5x+4y=7\end{array}$

Short answer

The solution of the system of equations is $\left(x,y\right)=\left(-5,8\right)$.

Step-by-step solution

– Apply elimination method to solve the system of equations.

Elimination method enables us to eliminate or get rid of one of the variable, so we can solve a more simplified equation.

– Evaluate the system of equations

Given system of equations:

$\begin{array}{c}4x+5y=20\\ 5x+4y=7\end{array}$

Multiply the first equation, $4x+5y=20$by 5 and second equation, $5x+4y=7$by 4.

Then subtract the equations to eliminate the x variable as follows:

localid="1648726261833" $\begin{array}{l}\underset{¯}{\begin{array}{c}4x+5y=20\\ 5x+4y=7\end{array}}\begin{array}{c}\text{multiplyby5}\to \\ \text{multiplyby4}\to \end{array}\underset{¯}{\begin{array}{c}20x+25y=100\\ \underset{-}{20x}+\underset{-}{16y}=\underset{-}{28}\end{array}}\\ 0x+9y=72\end{array}$

Solve the equation $9y=72$for y:

$\begin{array}{c}9y=72\\ \frac{9y}{9}=\frac{72}{9}\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}dividebothsidesby9}\\ y=8\end{array}$

– Find the value of x

Substitute $y=8$in $4x+5y=20$and solve for $x:$

$\begin{array}{c}4x+5y=20\\ 4x+5\left(8\right)=20\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}substitute8for}y\\ 4x+40=20\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}simplify}\\ 4x=-20\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}subtract40frombothsides}\\ x=-5\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}dividebothsidesby4}\end{array}$

So, the solution of the given system is $\left(x,y\right)=\left(-5,8\right)$.