Q48. - Algebra 2

Question

Solve each system of equations by using either substitution or elimination

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

Step-by-Step Solution

Answer

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

Step 1 – 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.

Step 2 – 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:

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

Solve the equation $9y=72$ for y:

 $\begin{array}{c}9y=72\\ \frac{9y}{9}=\frac{72}{9}\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}divide both sides by 9}\\ y=8\end{array}$

Step 3 – Find the value of x

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

$\begin{array}{c}4x+5y=20\\ 4x+5\left(8\right)=20\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}substitute 8 for }y\\ 4x+40=20\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}simplify}\\ 4x=-20\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}subtract 40 from both sides}\\ x=-5\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}divide both sides by 4}\end{array}$

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