Question

Solve the system by substitution:

$\left\{\begin{array}{l}4x-y=0\\ 2x-3y=5\end{array}\right.$

Short answer

The solution of the system of equations is$(-\frac{1}{2},-2)$.

Step-by-step solution

Step 1. Given Information.   

System of equations is$\left\{\begin{array}{l}4x-y=0\\ 2x-3y=5\end{array}\right.$.

Step 2. Solve one of the equations for either variable.

We'll solve for the first equation for y.

$\begin{array}{rcl}4x-y& =& 0\\ y& =& 4x\end{array}$

Step 3. Substitute the expression from Step 2 into the other equation. 

Replace the $y$with $4x$,

$2x-3y=5$

$2x-3\left(4x\right)=5$

Step 4. Solve the resulting equation.

Now we have an equation with just one variable.

$\begin{array}{rcl}2x-3\left(4x\right)& =& 5\\ 2x-12x& =& 5\\ -10x& =& 5\\ x& =& -\frac{1}{2}\end{array}$

Step 5. Substitute the solution from Step 4 into one of the original equations to find the other variable. 

We'll use the first equation and replace $x$with $-\frac{1}{2}$.

localid="1644477708322" $\begin{array}{rcl}4x-y& =& 0\\ 4(-\frac{1}{2})-y& =& 0\\ -2-y& =& 0\\ y& =& -2\end{array}$

Step 6. Write the solution as an ordered pair.

The ordered pair is $(x,y).$

$(-\frac{1}{2},-2)$

Step 7. Check that the ordered pair is a solution to both original equations.   

Substitute $x,y=-\frac{1}{2},-2$into both equations and make sure they are both true.

$4x-y=0$

$4(-\frac{1}{2})-(-2)=0$

$-2+2=0$

$0=0$

And,

$2x-3y=5$

$2(-\frac{1}{2})-3(-2)=5$

$-1+6=5$

$5=5$

Both equations are True.

The point $(-\frac{1}{2},-2)$is the solution to the system.