Question

Solve the system by substitution:$\left\{\begin{array}{l}x-4y=-4\\ -3x+4y=0\end{array}\right.$

Short answer

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

Step-by-step solution

Step 1. Given Information.  

The system of equations is:

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

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

We'll solve for the second equation for $y$,

$-3x+4y=0$

$4y=3x$

$y=\frac{3}{4}x$

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

Replace the $y$with $\frac{3}{4}x$,

$x-4y=-4$

$x-4\left(\frac{3}{4}x\right)=-4$

Step 4. Solve the resulting equation.

Now we have an equation with just one variable.

$x-4\left(\frac{3}{4}x\right)=-4$

$x-3x=-4$

$2x=4$

$x=2$

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

We'll use the second equation and replace $x$with 2.

$-3x+4y=0$

$-3\left(2\right)+4y=0$

$4y=6$

$y=\frac{3}{2}$

Step 6. Write the solution as an ordered pair.  

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

$(2,\frac{3}{2})$

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

Substitute $x,y=2,\frac{3}{2}$into both equations and make sure they are both true.

$x-4y=-4$

$2-4\left(\frac{3}{2}\right)=-4$

$2-6=-4$

$-4=-4$

And,

$-3x+4y=0$

$-3\left(2\right)+4\left(\frac{3}{2}\right)=0$

$-6+6=0$

$0=0$

Both equations are True.

The point$(2,\frac{3}{2})$is the solution to the system.