Answer
The solution is $(2,5)$.
Work Step by Step
The given system of equations is
$x+4y=22$ ...... (1)
$4x+y=13$ ...... (2)
Multiply equation (2) by $-4$.
$-4(4x+y)=-4(13)$
Use distributive property.
$-16x-4y=-52$ ...... (3)
Add equation (1) and (3).
$\Rightarrow x+4y-16x-4y=22-52$
Add like terms.
$\Rightarrow -15x=-30$
Divide each side by $-15$.
$\Rightarrow \frac{-15x}{-15}=\frac{-30}{-15}$
Simplify.
$\Rightarrow x=2$
Substitute $2$ for $x$ equation (1).
$\Rightarrow 2+4y=22$
Subtract $2$ from each side.
$\Rightarrow 2+4y-2=22-2$
Simplify.
$\Rightarrow 4y=20$
Divide each side by $4$.
$\Rightarrow \frac{4y}{4}=\frac{20}{4}$
Simplify.
$\Rightarrow y=5$
Check $(x,y)=(2,5)$
Equation (1):
$\Rightarrow x+4y=22$
$\Rightarrow 2+4(5)=22$
$\Rightarrow 2+20=22$
$\Rightarrow 22=22$
True.
Equation (2):
$\Rightarrow 4x+y=13$
$\Rightarrow 4(2)+5=13$
$\Rightarrow 8+5=13$
$\Rightarrow 13=13$
True.
Hence, the solution is $(2,5)$.
