Question

Solve for x and y. Give exact values, do not round

Answer 1

First, we express the following by supplementary angles.

$\begin{gathered} 8x+58+3x+2y=180 \\ 11x+2y=180-58 \\ 11x+2y=122 \end{gathered}$

Then, we use the interior angles theorem which states that the sum of all three interior angles of a triangle is 180°.

$\begin{gathered} 10y+27+6y+3x+2y=180 \\ 18y+3x=180-27 \\ 18y+3x=153 \end{gathered}$

Now, we form a system of linear equations with the equations we found.

$\mleft\{\begin{aligned}11x+2y=122 \\ 18y+3x=153\end{aligned}\mright.$

Let's multiply the first equation by -9 to sum them and eliminate y.

$\begin{gathered} \mleft\{\begin{aligned}-99x-18y=-1098 \\ 18y+3x=153\end{aligned}\mright. \\ -99x+18y-18y+3x=153-1098 \\ -96x=-945 \\ x=-\frac{945}{-96}=\frac{315}{32} \end{gathered}$

Then, we find y.

$\begin{gathered} 11x+2y=122 \\ 11\cdot\frac{315}{32}+2y=122 \\ \frac{3465}{32}+2y=122 \\ 2y=122-\frac{3465}{32} \\ 2y=\frac{3904-3465}{32} \\ 2y=\frac{439}{32} \\ y=\frac{439}{64} \end{gathered}$

Hence, x is equal to 315/32 and y is equal to 439/64.