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=-(945)/(-96)=(315)/(32) \end{gathered}`

Then, we find y.

`\begin{gathered} 11x+2y=122 \\ 11\cdot(315)/(32)+2y=122 \\ (3465)/(32)+2y=122 \\ 2y=122-(3465)/(32) \\ 2y=(3904-3465)/(32) \\ 2y=(439)/(32) \\ y=(439)/(64) \end{gathered}`

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