Question

1. find the value of y.
2. find the value of x.

Explanation:

Step1: Analyze left - hand side parallel lines

For the parallel lines on the left - hand side, the corresponding angles are equal. So we have the equation \(27x + 4=8x + 1\).
\[

$$\begin{align*} 27x+4&=8x + 1\\ 27x-8x&=1 - 4\\ 19x&=- 3\\ x&=-\frac{3}{19} \end{align*}$$

\]
Also, \(y + 10\) and \(27x+4\) are alternate interior angles. Substitute \(x =-\frac{3}{19}\) into \(27x + 4\):
\[27\times(-\frac{3}{19})+4=\frac{-81 + 76}{19}=-\frac{5}{19}\]
Then \(y+10=-\frac{5}{19}\), so \(y=-\frac{5}{19}-10=-\frac{5+190}{19}=-\frac{195}{19}\)

Step2: Analyze right - hand side parallel lines

For the parallel lines on the right - hand side, the sum of the interior angles on the same side of the transversal is \(180^{\circ}\). So we have the equation \((x + 8)+(6x-58)=180\).
\[

$$\begin{align*} x + 8+6x-58&=180\\ 7x-50&=180\\ 7x&=180 + 50\\ 7x&=230\\ x&=\frac{230}{7} \end{align*}$$

\]

Answer:

For the left - hand side problem: \(y =-\frac{195}{19}\)
For the right - hand side problem: \(x=\frac{230}{7}\)