Question

find the values of x and y.
x =
y =

Explanation:

Step1: Use the property of vertical - angles

Vertical angles are equal. The angle with measure $x + 52^{\circ}$ and the angle with measure $7x^{\circ}$ are vertical angles. So we set up the equation $x + 52=7x$.
$x + 52=7x$
$52=7x - x$
$52 = 6x$
$x=\frac{52}{6}=\frac{26}{3}$

Step2: Use the property of linear - pair

The angles $(4y)^{\circ}$ and $(7x)^{\circ}$ form a linear - pair, so their sum is $180^{\circ}$. Substitute $x = \frac{26}{3}$ into the equation.
$4y+7x=180$
$4y+7\times\frac{26}{3}=180$
$4y=180-\frac{182}{3}$
$4y=\frac{540 - 182}{3}=\frac{358}{3}$
$y=\frac{358}{3\times4}=\frac{179}{6}$

Answer:

$x=\frac{26}{3}$, $y = \frac{179}{6}$