Question

find the values of the variables. then find the side lengths of the square.
x=\square, y=\square

Explanation:

Step1: Solve for y (using square's equal sides)

In a square, adjacent sides are equal. So, \( y - 2 = 2y - 6 \) (or \( y - 2 = 3y - 10 \), either pair works). Let's use \( y - 2 = 2y - 6 \). Subtract \( y \) from both sides: \( -2 = y - 6 \). Add 6 to both sides: \( y = 4 \).

Step2: Verify y with another side

Check with \( y - 2 = 3y - 10 \). Substitute \( y = 4 \): \( 4 - 2 = 3(4) - 10 \) → \( 2 = 12 - 10 \) → \( 2 = 2 \). Correct.

Step3: Solve for x (using square's side length)

First, find the side length from y: \( y - 2 = 4 - 2 = 2 \)? Wait, no, wait: Wait, \( 2y - 6 = 2(4) - 6 = 8 - 6 = 2 \), \( 3y - 10 = 3(4) - 10 = 12 - 10 = 2 \), \( y - 2 = 4 - 2 = 2 \). Wait, but the top side is \( 2x - 10 \), which should equal the side length (2). So \( 2x - 10 = 2 \). Add 10: \( 2x = 12 \). Divide by 2: \( x = 6 \). Wait, wait, no—wait, maybe I messed up. Wait, if \( y = 4 \), then \( y - 2 = 2 \), \( 2y - 6 = 2 \), \( 3y - 10 = 2 \). Then the top side is \( 2x - 10 \), which must equal 2. So \( 2x - 10 = 2 \) → \( 2x = 12 \) → \( x = 6 \). Let's check: \( 2x - 10 = 12 - 10 = 2 \), which matches the other sides. So x=6, y=4.

Wait, let's recheck:

For y: Take two vertical sides: \( y - 2 \) and \( 2y - 6 \). Set equal: \( y - 2 = 2y - 6 \) → \( -2 + 6 = 2y - y \) → \( y = 4 \). Correct.

Then, horizontal sides: \( 2x - 10 \) and \( 3y - 10 \). Substitute y=4: \( 3(4) - 10 = 12 - 10 = 2 \). So \( 2x - 10 = 2 \) → \( 2x = 12 \) → \( x = 6 \). Correct.

Answer:

x = \(\boxed{6}\), y = \(\boxed{4}\)