Question

Isaiah draws a quadrilateral WXYZ. If side WX = x - 20, side XY = 6y + 4, side YZ = 4 - 3x, and WZ = 4y + 22, for what values of x and y can Isaiah be sure his quadrilateral is a parallelogram?(1 point)
Responses

x = 1 and y = 4
x = 1 and y = 4

x = 16 and y = 9
x = 16 and y = 9

x = 9 and y = 6
x = 9 and y = 6

x = 6 and y = 9

Answer 1

In a parallelogram, opposite sides must be equal. So, we have the following equations based on the sides of the quadrilateral WXYZ:

1. WX = YZ
2. XY = WZ

Based on the given expressions for the sides, we can set up the following equations:

1. \( x - 20 = 4 - 3x \)
2. \( 6y + 4 = 4y + 22 \)

Let's solve these equations one by one.

### For the first equation:

\[
x - 20 = 4 - 3x
\]

Add \( 3x \) to both sides:

\[
x + 3x - 20 = 4
\]

Combine like terms:

\[
4x - 20 = 4
\]

Add \( 20 \) to both sides:

\[
4x = 24
\]

Now, divide by \( 4 \):

\[
x = 6
\]

### For the second equation:

\[
6y + 4 = 4y + 22
\]

Subtract \( 4y \) from both sides:

\[
2y + 4 = 22
\]

Subtract \( 4 \) from both sides:

\[
2y = 18
\]

Now divide by \( 2 \):

\[
y = 9
\]

So, \( x = 6 \) and \( y = 9 \).

Thus, the values of \( x \) and \( y \) that ensure the quadrilateral WXYZ is a parallelogram are:

**x = 6 and y = 9**.