Question

in each figure, for what values of x and y is the figure a parallelogram?
a.
the figure is a parallelogram if x = \boxed{} and y = \boxed{}.
(simplify your answers.)
b.

Explanation:

Step1: Recall parallelogram side property

In a parallelogram, opposite sides are equal. So, for figure a, we have two pairs of opposite sides: \(3x + 8\) and \(4 - x\), \(y + 2\) and \(2y - 3\).
Set \(3x + 8 = 4 - x\) (opposite sides equal) and \(y + 2 = 2y - 3\) (opposite sides equal).

Step2: Solve for x

Solve \(3x + 8 = 4 - x\):
Add \(x\) to both sides: \(3x + x + 8 = 4\) → \(4x + 8 = 4\)
Subtract 8: \(4x = 4 - 8\) → \(4x = -4\)
Divide by 4: \(x = -1\)

Step3: Solve for y

Solve \(y + 2 = 2y - 3\):
Subtract \(y\) from both sides: \(2 = 2y - y - 3\) → \(2 = y - 3\)
Add 3: \(y = 2 + 3\) → \(y = 5\)

Answer:

\(x = -1\) and \(y = 5\)