Question

solve each problem. 13. a square and a rectangle have equal perimeters. the length of a side of the square is 4x − 1. the length of the rectangle is 2x + 2 and the width is 2x. write and solve an equation to find x.

Explanation:

Step1: Recall perimeter formulas

Perimeter of square: \( P_{square} = 4\times \text{side} \), so \( P_{square}=4(4x - 1) \).
Perimeter of rectangle: \( P_{rectangle}=2\times(\text{length}+\text{width}) \), so \( P_{rectangle}=2((2x + 2)+2x) \).

Step2: Set perimeters equal

Since perimeters are equal: \( 4(4x - 1)=2((2x + 2)+2x) \).

Step3: Simplify both sides

Left side: \( 4(4x - 1)=16x - 4 \).
Right side: \( 2(4x + 2)=8x + 4 \).

Step4: Solve for \( x \)

\( 16x - 4 = 8x + 4 \)
Subtract \( 8x \): \( 8x - 4 = 4 \)
Add 4: \( 8x = 8 \)
Divide by 8: \( x = 1 \).

Answer:

\( x = 1 \)