Question

the perimeter of a rectangle is 12 meters and the length is 2 meters longer than the width. find the dimensions of the rectangle. let ( x = ) the length and ( y = ) the width. the corresponding modeling system is ( \begin{cases} 2x + 2y = 12 \\ x - y = 2 end{cases} ). use the method of substitution to solve this system. length of the rectangle = (square) m. width of the rectangle = (square) m

Explanation:

Step1: Solve the second equation for x

From \( x - y = 2 \), we can express \( x \) in terms of \( y \) as \( x = y + 2 \).

Step2: Substitute x into the first equation

Substitute \( x = y + 2 \) into \( 2x + 2y = 12 \). We get \( 2(y + 2) + 2y = 12 \).

Step3: Simplify and solve for y

First, expand the left - hand side: \( 2y+4 + 2y=12 \).
Combine like terms: \( 4y+4 = 12 \).
Subtract 4 from both sides: \( 4y=12 - 4=8 \).
Divide both sides by 4: \( y=\frac{8}{4}=2 \).

Step4: Find the value of x

Substitute \( y = 2 \) into \( x = y + 2 \), so \( x=2 + 2 = 4 \).

Answer:

Length of the rectangle = \( 4 \) m.
Width of the rectangle = \( 2 \) m.