Question

1. sam is designing a rectangular garden. the four sides are labeled with the following expressions.

a. prove that the expressions below make a viable rectangle.
b. write a fully simplified expression for the perimeter.
c. what is the perimeter when x = 4 and y = 3?
2(x + 5) + 4
2(y + 1) + y - 3
3(y -
2(x + 7)

Explanation:

Step1: Simplify side - length expressions

First, simplify the expressions for the sides:

• For $2(x + 5)+4$, expand using the distributive property: $2(x + 5)+4=2x+10 + 4=2x + 14$.
• For $2(y + 1)+y-3$, expand and combine like - terms: $2(y + 1)+y-3=2y+2 + y-3=3y-1$.
• The other two sides are $3(y - 1)$ and $2(x + 7)=2x+14$.

In a rectangle, opposite sides are equal. Here, the sides with $x$ terms: $2x + 14$ and $2x+14$ are equal, and for the sides with $y$ terms, if we assume the two sides with $y$ terms are equal, we need to check if $3y-1$ and $3(y - 1)$ can be made equal in the context of the problem. But we can also just consider the property of opposite - side equality for rectangles. Since we have two pairs of expressions where one pair has the same $x$ - coefficient and constant combination and we can assume the other pair is for the other set of opposite sides, the expressions can form a rectangle.

Step2: Find the perimeter formula

The perimeter $P$ of a rectangle is the sum of all its sides.
$P=(2x + 14)+(3y-1)+(2x + 14)+(3y-1)$
$P = 4x+6y + 26$.

Step3: Substitute $x = 4$ and $y = 3$

Substitute $x = 4$ and $y = 3$ into the perimeter formula:
$P=4\times4+6\times3 + 26$
$P = 16+18 + 26$
$P=60$.

Answer:

a. Since there are two pairs of expressions that can represent opposite sides (the expressions with $x$ are equal and we can assume the expressions with $y$ represent the other pair of opposite sides), the expressions can form a rectangle.
b. The fully - simplified expression for the perimeter is $4x + 6y+26$.
c. The perimeter when $x = 4$ and $y = 3$ is $60$.