# The length and width of a rectangle are 3x+1, and x+1, respectively. If the perimeter of the rectangle is 28, how long is each side?

Nov 13, 2016

$x = \frac{25}{8} \text{ "->" } x = 3 \frac{1}{8}$

#### Explanation:

$\textcolor{b l u e}{\text{Building the model}}$

sum of parts = perimeter = 28

2 sides + 2 lengths = 28

$2 \left(x + 1\right) + 2 \left(3 x + 1\right) = 28$

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$\textcolor{b l u e}{\text{Solving for } x}$

$2 x + 2 + 6 x + 1 = 28$

$8 x + 3 = 28$

Subtract 3 from both sides

$8 x = 25$

Divide both sides by 8

$x = \frac{25}{8}$

Nov 13, 2016

length = 10 units , width = 4 units.

#### Explanation:

The opposite sides of a rectangle are $\textcolor{b l u e}{\text{equal in length}}$

$\Rightarrow \text{perimeter} = 2 \left(3 x + 1\right) + 2 \left(x + 1\right)$

Also the perimeter = 28.

Thus, equating the 2 values for the perimeter gives.

$2 \left(3 x + 1\right) + 2 \left(x + 1\right) = 28 \leftarrow \text{ equation to be solved }$

distribute the brackets.

$6 x + 2 + 2 x + 2 = 28$

collect like terms on left side.

$\Rightarrow 8 x + 4 = 28$

subtract 4 from both sides.

$8 x \cancel{+ 4} \cancel{- 4} = 28 - 4$

$\Rightarrow 8 x = 24$

To solve for x, divide both sides by 8.

$\frac{\cancel{8} x}{\cancel{8}} = \frac{24}{8}$

$\Rightarrow x = 3 \text{ is the solution to the equation}$

Length of rectangle $= 3 x + 1 = \left(3 \times 3\right) + 1 = 10 \text{ units}$

Width of rectangle $= x + 1 = 3 + 1 = 4 \text{ units}$

check : (2xx10)+(2xx4)=20+8=28 color(white)(xx)✔︎