# How to find the length and width of a rectangle when the sides are given in terms of x (making equations).

Sometimes you will be asked to work out the length and width of a rectangle when the sides are given in terms of x and all that you are given is the value of the perimeter. To solve these types of perimeter questions you will need to set up an equation and solve it.

Since you are given the perimeter you need to make an equation for the perimeter of the rectangle. To do this add up the four sides of the rectangle so that you have an expression that involves x and make this equal the perimeter that is stated in the question. All you need to do now is solve this equation and this will tell you the value of x. Finally, sub this value of x back into the expressions for the length and width to give the lengths of the sides that are needed. To summarise:

1. First add up all the lengths of the rectangle to give an expression in terms of x.

2. Next make this equal to the perimeter of the rectangle to give an equation in terms of x.

3. Solve this equation to find x.

4. Sub this value of x into the expressions for the length and width.

5. Check your values by finding the perimeter of the rectangle using your calculated values (this step is optional).

Example

A rectangle has side lengths of 5x + 3 and x – 1. If the perimeter of the rectangle is 64cm work out the actual length and width of the rectangle.

First add up all the lengths of the rectangle to give an expression in terms of x:

5x + 3 + x – 1 + 5x + 3 + x – 1

= 12x + 4

Next make this equal to the perimeter of the rectangle to give an equation in terms of x:

12x + 4 = 64

Next solve this equation to find x:

12x + 4 = 64

12x = 64 -4

12x = 60

x = 60/12

x = 5

Now sub x =5 into the expressions 5x + 3 and x – 1 to give the values of the length and width of the rectangle:

Length = 5x + 3 = 5 × 5 + 3 = 28 cm

Width = x – 1 = 5 -1 = 4 cm

You can also check that these values are correct by adding them up and seeing if they give the perimeter of the rectangle given in the question. If they don’t give the perimeter in the question then you have probably set up the wrong equation or solved the equation incorrectly.

28 + 4 + 28 + 4 = 64cm (which is the perimeter of the rectangle)

So the length of the rectangle is 28 cm and the width of the rectangle is 4 cm.