if the width of a rectangle is 5 inches shorter than the length and the perimeter of the rectangle is 80 inches.
how do you set up the equation for the perimeter involving only L and how do you solve this in a linear equation algebraically to find the length and the width
let the length be L
"..the width of a rectangle is 5 inches shorter than the length"
so width = L - 5
Twice the width + twice the length = 80
2L + 2(L-5) = 80
take it from there.
once you have L, evaluate L-5 for the width
Let's denote the length of the rectangle as L and the width as W.
According to the given information, the width of the rectangle is 5 inches shorter than the length. So, we can write the equation for the width as:
W = L - 5
The formula for the perimeter of a rectangle is given by:
Perimeter = 2(L + W)
Substituting the value of W in terms of L in the above equation, we can write the equation for the perimeter involving only L as:
80 = 2(L + (L - 5))
Simplifying this equation will allow us to solve for the length and width.
Let's solve this equation algebraically:
80 = 2(2L - 5)
80 = 4L - 10
4L = 90
L = 22.5
Now, we can substitute the value of L back into the equation for the width:
W = L - 5
W = 22.5 - 5
W = 17.5
Therefore, the length of the rectangle is 22.5 inches, and the width is 17.5 inches.
To set up the equation for the perimeter involving only the length (L), you need to understand that the perimeter is the sum of all sides of a rectangle. A rectangle has two pairs of equal sides, so the formula for the perimeter is:
Perimeter = 2 × (length + width)
In this case, the width is 5 inches shorter than the length, which means the width can be represented as (L - 5) inches. So, the equation for the perimeter can be written as:
80 = 2 × (L + (L - 5))
Now, we can solve this linear equation algebraically to find the length (L) and the width.
To solve it, we can start by simplifying the equation by distributing the 2 on the right side:
80 = 2L + 2(L - 5)
Next, we can simplify further by combining like terms:
80 = 2L + 2L - 10
Now, we can continue simplifying:
80 = 4L - 10
To isolate the variable, we need to move the constant term to the other side. We can do this by adding 10 to both sides:
80 + 10 = 4L
90 = 4L
Finally, we can solve for L by dividing both sides by 4:
90/4 = L
L = 22.5
So, the length of the rectangle is 22.5 inches.
To find the width, we can substitute the value of L into the expression for the width:
Width = L - 5 = 22.5 - 5 = 17.5
Therefore, the width of the rectangle is 17.5 inches.