When we have an equation, it does not matter if we know the value of a variable or not. We can use the Properties of Equality to rearrange the equation any which way we need and isolate any variable we would like to have isolated. This can help us greatly in real life applications of various formulas.

Example

We have been tasked with creating a triangular shaped garden for your friend's father's uncle's purple backyard. He has $20$ square feet of sod and the area of the garden will, therefore, be $20$ square feet. He wants one side lined up with his $5$ feet of patio and the other side of the right triangle will be running along the edge of the yard. We have the following basic sketch of the garden.

We know that the base

b

is

5

and the area

A

is

20 .

We need to find the height

h

that will allow this configuration. Let's remember the formula for the area of a triangle.

A = \frac{1}{2} b h

Let's solve it for height.

A = \frac{1}{2} b h

MultEqn

$LHS \cdot 2 = RHS \cdot 2$

2 A = b h

DivEqn

$LHS / b = RHS / b$

\frac{2 A}{b} = h

RearrangeEqn

Rearrange equation

h = \frac{2 A}{b}

Now that we have a formula for height of a triangle, we can find the height (length) of the garden side by substituting

A = 20

into the formula.

h = \frac{2 \cdot 2 0}{5} \Rightarrow h = 8

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