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# Rewriting Formulas and Equations

Perhaps you need to find the width of your pool, you've been given the volume, height, and length. These are all variables that are used in the formula to find the volume of a rectangular prism. You can still find the width of the pool, you just need to rewrite your formula! Continue reading to find out how you can do this.

Formulas are used to calculate something specific, they contain an equals sign and at least two variables. A formula is considered to always be true.

Let's look at some examples of formulas.

Some examples of formulas that you may be familiar with include:

• Area of a circle = $$\pi r^2$$
• Area of a triangle = $$\frac{1}{2}bh$$
• Distance = $$rt$$

Formulas involve two or more quantities, and variables and are used to find different things based on the formulas themselves.

An equation involves an equals sign and can contain missing values. It shows you that what is on one side of the equal sign is equal to what is on the right.

Here are some examples.

Some examples of equations include:

• $$3x+7=13$$
• $$x-6=2$$

Since equations do not have to always be true, the above equations will only be true if the missing variable allows for the calculation to be correct. For example in the equation $$3x + 7 = 13$$, $$x$$ must equal 2 in order for the equation to be true.

It can be helpful to know how to rearrange formulas and equations. For example you may be given the area of a circle and asked to find the radius.

## Rules for Rewriting Formulas and Equations

It will be helpful to know how to rewrite formulas so that it is possible to solve for a variable. When you rewrite a formula the aim is to create an equation that is equivalent to the formula but with the missing variable by itself. There are two main rules to follow to ensure that you are rewriting the formula correctly:

• Whether you are adding, subtracting, multiplying, or dividing make sure that you do it to both sides of the equation, this means both sides of the $$=$$ sign.

• When multiplying or dividing, make sure you apply it to all of the terms in the equation, don't leave any out.

Solve for $$H$$ in the equation, $$H+3=b$$.

To do this you can subtract 3 from both sides of the equation:

$H+3-3=b-3.$

Then you can simplify to get

$H=b-3.$

Rewriting formulas and equations can be helpful when you are trying to find specific answers.

Find the length of the rectangle with an area of $$20\,cm^2$$ and width of $$8\,cm$$.

To solve this you would first need to think about the correct formula to use. It can be helpful to first list all of the information you know:

• the shape is a rectangle
• the area is $$20\,cm^2$$
• the width is $$8\,cm$$ .

Now you can tell that you are going to need the formula for the area of a rectangle, which is

$A=lw$

where

\begin{align} A &= \text{ Area of rectangle} \\ l &= \text{ length of rectangle} \\ w &= \text{ width of rectangle} \end{align}

There are two ways you can go from here. Either one will give you exactly the same answer.

Method one: solve for the variable you need in the formula, then plug in values.

Lets have a look how to do this, you need to get $$l$$ by itself. First, the formula for the area is

$A=lw.$

Since you want to isolate the length, divide both sides by $$w$$ to get:

$\frac{A}{w}=\frac{lw}{w}.$

Then you can cancel on the right hand side to get:

$\frac{A}{w}=l.$

Now that you have rewritten the formula, you can input the variables and find the value of $$l$$:

$\frac{20}{8}=l$

so including the units $$l=2.5\,cm$$.

Method two: first plug in the information you have, then solve for the variable you need.

Plugging in the numbers you found above into the formula $$A=lw$$,

$20 = l\cdot 8.$

Then you can solve for $$l$$ by dividing both sides by $$8$$ and simplifying:

$\frac{20}{8} = l$

so

$l = 2.5.$

Don't forget to include the units! The length of the rectangle is $$l=2.5\,cm$$.

When rewriting formulas, it is important to remember, if you divide on one side, you need to do the same to the other side of the equation.

## Rewrite Algebraic Equations

You can rewrite algebraic equations too to help you find values of $$x$$ or $$y$$.

Solve $$5x+y=18$$ for $$y$$ when $$x=2$$.

To begin with, you need to rewrite the algebraic equation to make $$y$$ the subject, to do this you can subtract $$5x$$ from both sides of the equation

$5x+y=18$

giving you

$-5x+5x+y=18-5x.$

Then cancelling on the left side gives you

$y=18-5x.$

Now you can substitute the value of $$x = 2$$ to solve for $$y$$, which gives you

\begin{align} y&=18-5(2)\\ &=8. \end{align}

## Rewriting Equation in a Standard Form

In algebra, the standard form is a way of writing out an equation, for a line it follows the form $$Ax+By=C$$.

To rewrite an equation using standard form, you can simply move the terms of the equation around. Let's take a look at an example.

Rewrite the equation $$y=12-2x$$.

You can start by looking at the standard form:

$Ax+By=C$

To rewrite this in the standard form you need to move the $$x$$ value to the same side of the equation as the $$y$$ term. So adding $$2x$$ to both sides of the equation gives you

$2x+y=12.$

This is now written in standard form!

It is important to remember that the form $$Ax+By=C$$ is the standard form for a line, things like circles will have a different standard form.

## Rewriting Equation of a Function

A function takes an input and creates an output.

Let's look at some examples.

Functions are often written as:

• $$f(x)=x^2$$
• $$f(x)=2x^3$$

If you want to rewrite a function and turn it into an equation, you can simply replace the $$f(x)$$ with a $$y$$. There are lots of other properties of functions and things you can do with them. Take a look at the Functions article for more details.

## Examples of Rewriting Equations and Formulas

Find the radius of the circle with a circumference of $$35\, in$$.

Let's begin by looking at an appropriate formula;

$C=2 \pi r$

where

\begin{align} C &= \text{ circumference of the circle} \\ r &= \text{ radius of the circle}.\end{align}

Now you can rewrite the formula to help you find the radius. To do this you can divide both sides by $$2\pi$$;

$\frac{C}{2 \pi}=r.$

Now you can input your variables into the formula to find the radius;

$r=\frac{35}{2 \pi},$

so

$r \approx 5.57$

Don't forget units! The radius of the circle is approximately $$5.57\, in$$.

Let's take a look at another example.

Solve $$2x+2y=22$$ for $$y$$ when $$x=4$$.

Let's start by rewriting the formula in terms of $$y$$. Since

$2x+2y=22,$

you get

$2y=22-2x,$

and then dividing both sides by $$2$$

\begin{align} y &=\frac{22-2x}{2} \\ &= 11-x. \end{align}

Now you can substitute in the value of $$x=4$$, which gives you

\begin{align} y&=11 - 4 \\ &=7. \end{align}

## Rewrite Formulas and Equations - Key takeaways

• You can rewrite formulas and equations to help you find missing variables.
• It is important to remember that when you are subtracting, adding, multiplying or dividing, you need to do it to both sides of the equation.
• When you are multiplying or dividing, you need to remember to do it to all of the terms!

It is important to understand how to rewrite formula and equations as you may have a problem where the area of a triangle is given but you need to find its height. Here you can still use the area of a triangle formula, you simply need to rewrite it.

When rewriting an equation or a formula there are two important things to remember;

• When adding, subtracting, multiplying or dividing, you need to make sure you do it to both sides of the = sign.
• When multiplying or dividing, you need to remember to do it to all of the terms.

## Final Rewriting Formulas and Equations Quiz

Question

What does a formula include?

A formula includes is an equals sign and at least two variables.

Show question

Question

Why do you rewrite formulas?

Formulas are rewritten so that you can find the value of a missing variable.

Show question

Question

What does an equation show?

An equation shows you that what is on one side of the equal sign is equal to what is on the right.

Show question

Question

What are the important rules to remember when rewriting formulas and equations?

There are two rules to remember:

• Make sure that you complete operations on both sides of the equation.
• When multiplying or dividing, make sure you apply it to all of the terms in the equation.

Show question

Question

What is standard form for a line?

$$Ax + By = C$$

Show question

Question

To find the area of a rectangle you use the formula $$A=lw$$.

Which formula has been rewritten correctly to find the width of a rectangle when you have been given the area?

$$\frac{A}{l}=w$$

Show question

Question

$$3x+y=21$$

Which equation has been correctly rewritten in terms of $$y$$?

$$y=21-3x$$

Show question

Question

The equation $$y=15-5x$$ can be written in the standard form. Which of the below options is correct?

$$5x+y=15$$

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Question

When is an equation true?

When the missing variable allows the calculation to be correct.

Show question

Question

Why is it useful to rewrite equations?

It can be useful to rewrite equations so that you can find missing values.

Show question

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