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Rearranging equations

# Rearranging equations

Here you will learn about rearranging equations, including how to solve equations for a particular variable as well as change the subject of formulas.

Students first learn how to rearrange equations in 8 th grade math when they learn how to put linear equations in slope intercept form, y=mx+b. They expand this knowledge as they progress through Algebra I and Algebra II.

## What is rearranging equations?

Rearranging equations means to solve an equation for a particular variable or change the subject of the equation to write it in a different way.

The strategies used to rearrange an equation are the same as when solving an equation for an unknown variable, which means using inverse operations (in the opposite order of the order of operations) to change the subject of the formula.

Let’s look at a side-by-side comparison of solving an equation for the unknown variable and rearranging an equation for a particular variable (changing the subject).

Remember, when solving any equation, be sure to do the same exact thing to both sides of the equal sign.

Step-by-step guide: Solving equations

## Common Core State Standards

How does this apply to high school math?

• High School Algebra – Creating Equations (HSA-CED.A.4)
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V=IR to highlight resistance R.

## How to rearrange equations

In order to rearrange equations:

1. Identify the variable you need to isolate.
2. Use inverse operations and/or the distributive property to isolate the variable.
3. State the solution and simplify if possible.

## Rearranging equations examples

### Example 1: two step

Solve the equation for y.

5x+5y=10

1. Identify the variable you need to isolate.

In this case, solve the equation for y.

2Use inverse operations and/or the distributive property to isolate the variable.

First, subtract 5x from both sides of the equation because subtraction is the inverse operation to addition.

Then, divide both sides of the equation by 5 because division is the inverse operation of multiplication. Remember to divide every term by 5 .

3State the solution and simplify if possible.

y=2-x

### Example 2: distributive property

Rearrange the equation for x.

p=2(x-3)

Identify the variable you need to isolate.

Use inverse operations and/or the distributive property to isolate the variable.

State the solution and simplify if possible.

### Example 3: change the subject of a formula

Solve the formula for h.

A=bh

Identify the variable you need to isolate.

Use inverse operations and/or the distributive property to isolate the variable.

State the solution and simplify if possible.

### Example 4: equations with x²

Solve the equation for x.

y=x^{2}-4

Identify the variable you need to isolate.

Use inverse operations and/or the distributive property to isolate the variable.

State the solution and simplify if possible.

### Example 5: questions involving √x

Solve the equation for x.

y=\sqrt{3x}+n

Identify the variable you need to isolate.

Use inverse operations and/or the distributive property to isolate the variable.

State the solution and simplify if possible.

## How to rearrange equations

In order to rearrange equations with more than 1 of the same variable:

1. Identify the variables you need to isolate.
2. Use inverse operations to isolate the variables.
3. Apply the strategy of factoring.
4. State the solution and simplify if possible.

### Example 6: more than 1 of the same variable

Rearrange \cfrac{a}{3}=\cfrac{2-7x}{x-5} to make x the subject.

Identify the variables you need to isolate.

Use inverse operations to isolate the variables.

Apply the strategy of factoring.

State the solution and simplify if possible.

### Example 7: factoring of the variable is required

Solve the equation for z.

y=\cfrac{2xz}{z-x}

Identify the variables you need to isolate.

Use inverse operations to isolate the variables.

Apply the strategy of factoring.

State the solution and simplify if possible.

### Teaching tips for rearranging equations

• Demonstrate side-by-side examples of solving equations for a numerical solution with an equation that requires rearrangement of variables so that students can conceptualize that the steps are the same.

• Rearranging equations is a skill that is seen on exam questions such as the SAT or college placement tests, so infuse practice SAT exam questions as warm up questions or “do nows.”

### Easy mistakes to make

• Forgetting to apply the inverse operation to both sides of the equation
For example, with the equation y=5x-3 , forgetting to add 3 to both sides of the equation.

y 5x-3+3

y 5x

\begin{aligned}y+3&=5x-3+3 \\\\ y+3&=5x \\\\ \cfrac{y+3}{5}&=\cfrac{5x}{5} \\\\ \cfrac{y+3}{5}&=x \end{aligned}

• Using the incorrect inverse operation
For example, changing the subject from A to b formula:

A=bh (multiplying both sides of the formula by h instead of dividing by h )

A\times{h}=bh\times{h}

\begin{aligned}A&=bh \\\\ \cfrac{A}{h}&=\cfrac{bh}{h} \end{aligned}

• Forgetting to include the negative root of a square root
For example, when taking the square root to undo a squared term, remember to include the positive root and the negative root.
\sqrt{x} should be written as \pm\sqrt{x} .

• Forgetting to factor when the subject is in more than one term
For example, when solving the equation 2x+3xy=3y for x, you must factor the x from the left hand side of the equation so that it is only in one term and not two terms.

\begin{aligned}x(2+3y)&=3y \\ x&=\cfrac{3y}{2+3y} \end{aligned}

### Rearranging equations practice problems

1. Make a the subject of the formula h=3(a+7).

h-7=a

\cfrac{h}{3}-7=a

\cfrac{h}{3}-21=a

h-21=a

Solve the equation, h=3(a+7) for the variable a by first distributing the 3 into the binomial (a+7) then isolating a.

2. Solve the formula for h.

A=\cfrac{1}{2} \, bh

\cfrac{Ab}{2}=h

\cfrac{2A}{h}=b

\cfrac{2A}{b}=h

2Ab=h

Solving the formula A=\cfrac{1}{2} \, bh for h means to change the subject from A to h.

3. Make c the subject of the formula g=\sqrt{5 c-r}

g^{2}+r=c

\cfrac{g^{2}-r}{5}=c

\cfrac{g^{2}+r}{5}=c

g^{2}-r=c

Solve the equation for c\text{:}

4. Solve p=b^{2}-9k for b.

\pm\sqrt{p+9k}=b

p+9k=b

\pm\sqrt{p-9 k}=b

\pm\sqrt{9k-p}=b

Isolate the variable b\text{:}

5. Make d the subject of the formula y=\cfrac{3d+1}{4d}.

d=\cfrac{1}{4y}-3

d=\cfrac{1}{4y+3}

d=\cfrac{1}{4y}+3

d=\cfrac{1}{4y-3}

Rearrange the equation so that only one term contains the variable d then isolate this on one side of the equals sign:

6. Make e the subject of the formula.

\cfrac{q}{3}=\cfrac{6-2e}{e+1}

e=\cfrac{18-q}{q-6}

e=\cfrac{18-q}{q+6}

e=\cfrac{18+q}{q+6}

e=\cfrac{18-q}{6}

Rearrange the equation so that only one term contains the variable e then isolate this on one side of the equals sign:

## Rearranging equations FAQs

Yes, you can rearrange any algebraic equation by changing the subject.

Does rearranging algebraic equations help to graph them?

Yes, sometimes rearranging quadratic equations, linear equations, and/or any algebraic equation might be helpful when trying to graph them.

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