Mathematics » Introducing Fractions » Solve Equations with Fractions

Solving Equations with Fractions Using the Multiplication Property of Equality

Solving Equations with Fractions Using the Multiplication Property of Equality

Consider the equation \(\frac{x}{4}=3.\) We want to know what number divided by \(4\) gives \(3.\) So to “undo” the division, we will need to multiply by \(4.\) The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.

The Multiplication Property of Equality

For any numbers \(a,b,\) and \(c,\)

\(\text{if}\phantom{\rule{0.2em}{0ex}}a=b,\text{then}\phantom{\rule{0.2em}{0ex}}ac=bc.\)

If you multiply both sides of an equation by the same quantity, you still have equality.

Let’s use the Multiplication Property of Equality to solve the equation \(\frac{x}{7}=-9.\)

Example

Solve: \(\frac{x}{7}=-9.\)

Solution

 Solving Equations with Fractions Using the Multiplication Property of Equality
Use the Multiplication Property of Equality to multiply both sides by \(7\). This will isolate the variable.Solving Equations with Fractions Using the Multiplication Property of Equality
Multiply.Solving Equations with Fractions Using the Multiplication Property of Equality
Simplify.Solving Equations with Fractions Using the Multiplication Property of Equality
Solving Equations with Fractions Using the Multiplication Property of EqualitySolving Equations with Fractions Using the Multiplication Property of Equality 
The equation is true.Solving Equations with Fractions Using the Multiplication Property of Equality 

Example

Solve: \(\frac{p}{-8}=-40.\)

Solution

Here, \(p\) is divided by \(-8.\) We must multiply by \(-8\) to isolate \(p.\)

 Solving Equations with Fractions Using the Multiplication Property of Equality
Multiply both sides by \(-8\)Solving Equations with Fractions Using the Multiplication Property of Equality
Multiply.Solving Equations with Fractions Using the Multiplication Property of Equality
Simplify.Solving Equations with Fractions Using the Multiplication Property of Equality
Check:  
Substitute \(p=320\).Solving Equations with Fractions Using the Multiplication Property of Equality 
The equation is true.Solving Equations with Fractions Using the Multiplication Property of Equality 

Look at the equation \(-y=15.\) Does it look as if \(y\) is already isolated? But there is a negative sign in front of \(y,\) so it is not isolated.

There are three different ways to isolate the variable in this type of equation. We will show all three ways in the example below.

Example

Solve: \(-y=15.\)

Solution

One way to solve the equation is to rewrite \(-y\) as \(-1y,\) and then use the Division Property of Equality to isolate \(y.\)

 Solving Equations with Fractions Using the Multiplication Property of Equality
Rewrite \(-y\) as \(-1y\).Solving Equations with Fractions Using the Multiplication Property of Equality
Divide both sides by −1.Solving Equations with Fractions Using the Multiplication Property of Equality
Simplify each side.Solving Equations with Fractions Using the Multiplication Property of Equality

Another way to solve this equation is to multiply both sides of the equation by \(-1.\)

 Solving Equations with Fractions Using the Multiplication Property of Equality
Multiply both sides by −1.Solving Equations with Fractions Using the Multiplication Property of Equality
Simplify each side.Solving Equations with Fractions Using the Multiplication Property of Equality

The third way to solve the equation is to read \(-y\) as “the opposite of \(y\text{.”}\) What number has \(15\) as its opposite? The opposite of \(15\) is \(-15.\) So \(y=-15.\)

For all three methods, we isolated \(y\) is isolated and solved the equation.

Check:

 Solving Equations with Fractions Using the Multiplication Property of Equality
Substitute \(y=-15\).Solving Equations with Fractions Using the Multiplication Property of Equality
Simplify. The equation is true.Solving Equations with Fractions Using the Multiplication Property of Equality

Solving Equations with a Fraction Coefficient

When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to \(1.\)

For example, in the equation:

\(\frac{3}{4}x=24\)

The coefficient of \(x\) is \(\frac{3}{4}.\) To solve for \(x,\) we need its coefficient to be \(1.\) Since the product of a number and its reciprocal is \(1,\) our strategy here will be to isolate \(x\) by multiplying by the reciprocal of \(\frac{3}{4}.\) We will do this in the example below.

Example

Solve: \(\frac{3}{4}x=24.\)

Solution

 Solving Equations with Fractions Using the Multiplication Property of Equality
Multiply both sides by the reciprocal of the coefficient.Solving Equations with Fractions Using the Multiplication Property of Equality
Simplify.Solving Equations with Fractions Using the Multiplication Property of Equality
Multiply.Solving Equations with Fractions Using the Multiplication Property of Equality
Check:Solving Equations with Fractions Using the Multiplication Property of Equality 
Substitute \(x=32\).Solving Equations with Fractions Using the Multiplication Property of Equality 
Rewrite \(32\) as a fraction.Solving Equations with Fractions Using the Multiplication Property of Equality 
Multiply. The equation is true.Solving Equations with Fractions Using the Multiplication Property of Equality 

Notice that in the equation \(\frac{3}{4}x=24,\) we could have divided both sides by \(\frac{3}{4}\) to get \(x\) by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.

Example

Solve: \(-\frac{3}{8}w=72.\)

Solution

The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.

 Solving Equations with Fractions Using the Multiplication Property of Equality
Multiply both sides by the reciprocal of \(-\frac{3}{8}\).Solving Equations with Fractions Using the Multiplication Property of Equality
Simplify; reciprocals multiply to one.Solving Equations with Fractions Using the Multiplication Property of Equality
Multiply.Solving Equations with Fractions Using the Multiplication Property of Equality
Check:Solving Equations with Fractions Using the Multiplication Property of Equality 
Let \(w=-192\).Solving Equations with Fractions Using the Multiplication Property of Equality 
Multiply. It checks.Solving Equations with Fractions Using the Multiplication Property of Equality 

Optional Video: Solve One Step Equations With Fraction by Multiplying

[Attributions and Licenses]


This is a lesson from the tutorial, Introducing Fractions and you are encouraged to log in or register, so that you can track your progress.

Log In

Ask Question, Post Comment, Tip or Contribution

Do NOT follow this link or you will be banned from the site!