The Distributive Property and Equations with Fractional and Decimal Coefficients

Learning Objectives

  • Use the distributive property
    • Use the properties of equality and the distributive property to solve equations containing parentheses
    • Clear fractions and decimals from equations to make them easier to solve

The Distributive Property

As we solve linear equations, we often need to do some work to write the linear equations in a form we are familiar with solving. This section will focus on manipulating an equation we are asked to solve in such a way that we can use the skills we learned for solving multi-step equations to ultimately arrive at the solution.

Parentheses can make solving a problem difficult, if not impossible. To get rid of these unwanted parentheses we have the distributive property. Using this property we multiply the number in front of the parentheses by each term inside of the parentheses.

The Distributive Property of Multiplication

For all real numbers a, b, and c, [latex]a(b+c)=ab+ac[/latex].

What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually. Then, you can follow the steps we have already practiced to isolate the variable and solve the equation.

Example

Solve for [latex]a[/latex]. [latex]4\left(2a+3\right)=28[/latex]

In the video that follows, we show another example of how to use the distributive property to solve a multi-step linear equation.

In the next example, you will see that there are parentheses on both sides of the equal sign, so you will need to use the distributive property twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!

Example

Solve for [latex]t[/latex].  [latex]2\left(4t-5\right)=-3\left(2t+1\right)[/latex]

In the following video, we solve another multi-step equation with two sets of parentheses.

Sometimes, you will encounter a multi-step equation with fractions. If you prefer not working with fractions, you can use the multiplication property of equality to multiply both sides of the equation by a common denominator of all of the fractions in the equation. This will clear all the fractions out of the equation. See the example below.

Example

Solve  [latex]\frac{1}{2}x-3=2-\frac{3}{4}x[/latex] by clearing the fractions in the equation first.

Of course, if you like to work with fractions, you can just apply your knowledge of operations with fractions and solve.

In the following video, we show how to solve a multi-step equation with fractions.

Regardless of which method you use to solve equations containing variables, you will get the same answer. You can choose the method you find the easiest! Remember to check your answer by substituting your solution into the original equation.

Sometimes, you will encounter a multi-step equation with decimals. If you prefer not working with decimals, you can use the multiplication property of equality to multiply both sides of the equation by a a factor of 10 that will help clear the decimals. See the example below.

Example

Solve [latex]3y+10.5=6.5+2.5y[/latex] by clearing the decimals in the equation first.

In the following video, we show another example of clearing decimals first to solve a multi-step linear equation.

Here are some steps to follow when you solve multi-step equations.

Solving Multi-Step Equations

1. (Optional) Multiply to clear any fractions or decimals.

2. Simplify each side by clearing parentheses and combining like terms.

3. Add or subtract to isolate the variable term—you may have to move a term with the variable.

4. Multiply or divide to isolate the variable.

5. Check the solution.

Summary

Complex, multi-step equations often require multi-step solutions. Before you can begin to isolate a variable, you may need to simplify the equation first. This may mean using the distributive property to remove parentheses or multiplying both sides of an equation by a common denominator to get rid of fractions. Sometimes it requires both techniques.