Learning Outcomes
- Use the distributive property to solve equations containing parentheses
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. To get rid of these unwanted parentheses, we use 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 [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex], [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.
Simple distribution and two-step equations
Example
Solve for [latex]a[/latex].
[latex]4\left(2a+3\right)=28[/latex]
In our next example, we will use the distributive property of multiplication over addition first, simplify, then use the division property to finally solve.
example
Solve: [latex]-3\left(n - 2\right)-6=21[/latex]
Remember—always simplify each side first.
Now you can try a similar problem.
In the video that follows, we show another example of how to use the distributive property to solve a multi-step linear equation.
Distribution and combining like terms
example
Solve: [latex]3\left(n - 4\right)-2n=-3[/latex]
Now you can try a few problems that involve distribution.
Distribution and simplifying on both sides
The next example has expressions on both sides that need to be simplified.
example
Solve: [latex]2\left(3k - 1\right)-5k=-2 - 7[/latex]
Now, you give it a try!
In the following video, we present another example of how to solve an equation that requires simplifying before using the addition and subtraction properties.
Using the distribution property on both sides of the 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]
Try It
In the following video, we solve another multi-step equation with two sets of parentheses.