Learning Outcomes
- Use properties of equality to isolate variables and solve algebraic equations
- Use the distributive property
- Clear fractions and decimals from equations to make them easier to solve
Use properties of equality to isolate variables and solve algebraic equations
There are some equations that you can solve in your head quickly. For example—what is the value of y in the equation [latex]2y=6[/latex]? Chances are you didn’t need to get out a pencil and paper to calculate that [latex]y=3[/latex]. You only needed to do one thing to get the answer: divide 6 by 2.
Other equations are more complicated. Solving [latex]\displaystyle 4\left( \frac{1}{3}t+\frac{1}{2}\right)=6[/latex] without writing anything down is difficult. That’s because this equation contains not just a variable but also fractions and terms inside parentheses. This is a multi-step equation, one that takes several steps to solve. You’ve probably studied these equations previously and seen that, although multi-step equations take more time and require more operations, they can still be simplified and solved by applying basic algebraic rules.
Remember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The addition property of equality and the multiplication property of equality explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you’ll keep both sides of the equation equal.
If the equation is in the form [latex]ax+b=c[/latex], where x is the variable, you can solve the equation as before. First “undo” the addition and subtraction, and then “undo” the multiplication and division.
Example
Solve [latex]3y+2=11[/latex].
Watch the following video to see examples of solving two step linear equations.
Example
Solve [latex]3x+5x+4-x+7=88[/latex].
The following video shows an example of solving a linear equation that requires combining like terms.
Some equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[/latex].
To solve this equation, we need to “move” one of the variable terms. This can make it difficult to decide which side to work with. It doesn’t matter which term gets moved, [latex]4x[/latex] or [latex]2x[/latex], however, to avoid negative coefficients, you can move the smaller term.
Examples
Solve: [latex]4x-6=2x+10[/latex]
In this video, you can see an example of solving equations that have variables on both sides of the equal sign.
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, see 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.
If you are more comfortable working directly with the fractions, you can. It’s handy to have this technique available though, especially when the fractions are complicated or involve larger numbers.
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.
