7.1.c – Simplifying Equations Before Solving

Learning Outcomes

Solve equations that need to be simplified

Steps With an End In Sight

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 [latex]y[/latex] 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 [latex]6[/latex] by [latex]2[/latex].

Other equations are more complicated. Although multi-step equations take more time and more operations to solve, they can still be simplified and solved by applying basic algebraic rules. In this section, we will look at equations that require some additional steps before they can be solved.

Combining Like Terms

Many equations start out more complicated than the ones we’ve just solved in the previous section. Let’s work through some examples that will employ simplifying by combining like terms.

Example

Solve: [latex]8x+9x - 5x=-3+15[/latex]

Solution:

First, we need to simplify both sides of the equation as much as possible.

Start by combining like terms to simplify each side.

	[latex]8x+9x-5x=-3+15[/latex]
Combine like terms.	[latex]12x=12[/latex]
Divide both sides by 12 to isolate x.	[latex]\Large\frac{12x}{\color{red}{12}}\normalsize =\Large\frac{12}{\color{red}{12}}[/latex]
Simplify.	[latex]x=1[/latex]
Check your answer. Let [latex]x=1[/latex]
[latex]8x+9x-5x=-3+15[/latex]
[latex]8\cdot\color{red}{1}+9\cdot\color{red}{1}-5\cdot\color{red}{1}\stackrel{\text{?}}{=}-3+15[/latex]
[latex]8+9-5\stackrel{\text{?}}{=}-3+15[/latex]
[latex]12=12\quad\checkmark[/latex]

Example

Solve [latex]3x+5x+4-x+7=88[/latex]

Show Solution

Here is a similar problem for you to try.

Try it

In the following video, we show an example of solving a linear equation that requires combining “like” terms.

Solving equations when the variables are on the right side of the equation

You may not always have the variables on the left side of the equation, so we will show an example with variables on the right side. You will see that the properties used to solve this equation are exactly the same as the previous example.

example

Solve: [latex]11 - 20=17y - 8y - 6y[/latex]

Show Solution

Notice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side of the equal sign.

Now you can try solving a similar problem.

try it

Solving equations when the variables are on both sides of the equation

You may have noticed that in all the equations we have solved so far, we had variables on only one side of the equation. This does not happen all the time—so now we’ll see how to solve equations where there are variable terms on both sides of the equation, as in this equation: [latex]4x-6=2x+10[/latex]. We will start like we did above—choosing a variable side and a constant side, and then use the Subtraction and Addition Properties of Equality to collect all variables on one side and all constants on the other side. Remember, what you do to the left side of the equation, you must do to the right side as well.

To solve [latex]4x-6=2x+10[/latex], 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]

Show Solution

In this video, we show an example of solving equations that have variables on both sides of the equal sign.

In the next example, the variable, [latex]x[/latex], is on both sides, but the constants appear only on the right side, so we’ll make the right side the “constant” side. Then the left side will be the “variable” side.

ExampleS

Solve: [latex]5x=4x+7[/latex]

Show Solution

Solve: [latex]7x=-x+24[/latex].

Show Solution

Did you see the subtle difference between the two equations? In the first, the right side looked like this: [latex]4x+7[/latex], and in the second, the right side looked like this: [latex]-x+24[/latex]. Even though they look different, we still used the same techniques to solve both.

Now you can try solving an equation with variables on both sides, where it is beneficial to move the variable term to the left side.

try it

In our last examples, we moved the variable term to the left side of the equation. In the next example, you will see that it is beneficial to move the variable term to the right side of the equation. There is no “correct” side to move the variable term, but the choice can help you avoid working with negative signs.

example

Solve: [latex]5y - 8=7y[/latex]

Show Solution

Now you can try solving an equation where it is beneficial to move the variable term to the right side.

try it

Solve Equations with Variables and Constants on Both Sides

The next example will be the first to have variables and constants on both sides of the equation. As we did before, we’ll collect the variable terms to one side and the constants to the other side. You will see that as the number of variable and constant terms increases, so do the number of steps it takes to solve the equation.

Examples

Solve: [latex]7x+5=6x+2[/latex]

Show Solution

Solve: [latex]6n - 2=-3n+7[/latex]

Show Solution

In the following video we show an example of how to solve a multi-step equation by moving the variable terms to one side and the constants to the other side. You will see that it doesn’t matter which side you choose to be the variable side; you can get the correct answer either way.

In the next example, we move the variable terms to the right side, to keep a positive coefficient on the variable.

EXAMPLE

Solve: [latex]2a - 7=5a+8[/latex]

Show Solution

The following video shows another example of solving a multi-step equation by moving the variable terms to one side and the constants to the other side.

Try these problems to see how well you understand how to solve linear equations with variables and constants on both sides of the equal sign.

try it

We just showed a lot of examples of different kinds of linear equations you may encounter. There are some good habits to develop that will help you solve all kinds of linear equations. We’ll summarize the steps we took so you can easily refer to them.

Solve an equation with variables and constants on both sides

Choose one side to be the variable side, and then the other will be the constant side.
Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
Make the coefficient of the variable [latex]1[/latex], using the Multiplication or Division Property of Equality.
Check the solution by substituting it into the original equation.

In the next section, we will look at how to solve equations with parentheses by using the Distribution Property.

	[latex]11-20=17y-8y-6y[/latex]
Simplify each side.	[latex]-9=3y[/latex]
Divide both sides by 3 to isolate y.	[latex]\Large\frac{-9}{\color{red}{3}}\normalsize =\Large\frac{3y}{\color{red}{3}}[/latex]
Simplify.	[latex]-3=y[/latex]
Check your answer. Let [latex]y=-3[/latex]
[latex]11-20=17y-8y-6y[/latex]
[latex]11-20\stackrel{\text{?}}{=}17( \color{red}{-3})-8(\color{red}{-3})-6(\color{red}{-3})[/latex]
[latex]11-20\stackrel{\text{?}}{=}-51+24+18[/latex]
[latex]-9=-9\quad\checkmark[/latex]

[latex]5x[/latex] is the side containing only a variable.[latex]4x+7[/latex] is the side containing a constant.
We don’t want any variables on the right, so subtract the [latex]4x[/latex] .		[latex]5x\color{red}{-4x}=4x\color{red}{-4x}+7[/latex]
Simplify.		[latex]x=7[/latex]
We have all the variables on one side and the constants on the other. We have solved the equation.
Check:		[latex]5x=4x+7[/latex]
Substitute [latex]7[/latex] for [latex]x[/latex] .		[latex]5(\color{red}{7})\stackrel{\text{?}}{=}4(\color{red}{7})+7[/latex]
		[latex]35\stackrel{\text{?}}{=}28+7[/latex]
		[latex]35=35\quad\checkmark[/latex]

[latex]7x[/latex] is the side containing only a variable.[latex]-x+24[/latex] is the side containing a constant.
Remove the [latex]-x[/latex] from the right side by adding [latex]x[/latex] to both sides.	[latex]7x\color{red}{+x}=-x\color{red}{+x}+24[/latex]
Simplify.	[latex]8x=24[/latex]
All the variables are on the left and the constants are on the right. Divide both sides by [latex]8[/latex].	[latex]\Large\frac{8x}{\color{red}{8}}\normalsize =\Large\frac{24}{\color{red}{8}}[/latex]
Simplify.	[latex]x=3[/latex]
Check:	[latex]7x=-x+24[/latex]
Substitute [latex]x=3[/latex].	[latex]7(\color{red}{3})\stackrel{\text{?}}{=}-(\color{red}{3})+24[/latex]
	[latex]21=21\quad\checkmark[/latex]

[latex]5y-8[/latex] is the side containing a constant.[latex]7y[/latex] is the side containing only a variable.
Subtract [latex]5y[/latex] from both sides.	[latex]5y\color{red}{-5y}-8=7y\color{red}{-5y}[/latex]
Simplify.	[latex]-8=2y[/latex]
We have the variables on the right and the constants on the left. Divide both sides by [latex]2[/latex].	[latex]\Large\frac{-8}{\color{red}{2}}\normalsize =\Large\frac{2y}{\color{red}{2}}[/latex]
Simplify.	[latex]-4=y[/latex]
Rewrite with the variable on the left.	[latex]y=-4[/latex]
Check:	[latex]5y-8=7y[/latex]
Let [latex]y=-4[/latex].	[latex]5(\color{red}{-4})-8\stackrel{\text{?}}{=}7(\color{red}{-4})[/latex]
	[latex]-20-8\stackrel{\text{?}}{=}-28[/latex]
	[latex]-28=-28\quad\checkmark[/latex]

	[latex]7x+5=6x+2[/latex]
Collect the variable terms to the left side by subtracting [latex]6x[/latex] from both sides.	[latex]7x\color{red}{-6x}+5=6x\color{red}{-6x}+2[/latex]
Simplify.	[latex]x+5=2[/latex]
Now, collect the constants to the right side by subtracting [latex]5[/latex] from both sides.	[latex]x+5\color{red}{-5}=2\color{red}{-5}[/latex]
Simplify.	[latex]x=-3[/latex]
The solution is [latex]x=-3[/latex] .
Check:	[latex]7x+5=6x+2[/latex]
Let [latex]x=-3[/latex].	[latex]7(\color{red}{-3})+5\stackrel{\text{?}}{=}6(\color{red}{-3})+2[/latex]
	[latex]-21+5\stackrel{\text{?}}{=}-18+2[/latex]
	[latex]16=16\quad\checkmark[/latex]

Module 7: Linear Equations