CR.4: Solving Linear Equations

Learning Outcomes

Determine whether a number is a solution to an equation
Review the use of the subtraction and addition properties of equality to solve linear equations
Solve a linear equation that needs to be simplified before using the subtraction and addition properties of equality
Check your solution to a linear equation to verify its accuracy
Review and use the division and multiplication properties of equality to solve linear equations
Use a reciprocal to solve a linear equation that contains fractions
Solve a linear equation that requires simplification before using properties of equality
Solve a linear equation that requires a combination of the properties of equality
Solve linear equations by isolating constants and variables
Solve linear equations with variables on both sides that require several steps

Using the Subtraction and Addition Properties for Single-Step Equations

We began our work solving equations in previous chapters, where we said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.

Solution of an Equation

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.

Determine whether a number is a solution to an equation.

Substitute the number for the variable in the equation.
Simplify the expressions on both sides of the equation.
Determine whether the resulting equation is true.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.

In the following example, we will show how to determine whether a number is a solution to an equation that contains addition and subtraction. You can use this idea to check your work later when you are solving equations.

EXAMPLE

Determine whether [latex]y=\frac{3}{4}[/latex] is a solution for [latex]4y+3=8y[/latex].

Solution:

	[latex]4y+3=8y[/latex]
Substitute [latex]\color{red}{\frac{3}{4}}[/latex] for [latex]y[/latex]	[latex]4(\color{red}{\frac{3}{4}})+3\stackrel{\text{?}}{=}8(\color{red}{\frac{3}{4}})[/latex]
Multiply.	[latex]3+3\stackrel{\text{?}}{=}6[/latex]
Add.	[latex]6=6\quad\checkmark[/latex]

Since [latex]y=\frac{3}{4}[/latex] results in a true equation, [latex]\frac{3}{4}[/latex] is a solution to the equation [latex]4y+3=8y[/latex].

Now it is your turn to determine whether a fraction is the solution to an equation.

TRY IT

Subtraction Property of Equality

For all real numbers [latex]a,b[/latex], and [latex]c[/latex], if [latex]a=b[/latex], then [latex]a-c=b-c[/latex].

Addition Property of Equality

For all real numbers [latex]a,b[/latex], and [latex]c[/latex], if [latex]a=b[/latex], then [latex]a+c=b+c[/latex].

The goal is to isolate the variable on one side of the equation.

Some people picture a balance scale, as in the image below, when they solve equations.

Three balance scales are shown. The top scale has one red weight on each side and is balanced. Beside it is

The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.

In the following example we review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. You can check your solutions by substituting the value into the equation to make sure you have a true statement.

EXAMPLE

Solve: [latex]x+11=-3[/latex].

Show Solution

Now you can try solving an equation that requires using the addition property.

TRY IT

In the original equation in the previous example, [latex]11[/latex] was added to the [latex]x[/latex] , so we subtracted [latex]11[/latex] to “undo” the addition. In the next example, we will need to “undo” subtraction by using the Addition Property of Equality.

EXAMPLE

Solve: [latex]m - 4=-5[/latex].

Show Solution

Now you can try using the addition property to solve an equation.

TRY IT

In the following video, we present more examples of solving equations using the addition and subtraction properties.

Using the Subtraction and Addition Properties for Multi-Step Equations

In the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.

Example

Solve:

[latex]3x - 7 - 2x - 4=1[/latex].

Solution:
The left side of the equation has an expression that we should simplify before trying to isolate the variable.

	[latex]3x-7-2x-4=1[/latex]
Rearrange the terms, using the Commutative Property of Addition.	[latex]3x-2x-7-4=1[/latex]
Combine like terms.	[latex]x-11=1[/latex]
Add [latex]11[/latex] to both sides to isolate [latex]x[/latex] .	[latex]x-11\color{red}{+11}=1\color{red}{+11}[/latex]
Simplify.	[latex]x=12[/latex]
Check. Substitute [latex]x=12[/latex] into the original equation. [latex]3x-7-2x-4=1[/latex] [latex]3(\color{red}{12})-7-2(\color{red}{12})-4=1[/latex] [latex]36-7-24-4=1[/latex] [latex]29-24-4=1[/latex] [latex]5-4=1[/latex] [latex]1=1\quad\checkmark[/latex] The solution checks.

Now you can try solving a couple equations where you should simplify first.

TRY IT

The last few examples involved simplifying using addition and subtraction. Let’s look at an example where we need to distribute first in order to simplify the equation as much as possible.

example

Solve: [latex]3\left(n - 4\right)-2n=-3[/latex].

Show Solution

Now you can try a few problems that involve distribution.

TRY IT

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].

Show Solution

Now, you give it a try!

TRY IT

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 Division and Multiplication Properties of Equality for Single-Step Equations

Let’s review the Division and Multiplication Properties of Equality as we prepare to use them to solve single-step equations.

Division Property of Equality

For all real numbers [latex]a,b,c[/latex], and [latex]c\ne 0[/latex], if [latex]a=b[/latex], then [latex]\frac{a}{c}=\frac{b}{c}[/latex].

Multiplication Property of Equality

For all real numbers [latex]a,b,c[/latex], if [latex]a=b[/latex], then [latex]ac=bc[/latex].

Stated simply, when you divide or multiply both sides of an equation by the same quantity, you still have equality.

Let’s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to “undo” the operation on the variable. In the example below the variable is multiplied by [latex]4[/latex], so we will divide both sides by [latex]4[/latex] to “undo” the multiplication.

example

Solve: [latex]4x=-28[/latex].

Solution:

To solve this equation, we use the Division Property of Equality to divide both sides by [latex]4[/latex].

[latex]4x=-28[/latex]
Divide both sides by 4 to undo the multiplication.	[latex]\frac{4x}{\color{red}4}=\frac{-28}{\color{red}4}[/latex]
Simplify.	[latex]x =-7[/latex]
Check your answer.	[latex]4x=-28[/latex]
Let [latex]x=-7[/latex]. Substitute [latex]-7[/latex] for x.	[latex]4(\color{red}{-7})\stackrel{\text{?}}{=}-28[/latex]
	[latex]-28=-28[/latex]

Since this is a true statement, [latex]x=-7[/latex] is a solution to [latex]4x=-28[/latex].

Now you can try to solve an equation that requires division and includes negative numbers.

try it

In the previous example, to “undo” multiplication, we divided. How do you think we “undo” division? Next, we will show an example that requires us to use multiplication to undo division.

example

Solve: [latex]\frac{a}{-7}=-42[/latex].

Show Solution

Now see if you can solve a problem that requires multiplication to undo division. Recall the rules for multiplying two negative numbers – two negatives give a positive when they are multiplied.

try it

As you begin to solve equations that require several steps you may find that you end up with an equation that looks like the one in the next example, with a negative variable. As a standard practice, it is good to ensure that variables are positive when you are solving equations. The next example will show you how.

example

Solve: [latex]-r=2[/latex].

Show Solution

Now you can try to solve an equation with a negative variable.

try it

In our next example, we are given an equation that contains a variable multiplied by a fraction. We will use a reciprocal to isolate the variable.

example

Solve: [latex]\frac{2}{3}x=18[/latex].

Show Solution

Notice that we could have divided both sides of the equation [latex]\frac{2}{3}x=18[/latex] by [latex]\frac{2}{3}[/latex] to isolate [latex]x[/latex]. While this would work, multiplying by the reciprocal requires fewer steps.

try it

The next video includes examples of using the division and multiplication properties to solve equations with the variable on the right side of the equal sign.

Using the Division and Multiplication Properties of Equality for Multi-Step Equations

Many equations start out more complicated than the ones we’ve just solved. Our goal has been to familiarize you with the many ways to apply the addition, subtraction, multiplication, and division properties that are used to solve equations algebraically. Let’s work through an example that will employ the following techniques:

simplify by combining like terms
isolate x by using the division property of equality

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]\frac{12x}{\color{red}{12}}=\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]

Here is a similar problem for you to try.

Try it

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

In our next example, we have an equation that contains a set of parentheses. 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.

Show Solution

Now you can try a similar problem.

try it

In the following video you will see another example of using the division property of equality to solve an equation as well as another example of how to solve a multi-step equation that includes a set of parentheses.

Solving Equations With Variables on Both Sides

The equations we solved in the last section simplified nicely so that we could use the division property to isolate the variable and solve the equation. Sometimes, after you simplify you may have a variable and a constant term on the same side of the equal sign.

Our strategy will involve choosing one side of the equation to be the variable side, and the other side of the equation to be the constant side. This will help us with organization. Then, we will use the Subtraction and Addition Properties of Equality, step by step, to isolate the variable terms on one side of the equation.

Read on to find out how to solve this kind of equation.

Examples

Solve: [latex]4x+6=-14[/latex].

Solution:

In this equation, the variable is only on the left side. It makes sense to call the left side the variable side. Therefore, the right side will be the constant side.

Since the left side is the variable side, the 6 is out of place. We must “undo” adding [latex]6[/latex] by subtracting [latex]6[/latex], and to keep the equality we must subtract [latex]6[/latex] from both sides. Use the Subtraction Property of Equality.		[latex]4x+6\color{red}{-6}=-14\color{red}{-6}[/latex]
Simplify.		[latex]4x=-20[/latex]
Now all the [latex]x[/latex] s are on the left and the constant on the right.
Use the Division Property of Equality.		[latex]\frac{4x}{\color{red}{4}}=\frac{-20}{\color{red}{4}}[/latex]
Simplify.		[latex]x=-5[/latex]
Check:		[latex]4x+6=-14[/latex]
Let [latex]x=-5[/latex] .		[latex]4(\color{red}{-5})+6=-14[/latex]
		[latex]-20+6=-14[/latex]
		[latex]-14=-14\quad\checkmark[/latex]

Solve: [latex]2y - 7=15[/latex].

Show Solution

Now you can try a similar problem.

Try It

Solve Equations with Variables on Both Sides

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. 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.

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]2x+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.

		[latex]x+11=-3[/latex]
Subtract 11 from each side to “undo” the addition.		[latex]x+11\color{red}{-11}=-3\color{red}{-11}[/latex]
Simplify.		[latex]x=-14[/latex]
Check:	[latex]x+11=-3[/latex]
Substitute [latex]x=-14[/latex] .	[latex]\color{red}{-14}+11\stackrel{\text{?}}{=}-3[/latex]
	[latex]-3=-3\quad\checkmark[/latex]

		[latex]m-4=-5[/latex]
Add 4 to each side to “undo” the subtraction.		[latex]m-4\color{red}{+4}=-5\color{red}{+4}[/latex]
Simplify.		[latex]m=-1[/latex]
Check:	[latex]m-4=-5[/latex]
Substitute [latex]m=-1[/latex] .	[latex]\color{red}{-1}+4\stackrel{\text{?}}{=}-5[/latex]
	[latex]-5=-5\quad\checkmark[/latex]
		The solution to [latex]m - 4=-5[/latex] is [latex]m=-1[/latex] .

	[latex]3(n-4)-2n=-3[/latex]
Distribute on the left.	[latex]3n-12-2n=-3[/latex]
Use the Commutative Property to rearrange terms.	[latex]3n-2n-12=-3[/latex]
Combine like terms.	[latex]n-12=-3[/latex]
Isolate n using the Addition Property of Equality.	[latex]n-12\color{red}{+12}=-3\color{red}{+12}[/latex]
Simplify.	[latex]n=9[/latex]
Check. Substitute [latex]n=9[/latex] into the original equation. [latex]3(n-4)-2n=-3[/latex] [latex]3(\color{red}{9}-4)-2\cdot\color{red}{9}=-3[/latex] [latex]3(5)-18=-3[/latex] [latex]15-18=-3[/latex] [latex]-3=-3\quad\checkmark[/latex] The solution checks.

	[latex]2(3k-1)-5k=-2-7[/latex]
Distribute on the left, subtract on the right.	[latex]6k-2-5k=-9[/latex]
Use the Commutative Property of Addition.	[latex]6k-5k-2=-9[/latex]
Combine like terms.	[latex]k-2=-9[/latex]
Undo subtraction by using the Addition Property of Equality.	[latex]k-2\color{red}{+2}=-9\color{red}{+2}[/latex]
Simplify.	[latex]k=-7[/latex]
Check. Let [latex]k=-7[/latex]. [latex]2(3k-1)-5k=-2-7[/latex] [latex]2(3(\color{red}{-7}-1)-5(\color{red}{-7})=-2-7[/latex] [latex]2(-21-1)-5(-7)=-9[/latex] [latex]2(-22)+35=-9[/latex] [latex]-44+35=-9[/latex] [latex]-9=-9\quad\checkmark[/latex]

[latex]\frac{a}{-7}=-42[/latex]
Multiply both sides by [latex]-7[/latex] .	[latex]\color{red}{-7}(\frac{a}{-7})=\color{red}{-7}(-42)[/latex] [latex]\frac{-7a}{-7}=294[/latex]
Simplify.	[latex]a=294[/latex]
Check your answer.	[latex]\frac{a}{-7}=-42[/latex]
Let [latex]a=294[/latex] .	[latex]\frac{\color{red}{294}}{-7}\stackrel{\text{?}}{=}-42[/latex]
	[latex]-42=-42\quad\checkmark[/latex]

[latex]-r=2[/latex]
Rewrite [latex]-r[/latex] as [latex]-1r[/latex] .	[latex]-1r=2[/latex]
Divide both sides by [latex]-1[/latex] .	[latex]\frac{-1r}{\color{red}{-1}}=\frac{2}{\color{red}{-1}}[/latex]
Simplify.	[latex]r=-2[/latex]
Check.	[latex]-r=2[/latex]
Substitute [latex]r=-2[/latex]	[latex]-(\color{red}{-2})\stackrel{\text{?}}{=}2[/latex]
Simplify.	[latex]2=2\quad\checkmark[/latex]

[latex]\frac{2}{3}x=18[/latex]
Multiply by the reciprocal of [latex]\frac{2}{3}[/latex] .	[latex]\frac{\color{red}{3}}{\color{red}{2}}\cdot\frac{2}{3}x[/latex]
Reciprocals multiply to one.	[latex]1x=\frac{3}{2}\cdot\frac{18}{1}[/latex]
Multiply.	[latex]x=27[/latex]
Check your answer.	[latex]\frac{2}{3}x=18[/latex]
Let [latex]x=27[/latex].	[latex]\frac{2}{3}\cdot\color{red}{27}\stackrel{\text{?}}{=}18[/latex]
	[latex]18=18\quad\checkmark[/latex]

	[latex]11-20=17y-8y-6y[/latex]
Simplify each side.	[latex]-9=3y[/latex]
Divide both sides by 3 to isolate y.	[latex]\frac{-9}{\color{red}{3}}=\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]-3(n-2)-6=21[/latex]
Distribute.	[latex]-3n+6-6=21[/latex]
Simplify.	[latex]-3n=21[/latex]
Divide both sides by -3 to isolate n.	[latex]\frac{-3n}{\color{red}{-3}}=\frac{21}{\color{red}{-3}}[/latex] [latex]n=-7[/latex]
Check your answer. Let [latex]n=-7[/latex] .
[latex]-3(n-2)-6=21[/latex]
[latex]-3(\color{red}{-7}-2)-6\stackrel{\text{?}}{=}21[/latex]
[latex]-3(-9)-6\stackrel{\text{?}}{=}21[/latex]
[latex]27-6\stackrel{\text{?}}{=}21[/latex]
[latex]21=21\quad\checkmark[/latex]

[latex]2y-7[/latex] is the side containing a variable. [latex]15[/latex] is the side containing only a constant.
Add [latex]7[/latex] to both sides.		[latex]2y-7\color{red}{+7}=15\color{red}{+7}[/latex]
Simplify.		[latex]2y=22[/latex]
The variables are now on one side and the constants on the other.
Divide both sides by [latex]2[/latex].		[latex]\frac{2y}{\color{red}{2}}=\frac{22}{\color{red}{2}}[/latex]
Simplify.		[latex]y=11[/latex]
Check:		[latex]2y-7=15[/latex]
Let [latex]y=11[/latex] .		[latex]2\cdot\color{red}{11}-7\stackrel{\text{?}}{=}15[/latex]
		[latex]22-7\stackrel{\text{?}}{=}15[/latex]
		[latex]15=15\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]\frac{8x}{\color{red}{8}}=\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]\frac{-8}{\color{red}{2}}=\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]6n-2=-3n+7[/latex]
We don’t want variables on the right side—add [latex]3n[/latex] to both sides to leave only constants on the right.	[latex]6n\color{red}{+3n}-2=-3n\color{red}{+3n}+7[/latex]
Combine like terms.	[latex]9n-2=7[/latex]
We don’t want any constants on the left side, so add [latex]2[/latex] to both sides.	[latex]9n-2\color{red}{+2}=7\color{red}{+2}[/latex]
Simplify.	[latex]9n=9[/latex]
The variable term is on the left and the constant term is on the right. To get the coefficient of [latex]n[/latex] to be one, divide both sides by [latex]9[/latex].	[latex]\frac{9n}{\color{red}{9}}=\frac{9}{\color{red}{9}}[/latex]
Simplify.	[latex]n=1[/latex]
Check:	[latex]6n-2=-3n+7[/latex]
Substitute [latex]1[/latex] for [latex]n[/latex].	[latex]6(\color{red}{1})-2\stackrel{\text{?}}{=}-3(\color{red}{1})+7[/latex]
	[latex]4=4\quad\checkmark[/latex]

[latex]2a-7=5a+8[/latex]
Subtract [latex]2a[/latex] from both sides to remove the variable term from the left.	[latex]2a\color{red}{-2a}-7=5a\color{red}{-2a}+8[/latex]
Combine like terms.	[latex]-7=3a+8[/latex]
Subtract [latex]8[/latex] from both sides to remove the constant from the right.	[latex]-7\color{red}{-8}=3a+8\color{red}{-8}[/latex]
Simplify.	[latex]-15=3a[/latex]
Divide both sides by [latex]3[/latex] to make 1[/latex] the coefficient of [latex]a[/latex] .	[latex]\frac{-15}{\color{red}{3}}=\frac{3a}{\color{red}{3}}[/latex]
Simplify.	[latex]-5=a[/latex]
Check:	[latex]2a-7=5a+8[/latex]
	[latex]-10-7\stackrel{\text{?}}{=}-25+8[/latex]
	[latex]-17=-17\quad\checkmark[/latex]

Chapter CR: Corequisite Review

Learning Outcomes

Using the Subtraction and Addition Properties for Single-Step Equations

Solution of an Equation

EXAMPLE

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Subtraction Property of Equality

Addition Property of Equality

EXAMPLE

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EXAMPLE

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Using the Subtraction and Addition Properties for Multi-Step Equations

Example

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example

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example

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Using the Division and Multiplication Properties of Equality for Single-Step Equations

Division Property of Equality

Multiplication Property of Equality

example

try it

example

try it

example

try it

example

try it

Using the Division and Multiplication Properties of Equality for Multi-Step Equations

Example

Try it

example

try it

example

try it

Solving Equations With Variables on Both Sides

Examples

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Solve Equations with Variables on Both Sides

ExampleS

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example

try it

Solve Equations with Variables and Constants on Both Sides

Examples

EXAMPLE

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Solve an equation with variables and constants on both sides