Solving for a Variable:

Solving for a Variable on One Side Using Addition and Subtraction

There are many times when not all of the information is known. In math, we represent those unknown values as variables. The videos in this lesson will be a series that will teach you, step by step, how to solve for variables in equations. Remember, the = sign means that each side is equal to the other. Anything you change on one side must be changed on the other, or they will no longer be equal. Here are some math vocabulary words that will help you to understand this lesson better:

Variable = a letter in an equation that represents an unknown value
Additive Inverse = The number when added to another number equals zero
Isolate = To get the variable alone

The following video will show you how to begin solving for variables.

Video Source (06:49 mins) | Transcript

The goal when finding a variable is to isolate it (or get it by itself) in the equation. To get rid of any addition or subtraction is to add the additive inverse of whatever is being added or subtracted. This step must be done to both sides of the equation for it to stay equal.

Additional Resources

Khan Academy: What is a Variable? (03:17 mins, Transcript)
Khan Academy: Inverse Property of Addition (02:59 mins, Transcript)
Khan Academy: One-step Addition & Subtraction Equations (02:08 mins, Transcript)
Khan Academy: Same Thing to Both Sides (02:31 mins, Transcript)

Practice Problems

Solve for the variable:

\( {\text{n}} {-} 4 = 14\)

\( 12 + {\text{D}} = -6\)

\( 14 = {\text{y}} {-} 10\)

\( {\text{F}} + 19 = 1\)

\( -10 = {\text{J}} + 30\)

\( 7 = {\text{B}} {-} 3\)

\( -30 + {\text{Y}} = 40\)

Solutions

\(18\) (Written Solution)

In this example, we want to get the variable \({\text{n}}\) all by itself on one side of the equal sign. Right now there is a negative \(4\) on the same side of the equal sign.

(Remember subtraction is the same as adding a negative.)

We can rewrite this equation as \({\text{n}} + (-4) = 14\).

We need to remove the \(-4\). To do this we add the additive inverse of \(-4\) which is positive \(4\).

We add positive \(4\) to both sides of the equation.

In the following picture, the dotted line represents the separation of the two sides of the equation at the equal sign.

Since \(-4 + 4 = 0\), we are only left with the variable \({\text{n}}\) on the left side of the equal sign.

Since \(14 + 4 = 18\), we are left with \(18\) on the right side of the equal sign.

Our final solution: \({\text{n}} = 18\)
\(-18\) (Solution Video | Transcript)
\(24\)
\(-18\)
\(-40\) (Written Solution)

In this example, we want to get the variable \({\text{j}}\) all alone on one side of the equal sign. This will then tell us what \({\text{j}}\) is equal to.

Our variable \({\text{j}}\) currently also has a positive \(30\) with it on the right side of the equal sign. We want to get rid of this to get \({\text{j}}\) all by itself. To do this we add the additive inverse of positive \(30\) to both sides of the equation.

The additive inverse of positive \(30\) is \(-30\), so we add \(-30\) to both sides of the equation.

\(-10 {\color{Cyan} + (-30)} = {\text{j}} + 30 {\color{Cyan} + (-30)}\)

Next, we simplify both sides of the equation.

\(30 + (-30) = 0\)

\(-10 + (-30) = -40\)

This leaves us with just the variable \({\text{j}}\) on the right-hand side of the equal sign and \(-40\) on the left-hand side.

\(-40 = {\text{j}}\)

It doesn’t matter if our variable is on the right side or the left side of the equal sign. The only thing that matters is that it is all by itself.

Our final solution: \({\text{j}} = -40\)
\(10\) (Solution Video | Transcript)
\(70\)