Introduction
In this lesson, you will learn how to solve for a variable on one side of an equation by using addition and subtraction.
There are many times in life when not all of the information is known. In math, those unknown values are represented as variables. This lesson will be a series that will teach you, step by step, how to solve for variables in equations. Remember, the equal (=) 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.
This video illustrates the lesson material below. Watching the video is optional.
Solving for a Variable on One Side Using Addition and Subtraction
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.
You solve for variables all the time in your daily lives.
Example 1
You want to purchase something that costs $8, but you have $3. How much more money do you need?
You can set up this equation and work it out mentally. You would need $5 more to purchase the item.
\begin{align*}&3 +\_\_ = 8 &\color{red}\small\text{3 plus what is equal to 8?}\\\\&3 + 5 = 8 &\color{red}\small\text{3 plus 5 is equal to 8}\\\end{align*}
This example is straightforward, but it illustrates why you use variables and how they are helpful. Another way to look at the example above is by writing:
\begin{align*}&3 + x = 8 &\color{red}\small\text{Variable x represents an unknown number}\\\end{align*}
When you perform algebraic equations, you can use any letter to represent a variable, but x is most commonly used. If you were to solve this equation, you would find that \(x=5\). How would you solve \(3 + x = 8\)?
By looking at the equation, you can see that if you have isolated x on one side, it means you have removed 3 from both sides. To remove the 3, you would use the additive inverse of 3 which is -3 to keep the equation balanced.
\begin{align*}3 + x &= 8 &\color{red}\small\text{Solve for x}\\\\x + 3 &= 8 &\color{red}\small\text{Rearrange the order}\\\\x + 3 \color{red}\mathbf{-3} &= 8 \color{red}\mathbf{-3} &\color{red}\small\text{Additive inverse of 3 is -3}\\\\x &= 5 &\color{red}\small\text{Simplify both sides}\\\end{align*}
You will get the same value for x, which is 5. You can always check your work by putting the number you calculated for the variable into the equation and making sure the equation works. In this case, it does work: \(3+5=8\).
Example 2
Solve for \(x\): \(4 + x = 10\)
In order to solve for x, you will need to isolate it, and that means that you will need to add the inverse of 4 to both sides of the equation. The inverse of 4 is -4.
\begin{align*}4 + x &= 10 &\color{red}\small\text{Solve for x}\\\\x + 4 &= 10 &\color{red}\small\text{Rearrange the order}\\\\x + 4 \color{red}\mathbf{-4} &= 10 \color{red}\mathbf{-4} &\color{red}\small\text{Additive inverse of 4 is -4}\\\\x &= 6 &\color{red}\small\text{Simplify both sides}\\\end{align*}
The answer is \(x=6\). So \(4+6=10\), which is a true statement.
Example 3
Solve for \(x\): \(x - 3 = 9\)
\begin{align*} x - 3 &= 9 &\color{red}\small\text{Solve for x}\\\\x -3 \color{red}\mathbf{+3} &= 9 \color{red}\mathbf{+3} &\color{red}\small\text{Additive inverse of -3 is +3}\\\\x &= 12&\color{red}\small\text{Simplify both sides}\\\end{align*}
When you add 3 to both sides, you find that \(x=12\).
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, 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.
Things to Remember
- A variable is a number that represents an unknown value.
- Any number plus its inverse is equal to 0.
- You can always check your work by plugging the number you calculated for x into the equation and check if both sides are equivalent.
- If both sides are equivalent, then the answer is correct.
- If both sides are not equivalent, then the answer is incorrect.
Practice Problems
Solve for the variable:- \(n−4=14\) (
Details:
In this example, you 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.)
You can rewrite this equation as \({\text{n}} + (-4) = 14\).
You need to remove the \(-4\). To do this, add the additive inverse of \(-4\) which is positive \(4\).
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 \(-4 + 4 = 0\), you are only left with the variable \({\text{n}}\) on the left side of the equal sign.
Since \(14 + 4 = 18\), you are left with \(18\) on the right side of the equal sign.
The final solution is: \({\text{n}} = 18\). - \(12+D=−6\) (
- \(14=y−10\) (
- \(F+19=1\) (
- \(−10=J+30\) (
Details:
In this example, you want to get the variable \({\text{j}}\) all alone on one side of the equal sign. This will then tell you what \({\text{j}}\) equals.
The variable \({\text{j}}\) currently also has a positive \(30\) with it on the right side of the equal sign. You want to get rid of this to get \({\text{j}}\) all by itself. To do this you add the additive inverse of positive \(30\) to both sides of the equation.
The additive inverse of positive \(30\) is \(-30\), so add \(-30\) to both sides of the equation.
\(-10 {\color{TealBlue} + (-30)} = {\text{j}} + 30 {\color{TealBlue} + (-30)}\)
Next, simplify both sides of the equation.
\(30 + (-30) = 0\)
\(-10 + (-30) = -40\)
This leaves you 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 the 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.
The final solution is: \({\text{j}} = -40\). - \(7=B−3\) (
- \(-30+Y=40\) (