Solving One-Step Equations with Addition and Subtraction — Examples

Solving By Addition

Solving an equation means that you are trying to find the values of the variables in the equation. We do this by altering the problem to isolate the variable on one side of the equation.

If your equation has a subtraction operator, we are going to use addition to isolate the variable. Let’s look at an example. $$x-5=2$$ First, we can see that $x$ and $5$ are on one side of the equation, while $2$ is on the other. We want to figure out what $x$ is, so we want to isolate the variable. We'll do this by adding $5$ to both sides of the equation. $$\begin{align} x-5 &= 2\\ x-5 \color{#29A603}{+5} &= 2 \color{#29A603}{+5} \\ x &= 2+5 \\ x&= 7 \\ \end{align}$$ In summary, equations where a number is added to a variable can be solved by subtracting the number from both sides.

We can generalize this concept using a linear equation: $$x-a=b$$ where $a$ and $b$ are both constants and $x$ is the variable.

Now, we need to isolate the variable $x$. We're going to add $a$ to both sides of the equation. $$\begin{align} x-a&=b \\ x-a \color{#29A603}{ + a} &= b \color{#29A603}{ + a} \\ x &= b+a \\ x &= a +b \\ \end{align}$$ You can also draw a diagram to understand this concept.

Let’s start with the equation: $x-3=2$

Draw a diagram that illustrates this equation.

From the image, we know that $x=5$. We can check this using algebra. $$\begin{align} x-3 &= 2\\ x - 3 \color{#29A603}{+3} &= 2 \color{#29A603}{+3}\\ x &= 2 +3\\ x &= 5\\ \end{align}$$ We are now going to take a look at some of the types of problems you might encounter about solving by addition.

Before you start these problems, make sure to go over the basics of the introduction to equations: solving by addition.

$$\begin{align} x-13 &= -5\\ x - 13 \color{#29A603}{+13} &= -5 \color{#29A603}{+13} \\ x &= -5+13 \\ x &= 8 \\ \end{align}$$

First, solve for $x$ using the equation $3x-5=2x-10$. $$\begin{align} 3x - 5 &= 2x -10 \\ 3x - 5 \color{#29A603}{-2x} &= 2x -10 \color{#29A603}{-2x} \\ 3x - 2x -5 &= =-10 \\ x - 5 &=- 10\\ x -5 \color{#29A603}{ +5} &= -10 \color{#29A603}{ +5} \\ x &= -10 +5 \\ x &= -5\\ \end{align}$$ Now, solve for $y= x-10+2x+20$ using $x=-5$ $$\begin{align} y &= x-10+2x+20\\ &=(-5)-10+2(-5)+20 \\ &= -5-10-10+20 \\ &= -15-10+20 \\ &= -25+20 \\ &= -5\\ \end{align}$$

Let the amount of apples Mary picked equal $x$.

Since Dan pickled $5$ less than Mary, Dan picked a total of $x-5$ apples. Since Mary picked $2$ less than Luke, Luke picked a total of $x+2$ apples.

We know that $51$ apples were picked, so we can set up an equation and solve for $x$. $$\begin{align} (x-5)+(x+2)+x &= 51\\ x-5+x+2+x &= 51\\ 3x -5+2 &= 51\\ 3x -3 &= 51\\ 3x -3 \color{#29A603}{+3} &= 51 \color{#29A603}{+3}\\ 3x &= 54\\ x &= 18\\ \end{align}$$ Since $x=18$ and we know that Dan picked a total of $x-5$ apples, so Dan picked $18-5=13$ apples.

(Videos) Solving Equations with Addition

by mathman1024

Here's a video by mathman1024 that shows you how to solve equations written with addition. There's another video with more examples later in the explanation.

$\large{\text{Summary}}$

To solve means to get the variable by itself. This is called isolating the variable. Once we do this we have a solution. A solution is any replacement of the variable that makes the equation true.

The first example he does is: $x+5=17$. To isolate the variable, we have to get rid of the $+5$. If we remove $5$, $-5$, then the $x$ is left by itself. But since this is an equation, whatever we do to one side we have to do to the other. So if we subtract $5$ from the side with the variable, we also have to subtract $5$ from the side with the $17$. We call $-5$ the additive inverse of $5$, because $5$ and $-5$ cancel.

$$\begin{align} x+5&=17\\ \color{#29A603}{-5}&\ \ \ \color{#29A603}{-5}\\ \hline x+0&=12\\ x&=12 \end{align}$$

Try these next ones on your own and then watch the video to see the answers!

1. $w-18=-43$
2. $9+y=-4$

Here are some more examples. Try them on your own and then watch the video to see the answers.

1. $3x+19-2x=14$
2. $x-23=-47$

Use addition and/or subtraction to isolate the variable!

When you're solving equations, you're trying to find the value of the variables in the equations. To do this, you want to make sure that the variable is all alone on one side of the equals sign of the equation, while all the other information is on the other side.

If your equation only has addition in it (in other words, something is being added to the variable), we're going to have to use subtraction to move those numbers over!

Let's look at another equation: $$x + 5 = 7$$ First off, what do you see? $x$ and $5$ are on one side of the equation, while $7$ is on the other.

We want to get $5$ to the right of the equals sign so that $x$ is on its own on the left. Now, to do that we aren't really moving $5$ around, since we can't just pick numbers up. Instead, we're performing the same operation to both sides of the equation so that even though each side changes, they change in the same way and they keep the same relationship to each other. When we do this operation, we need to make sure that the $5$ on the left side of the equation ends up equal to $0$.

To cancel out adding $5$ on the left side of the equation, we need to subtract $5$ from both sides of the equation. This way, the equations are still equal but we aren't adding anything to $x$ anymore. Now, the equation should look like this:

$$\begin{align} x + 5 &= 7\\\ x + 5 - 5 &= 7 - 5\\\ x &= 7 - 5 \end{align}$$

$$7 - 5 = 2\\ \text{so}\\ x = 2$$

In summary: If you have to solve an equation where something is added to $x$, you have to subtract that number from both sides of the equation to solve it.

Use addition to get the variable alone on one side of the equation

To solve algebraic equations, you have to figure out what value of the variable makes the equation true. To do that, you have to make it so that the variable is alone on one side of the equation. If you're trying to solve a subtraction equation (something is being subtracted from your variable), you're going to have to use addition.

Our first step is to add $14$ to both sides of the equation. This will cancel out the $14$ that's being subtracted from $x$ on the left, but still make sure to change both sides of the equation in the same way so that they're still equal. Let's take a look at the process of adding $14$: $$\begin{align} x - 14 &= 10\\ x - 14 \,\color{#29A603}{+\,14} &= 10\, \color{#29A603}{+\,14}\\ x &= 10 + 14 \end{align}$$ During the process, the $-14$ on the left gets cancelled out, and we're now adding $14$ to the right side. All that's left is arithmetic!

If we just add $10$ and $14$, we discover that $x = 24$ because $10 + 14 = 24$.

In summary: If you have to solve an equation where something is subtracted from $x$, you have to add that to from both sides of the equation to solve it.

Solving a Subtraction Equation Using Inverse Operations

To easily solve one-step equations, the one rule you should always remember is to isolate the variable alone onto one side of the equation.

You’re probably wondering, "How do I isolate a variable in an equation?". Let’s look at an example.

$$x - 4 = 8$$

To isolate the variable $x$ in this equation, you are going to add $4$ to the left side, since addition is the inverse operation of subtraction. Doing so will ‘cancel out’ the $-4$ on the left side of the equation ($-4 +4 =0$) and leave us with just the variable, $x$, on the left side. But wait, what about the right side of the equation?

Another rule to remember is that whatever we do to one side of the equation, we have to do the same for the other side of the equation. We must always balance the left and right side of the equation.

Now back to the previous example.

Since we added $4$ to the left side of the equation, we now have to add $4$ to the right side of the equation. Doing so would leave us with $12$ ($8+4$) on the right side of the equation.

$$\begin{align} x - 4 &= 8\\ + 4 &= + 4 \quad\text{Add 4 to each side}\\ x &= 12 \quad\text{Simplify} \end{align}$$