Solving One-Step Equations

Hello! Today we’re going to take a look at how to solve one-step equations. This is one of the most fundamental tools for solving equations. So, let’s start with an example.

\(x+4=7\)

 

One thing that’s important to remember is that an equation means that the left side is equal to the right side. So in this case, \(x+4\) is equal to 7. To “solve an equation” means to find the value of \(x\) that makes the equation true. So, if we think about this logically for just a minute, we know that \(x\) has to be equal to 3, because \(3+4=7\).

This works well for simple equations like this, but once we get to more complex equations, it won’t work quite as well. So we’re going to take a look at a process that will work every time for solving equations. So, let’s look back at our equation: \(x+4=7\). To find the value of \(x\), we need to undo the operations around the \(x\) to get \(x\) by itself on one side of the equation. So in this case, we have \(x+4\), and we want to undo that \(+4\). So, since 4 is being added to \(x\), the opposite of addition is subtraction, so we’re going to subtract 4.

\(x+4-4\)

 

So if we have \(x+4\), and then we subtract 4, we’ll be left with just \(x\).

Now, one thing to pay attention to that’s really important is that an equation means that both sides are balanced — remember, we said the left side is equal to the right side, so our equation, we call this being balanced. Now, if we do something to the left side, we then make it unbalanced because we haven’t done anything to the right side. So, in order to balance this equation, we have to do the exact same thing to the right side. So if we subtract 4 from the left side, we have to subtract 4 from the right side, in order for this still to be true.

\(x+4-4=7-4\)

 

So, if we subtract \(7-4\), we’ll get 3, so \(x=3\), which is what we said earlier.

To double check our work one last time, we can plug in our value for \(x\) into our original equation. So we had:

\((3)+4=7\)
 
\(7=7\)

 

\(7=7\) is a true statement, so we know that \(x=3\) is the correct answer.

Now let’s take a look at a problem that involves subtraction:

\(x-14=2\)

 

So remember, we said we want to undo the operation around the \(x\), so here we’re subtracting 14. The opposite of subtraction is addition, so we’re going to add 14. Remember, our equation must be balanced, meaning that whatever we do to the left side, we also have to do to the right side, so if we add 14 to the left side, we need to add 14 to the right side.

\(x-14+14=2+14\)

 

Our 14s here will cancel out because we’re subtracting 14 and then adding 14, so that means we won’t have to worry about these. And we are just left with \(x\) on the left side, which is what we’re wanting. And then \(2+14=16\), so:

\(x=16\)

 

If we want to double check our answer, we can plug in our value for \(x\) back into our original equation, right where we see an \(x\). So if \(x=16\), we have:

\((16)-14=2\)

\(2=2\)

 

\(2=2\) is a true statement, so that means that our answer, \(x=16\), is correct.

Now let’s move on to multiplication and division. Addition and subtraction are opposite operations of one another, and so are multiplication and division. So, if we have an equation like

\(3x=27\)

 

this means \(3x=27\). Remember, if you have a number right next to a letter, or variable, that means that they’re being multiplied, even if you don’t see a multiplication symbol in the middle. So this means, \(3x=27\).

We just said that the opposite of multiplication is division, so in order to get rid of this multiplication by 3, we’re going to divide by 3 on both sides.

\(\frac{3x}{3}=\frac{27}{3}\)

 

Notice that I wrote a little fraction bar right here. Remember that fractions mean divide, so I could have written a division symbol like this (\(\div\)), and that also would have been correct. But fractions always mean divide, so this is just one way to write it. So, now we have \(3\div 3=1\), so we have \(1\cdot x=27\div 3\), which is 9. Now, 1 times anything is that thing, so \(1\cdot x\) is just \(x\). So this gives us:

\(x=9\)

 

Again, if we want to check our answer, we can plug in \(x\) back into our original equation. So we have:

\(3(9)=27\)
 
\(27=27\)

 

So we have \(27=27\), which is a true statement. And that tells us that our answer, \(x=9\), is true.

Let’s look at one more example, this time using division.

\(\frac{x}{2}=24\)

 

Remember, we said earlier that a fraction bar means divide, so this is the same as saying, \(x\div 2=24\), they’re the exact same thing. So, since we have division, and we need to do the opposite operation of division to both sides, we’re going to multiply both sides by 2.

\(2\times \frac{x}{2}=24\times 2\)

 

If we divide by 2 and then multiply by 2, those will cancel out, and we’ll be left with 1. Because \(x\) divided by 2 and then multiplied by 2 gives us just \(x\), which is \(1x\). So \(1x\), or \(x\), it always means the same thing. And then over here we have, \(24\times 2=48\). So our answer is:

\(x=48\)

 

But let’s double check just to be sure. Let’s go back to this original equation right here (\(\frac{x}{2}=24\)).

\(\frac{48}{2}=24\)
 
\(24=24\)

 

\(24=24\), which is a true statement. And that means that, \(x=48\), is the correct answer. So, I hope that this video helped you understand how to solve one-step equations! This is super important for all of the equations that you will be solving in the future. Thanks for watching and happy studying!

Practice Questions

Question #1:

 
Solve the equation for \(x: x-12=36\).

\(x=21\)
\(x=48\)
\(x=3\)
\(x=432\)
Answer:

To solve the equation for \(x\), we want to undo the operation around \(x\) in order to isolate \(x\) on the left-hand side of the equation. Since the operation around the \(x\) is subtraction, the opposite operation is addition. We need to add 12 to both sides of the equation to keep our equation balanced.

\(x-12+12=36+12\)
\(x=48\)

We can check our answer by substituting \(x=48\) into the original equation. The result must be a true statement.

\((48)-12=36\)
\(36=36\)

Since \(36=36\) is a true statement, our answer is correct.

Question #2:

 
Solve the equation for \(x: 4x=52\).

\(x=56\)
\(x=48\)
\(x=208\)
\(x=13\)
Answer:

To solve the equation for \(x\), we want to undo the operation around \(x\) in order to isolate \(x\) on the left-hand side of the equation. The number next to \(x\) implies we are multiplying it by \(x\), so we need to divide \(4x\) by 4 to undo the operation. To keep our equation balanced, we also need to divide by 4 on the right-hand side of the equation.

\(\frac{4x}{\mathbf4}=\frac{52}{\mathbf4}
x=13\)

We can check our answer by substituting \(x=13\) back the original equation. The result must be a true statement.

\(4\left(13\right)=52
52=52\)

Since \(52=52\) is a true statement, \(x=13\) is the correct answer.

Question #3:

 
Solve the equation for \(x: \frac{x}{6}=7\).

\(x=21\)
\(x=42\)
\(x=3\)
\(x=4\)
Answer:

To solve the equation for \(x\), we want to undo the operation around \(x\) in order to isolate \(x\) on the left-hand side of the equation. The fraction bar on the left-hand side of the equation means we are dividing \(x\) by 6, so we need to multiply \(\frac{x}{6}\) by 6 to undo the operation. To keep our equation balanced, we also need to multiply by 6 on the right-hand side of the equation.

\(6∙\frac{x}{6}=6∙7\)
\(x=42\)

We can check our answer by substituting \(x=42\) into the original equation. The result must be a true statement.

\(\frac{(42)}{6}=7\)
\(7=7\)

Since \(7=7\) is a true statement, \(x=42\) is the correct answer..

Question #4:

 
A department store sold 35 fewer pairs of tennis shoes than dress shoes on a given weekend. If the store sold 87 pairs of tennis shoes, how many pairs of dress shoes were sold?

27 pairs

62 pairs

52 pairs

122 pairs

Answer:

Since the number of pairs of dress shoes sold is unknown, use the variable \(x\) to represent this value. There were 35 fewer pairs of tennis shoes sold than pairs of dress shoes. This statement can be written using the expression \(x-35\) to represent the number of tennis shoes sold. We also know there were 87 pairs of pairs of tennis shoes sold so we can write the equation below to find \(x\).

\(x-35=87\)

To solve the equation for \(x\), we want to undo the operation around \(x\) in order to isolate \(x\) on the left-hand side of the equation. Since the operation around the \(x\) is subtraction, the opposite operation will be addition. We need to add 35 to both sides of the equation.

\(x-35+35=87+35\)
\(x=122\)

We can check our answer by substituting x=122 into the original equation. The result must be a true statement.

\((122)-35=87\)
\(87=87\)

Since \(87=87\) is a true statement, \(x=122\) is a solution to the equation. Therefore, there were 122 pairs of dress shoes sold at the department store for the given weekend.

Question #5:

 
You mow lawns during the summer as a summer job. You charge your customers $11 an hour to mow their lawns. If you made $66 mowing lawns on one summer day, how many hours, \(x\), did you work?

\(x=6\)
\(x=4\)
\(x=8\)
\(x=5\)
Answer:

To find the amount of money you made that day, multiply your hourly rate by the number of hours worked. You made $11 an hour and worked \(x\) hours. So, we can use the expression \(6x\) to represent the amount of money you made that day. We also know that you made $66 that day so we can write the equation below to find \(x\).

\(11x=66\)

To solve the equation for \(x\), we want to undo the operation around \(x\) in order to isolate \(x\) on the left-hand side of the equation. The number next to \(x\) implies we are multiplying it to \(x\), so we need to divide \(11x\) by 11 to undo the operation. To keep our equation balanced, we need to divide by 11 on the right-hand side of the equation as well.

\(\frac{11x}{11}=\frac{66}{11}\)
\(x=6\)

We can check our answer by substituting \(x=6\) into the original equation. The result must be a true statement.

\(11(6)=66\)
\(66=66\)

Since \(66=66\) is a true statement, \(x=6\) is a solution to the equation. So, you worked 6 hours that day.

 

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by Mometrix Test Preparation | This Page Last Updated: April 6, 2022