Simplifying Equations by Combining Like Terms

Hello! Today we’re going to take a look at simplifying and solving equations by combining like terms. Remember, like terms are any terms that have the same variable to the same power. For example, \(3x\) and \(7x\) are like terms because they both have the same variable (\(x\)) to the same power. In this case, they’re both raised to the power of 1. \(x^{1}\) can just be written as \(x\), so since they have the same variable and the same power, these are like terms, and it means that we can add them or subtract them. Constants, like 6 and -8, are also considered like terms and can be combined. So, if we were to add these two terms, we would have \(6+(-8)\), which would give us -2. So, let’s take a look at an example where we’ll use this to solve an equation. So the equation is:

\(2x+3+5x=-11\)

 

So, to start, we’re going to combine like terms on the left side of the equation, because we only have one term on the right side. So, on the left side, we have to first find our like terms. In this case, our like terms are \(2x\) and \(+5x\).

When you’re looking for like terms, make sure you grab the sign in front of the term, that’ll tell you whether it’s positive or negative — if it needs it to be added or subtracted — so it’s important to look for that too. So now we can combine \(2x\) and \(5x\). \(2+5=7\), so we have \(7x\), and then this \(+3\) is going to stay the same, we didn’t do anything to it. And same with the -11 on the other side.

\(7x+3=-11\)

 

Now we can solve this like a normal two-step equation. So we’ll start by subtracting 3 from both sides.

\(7x+3-3=-11-3\)

 

That gives us \(7x\), and \(-11-3=-14\).

\(7x=-14\)

 

Now to solve for \(x\), all we have to do is divide both sides by 7.

\(\frac{7x}{7}=\frac{-14}{7}\)

 

And we get:

\(x=-2\)

 

So, let’s take a look at another one. This equation is:

\(3y-14+12=4\)

 

Remember, we always start by combining our like terms, so our like terms on this left side are -14 (or minus 14) and positive 12. So, to combine those, we want to do, \(-14+12\), which will give us -2. Our \(3y\) here stays the same, so we have:

\(3y-2=4\)

 

And now we solve like a two-step equation, by adding 2 to both sides.

\(3y-2+2=4+2\)

 

Which will leave us with \(3y\) on the left side, and \(4+2=6\).

\(3y=6\)

 

And then to solve for \(y\), all we have to do is divide both sides by 3.

\(\frac{3y}{3}=\frac{6}{3}\)

 

So we have, \(y\) equals, \(6\div 3=2\).

\(y=2\)

 

Before we go, let’s take a look at one more problem together.

\(6x+9-2x=13-2\)

 

Now, notice for this problem, we can combine like terms on both the left and the right sides. So, on the left side, our like terms are \(6x\) and \(-2x\), so \(6x-2x=4x\), and then our \(+9\) stays the same. Then on the right side, our like terms are 13 and -2. \(13-2=11\), so we combine those like terms to get 11 on the right side.

\(4x+9=11\)

 

Now we solve by subtracting 9 on both sides.

\(4x+9-9=11-9\)

 

So we have \(4x\) equals, \(11-9=2\).

\(4x=2\)

 

And we solve for \(x\) by dividing both sides by 4.

\(\frac{4x}{4}=\frac{2}{4}\)

 

So:

\(x=\frac{2}{4}\)

 

Now, this fraction isn’t in simplest form, so to have a proper answer we’ll want to do that. To simplify this fraction, we divide both the numerator and the denominator by 2. \(2\div 2=1\), and \(4\div 2=2\).

\(x=\frac{2\div 2}{4\div 2}=\frac{1}{2}\)

 

So \(x=\frac{1}{2}\).

And that’s all there is to it! I hope that this video was helpful. Thanks for watching and happy studying!

Practice Questions

Question #1:

 
Solve the equation: \(5x-8+2x=13\).

\(x=5\)
\(x=3\)
\(x=-5\)
\(x=-3\)
Answer:

To solve the equation for \(x\), we need to isolate \(x\) on one side of the equation. The first step is to simplify the equation by combining like terms on either side of the equation.

\(5x-8+2x=13\)
\(7x-8=13\)

We now have a two-step equation to solve. Start by adding 8 to both sides.

\(7x-8+8=13+8\)
\(7x=21\)

Then, divide both sides by 7.

\(\frac{7x}{7}=\frac{21}{7}\)
\(x=3\)

Question #2:

 
Solve the equation: \(4x+8-17=25\).

\(x=\frac{17}{2}\)
\(x=\frac{15}{2}\)
\(x=4\)
\(x=2\)
Answer:

To solve the equation for \(x\), we need to isolate \(x\) on one side of the equation. The first step is to simplify the equation by combining like terms on either side of the equation.

\(4x+8-17=25\)
\(4x-9=25\)

We now have a two-step equation to solve. Start by adding 9 to both sides.

\(4x-9+9=25+9\)
\(4x=34\)

Then, divide both sides by 4.

\(\frac{4x}{4}=\frac{34}{4}\)
 
\(x=\frac{34}{4}\)

Reduce the fraction by 2, which puts the answer in simplest form.

\(x=\frac{34\div2}{4\div2}=\frac{17}{2}\)

Question #3:

 
Solve the equation: \(7x+6-4x=18+12\).

\(x=8\)
\(x=13\)
\(x=-8\)
\(x=-13\)
Answer:

To solve the equation for \(x\), we need to isolate \(x\) on one side of the equation. The first step is to simplify the equation by combining like terms on either side of the equation.

\(7x+6-4x=18+12\)
\(3x+6=30\)

We now have a two-step equation to solve. Start by subtracting 6 from both sides.

\(3x+6-6=30-6\)
\(3x=24\)

Then, divide both sides by 3.

\(\frac{3x}{3}=\frac{24}{3}\)
\(x=8\)

Question #4:

 
You and your friend landscape yards on the weekends. On one weekend you and your friend made $187 landscaping a yard close to where you live. Your friend charges $8 per hour landscaping yards, but because of your experience, you are able to charge $10 per hour. If you and your friend charge a flat fee of $25 plus your hourly wages to landscape the yard and you both work the same amount of hours, how many hours did you work landscaping the yard?

6 hours

9 hours

12 hours

18 hours

Answer:

Let \(x\) be the number of hours you and your friend worked on the yard. Since your friend charges $8 per hour, he makes \(8x\) dollars. Similarly, since you charge $10 per hour, you make \(10x\) dollars. The sum of what you and your friend make plus the $25 flat fee equals to $187, so we have the equation:

\(25+8x+10x=187\)

To solve the equation for x, we need to isolate x on one side of the equation. The first step is to simplify the equation by combining like terms on either side of the equation.

\(25+8x+10x=187\)
\(25+18x=187\)

We now have a two-step equation to solve. Start by subtracting 25 from both sides.

\(25+18x-25=187-25\)
\(18x=162\)

Then, divide both sides by 18.

\(\frac{18x}{18}=\frac{162}{18}\)
\(x=9\)

Therefore, you and your friend worked 9 hours landscaping the yard.

Question #5:

 
Four less than the sum of three times a number and six equals seventy-four. What is the number?

48

24

36

16

Answer:

Let x be the number. The sum of 3 times a number and 6 can be written by the expression \(3x+6\). Taking 4 less than this sum gives us the expression \(3x+6-4\). Since this expression equals to seventy-four, we have the equation:

\(3x+6-4=74\)

To solve the equation for \(x\), we need to isolate \(x\) on one side of the equation. The first step is to simplify the equation by combining like terms on either side of the equation.

\(3x+6-4=74\)
\(3x+2=74\)

We now have a two-step equation to solve. Start by subtracting 2 from both sides.

\(3x+2-2=74-2\)
\(3x=72\)

Then, divide both sides by 3.

\(\frac{3x}{3}=\frac{72}{3}\)
\(x=24\)

Therefore, the number is 24.

 

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by Mometrix Test Preparation | This Page Last Updated: May 12, 2022