Solving Equations

Solving an equation can be confusing - but never fear! We're here to help.

Let's say you have an equation like this:

3x+10=163x+10 = 16

First thing we should notice is this equation mostly just has numbers, but we do have a value xx in there. This xx just represents another number, and our task is to figure out what it is.

So

solving

this equation means making it look like this:

x=x =

where we can easily tell what number xx represents. It's like a magic trick, but we have the tools to make it happen 🥳.

We need to get xx by itself on the left side of the equals sign. To start, what could we put in the blank below to get rid of that 10?

3x+103x+10
=16 = 16

Hmm, so if we add 10, then we'll have

3x+10+10=163x+20=\begin{aligned} 3x+10+10&= 16\\3x + 20 &= \end{aligned}

Doesn't get us closer to getting xx by itself...🥴

Right!! This way, we end up with:

3x+1010=163x+0=3x=\begin{aligned} 3x+10-10&= 16\\3x + 0 &= \\3x &= \end{aligned}

Now...you'll notice we've left the right side of the equals sign blank. That's because the equals sign is like a balance - whatever you do to one side, you have to do the other.

So...imagine a balance or see-saw in your head.

Kind of like this one ⬇️

We have everything from the left side of the equation on the left side and everything from the right side of the equation on the right side. Everything in balance.

=
3x
+ 10
+ 16
+8
+8
+4
+4
-10
-10
+0
+0

Try moving some of these ⬆️ onto each side and see what happens.

Good! So now, if we look back - we subtracted 10 on the left, so what do we need to do on the right side?

3x+1010=163x+10 -10= 16

Right again!! We end up with:

3x+1010=16103x+0=63x=6\begin{aligned} 3x+10-10&= 16 -10\\3x + 0 &= 6\\3x &= 6\end{aligned}

Now, we have 3x3x on the left side, which is close to xx but we've still got that pesky 33 in the way. What can we do to get rid of that 33?

Hint: remember that 33=1\frac{3}{3} = 1

Exactly right! We can divide 3x3x by 3 to get our xx by itself. But remember, whatever we do on the left side, we have to do on the right side, too:

3x=63x3=63x=2\begin{aligned} 3x&= 6\\ \frac{3x}{3}&= \frac{6}{3}\\x &= 2\end{aligned}

And that's it! That's how to solve an equation. Just remember:

Tip #1

When trying to get a variable like xx by itself, always start by removing numbers that are added or subtracted first. Once those are all gone, that's when you can start handling multiplication or division.

Tip #2

Whatever you do to one side of the equation, you have to do the other. It's just like a see-saw - add too much to one side without adding to the other, and it will fall out of balance.