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This lesson will discuss how to solve an equation with a variable on both sides. In this video, we will be learning how to solve for x (or another variable) in complex algebraic equations where x (or another variable) is present on both sides using inverse operations.

For Example: $5x-7=2(x+5)-5\leftarrow$ Distribute the 2 to the x and 5 $5x-7=2x+10-5\leftarrow$ Simplify the right side using addition $5x-7-2x=2x+5-2x\leftarrow$ Isolate the variable by subtracting 2x from both sides $3x-7+7=5+7\leftarrow$ Isolate the variable by adding 7 to both sides $\frac{3x}{3}=\frac{12}{3}\leftarrow$ Divide by 3 on both sides $x=4$ ## Video-Lesson Transcript

Now, let’s solve an equation with a variable on both sides.

For example: $3x + 12 = 5x + 2$

This equation has $x$ on both sides.

My advice is, if you can, get all the $x$-term on one side only. It doesn’t matter which side – left or right.

Now, let’s solve it.

First, let’s try and subtract $3x$ from both sides of the equation. We have $3x - 3x + 12 = 5x - 3x + 2$ $12 = 2x + 2$

Here, we have a variable equation that needs solving.

So, let’s start it over again and subtract $5x$ from both sides of the equation. $3x - 5x + 12 = 5x - 5x + 2$ $-2x + 12 = 2$

The same thing. We still have a variable equation that needs solving.

So either way, even if you choose the left or right side, it’s up to you.

Just follow the steps correctly and you will arrive at the correct answer.

Since we have this equation now, let’s move on solving.

Let’s isolate $x$ by subtracting $12$ on both sides. $-2x + 12 - 12 = 2 - 12$ $-2x = -10$

Then divide by $-2$ $\dfrac{-2x}{-2} = \dfrac{-10}{-2}$

The answer is $x = 5$.

Now, let’s have a different example. $5x - 7 = 2(x+5) - 5$

So since we have to get all the $x$-terms on one side, we have to simplify the equation on the right first because an equation is inside a parenthesis.

Do this by distributing $2$ on each term inside the parenthesis.

So, we’ll have $5x - 7 = 2x + 10 - 5$

Let’s combine like terms here to further simplify the equation $5x - 7 = 2x + 5$

At this point, I choose to bring the $2x$ to the left. $5x - 2x - 7 = 2x - 2x + 5$ $3x - 7 = 5$

Now, this is the normal equation we can solve.

So let’s solve by adding $7$ to both sides of the equation. $3x - 7 + 7 = 5 + 7$ $3x = 12$

Now, let’s divide both sides by $3$ $\dfrac{3x}{3} = \dfrac{12}{3}$

Now, we have $x = 4$

So as you can see, this is very similar to solving equations in multiple steps.

The most important step is to get all the $x$-terms on one side of the equation. Then follow the rules as you go.