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Solving an Equation (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


Video-Lesson Transcript

Now, let’s solve an equation with 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 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.