Solving equations with variables on both sides of the equation

What to do when you have variables on both sides of the equation

When you need to solve equations with variables on both sides of the equals sign, make sure to move all the variables to one side of the equation together.

A simple saying that may help you remember this is “get all your $x$’s to Texas.” In other words, you need to move all the $x$ terms so that they’re on the same side of the equation.

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Solving Equations

1. Simplify both sides of the equation as much as possible using the order of operations (distribute, combine like terms, etc.).

2. If the variable (letter) you’re trying to solve for appears on both sides of the equation move one to the other side. Get all your “$x$’s to Texas” or get all the $x$’s to one side of the equation.

3. Solve by working backwards from the order of operations. Use inverse operations until the variable is alone and remember to do the same thing to both sides of the equation so that it stays balanced.

How to solve equations with variables on both sides

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Using inverse operations to collect like terms and solve the equation

Example

Solve for the variable.

$2x-3=3x+1$

Both sides are as simplified as they can get so we’ll start by moving all the $x$ terms to one side. Use inverse operations and subtract $2x$ from both sides to keep the equation balanced. It’s easier to subtract $2x$ instead of $3x$ because this will give us a positive $x$ on the right side.

$2x-2x-3=3x-2x+1$

$-3=x+1$

We need to undo the $+1$ by subtracting $1$ from both sides of the equation.

$-3-1=x+1-1$

$-4=x$

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Let’s try another example of solving equations with variables on both sides.

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Both sides are as simplified as they can get so we’ll start by moving all the x terms to one side.

Example

Solve for the variable.

$10x-13=4x+x-6$

Start by combining like terms to simplify the right side of the equation.

$10x-13=5x-6$

Move all the $x$ terms to one side. In this case, subtract $5x$ from both sides of the equation.

$10x-5x-13=5x-5x-6$

$5x-13=-6$

We need to undo the $-13$ by adding $13$ to both sides of the equation.

$5x-13+13=-6+13$

$5x=7$

Divide both sides by $5$.

$\frac{5x}{5}=\frac{7}{5}$

$x=\frac{7}{5}$

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