Skill: Solve equations with variables on both sides

An equation with variables on both sides can be solved by treating both sides like a balanced scale. Whatever operation you perform on one side, perform on the other side too.

Example

Solve:

$$3(x-2)+4=2x+9$$

1. Use the distributive property.

Multiply $3$ by each term inside the parentheses:

$$3x-6+4=2x+9$$

2. Combine like terms.

Combine $-6$ and $4$:

$$3x-2=2x+9$$

3. Move the variable terms to one side.

Subtract $2x$ from both sides:

$$3x-2x-2=2x-2x+9$$ $$x-2=9$$

4. Move the constant to the other side.

Add $2$ to both sides:

$$x-2+2=9+2$$ $$x=11$$

5. Check the solution.

Substitute $11$ for $x$ in the original equation:

$$3(11-2)+4=2(11)+9$$ $$3(9)+4=22+9$$ $$31=31$$

Since the statement is true, the solution is:

$$\boxed{x=11}$$

If all variable terms cancel and a true statement remains, there are infinitely many solutions. If all variable terms cancel and a false statement remains, there is no solution.