Solving Equations with Variables on Both Sides

When solving linear equations, you will often encounter problems where the unknown variable appears on both sides of the equal sign, such as $3x + 2 = x + 10$.

The main goal remains the same: isolate the variable. To do this, you need to collect all the variable terms on one side of the equation and all the constant terms (plain numbers) on the other side.

The Core Strategy

Follow these four steps to solve any equation with variables on both sides:

1. Simplify each side first: If there are parentheses, use the distributive property. Combine any like terms on each side.
2. Move the variables to one side: Use addition or subtraction to move all terms containing the variable to the same side of the equal sign.
3. Move the constants to the other side: Use addition or subtraction to get all the plain numbers on the opposite side of the variables.
4. Isolate the variable: Multiply or divide to find the final value of the unknown.

Step-by-Step Examples

Example 1: Basic Variables on Both Sides

Solve for $x$: $3x + 2 = x + 10$

Step 1: Subtract $x$ from both sides to group the variables on the left. $3x - x + 2 = 10$ $2x + 2 = 10$

Step 2: Subtract $2$ from both sides to group the constants on the right. $2x = 10 - 2$ $2x = 8$

Step 3: Divide both sides by $2$ to isolate $x$. $x = 4$

Example 2: Distributing on One Side

Solve for $x$: $5(x - 1) = 2x + 7$

Step 1: Distribute the $5$ into the parentheses. $5x - 5 = 2x + 7$

Step 2: Subtract $2x$ from both sides. $3x - 5 = 7$

Step 3: Add $5$ to both sides. $3x = 12$

Step 4: Divide by $3$. $x = 4$

Example 3: Distributing on Both Sides

Solve for $x$: $2(3x + 4) = 4(x - 1)$

Step 1: Distribute on both sides of the equation. $6x + 8 = 4x - 4$

Step 2: Subtract $4x$ from both sides. $2x + 8 = -4$

Step 3: Subtract $8$ from both sides. $2x = -4 - 8$ $2x = -12$

Step 4: Divide by $2$. $x = -6$

A Helpful Tip

When deciding which way to move your variables, try to move the smaller variable term toward the larger one. For example, in $2x = 5x - 9$, subtracting $2x$ from both sides keeps the $x$ term positive ($0 = 3x - 9$), which helps prevent mistakes with negative signs!