Multi-Step Linear Equations

A multi-step linear equation is an equation that requires two or more steps to solve. Your ultimate goal is always the same: isolate the variable (get the letter by itself on one side of the equal sign).

To do this, you will use inverse operations (opposite operations) to "undo" the math step by step.

The Core Strategy

When solving multi-step equations, it helps to follow a standard order:

1. Simplify first: If there are parentheses, use the distributive property. If there are like terms on the same side, combine them.
2. Undo addition and subtraction: Move the constant numbers away from the variable term.
3. Undo multiplication and division: Isolate the variable completely.

Remember the golden rule of algebra: Whatever you do to one side of the equation, you must do to the exact same thing to the other side.

Step-by-Step Examples

Example 1: The Standard Two-Step Equation

Solve for $x$: $2x + 3 = 11$

Step 1: Undo the addition. Subtract $3$ from both sides to isolate the $2x$ term. $2x + 3 - 3 = 11 - 3$ $2x = 8$

Step 2: Undo the multiplication. The variable $x$ is being multiplied by $2$, so divide both sides by $2$. $\frac{2x}{2} = \frac{8}{2}$ $x = 4$

Example 2: Equations with Fractions

Solve for $x$: $\frac{x + 3}{2} = 5$

In this case, the entire expression $x + 3$ is being divided by $2$. We need to clear the fraction first.

Step 1: Undo the division. Multiply both sides by $2$. $2 \cdot \frac{x + 3}{2} = 5 \cdot 2$ $x + 3 = 10$

Step 2: Undo the addition. Subtract $3$ from both sides. $x + 3 - 3 = 10 - 3$ $x = 7$

Example 3: Using the Distributive Property

Solve for $x$: $3(x - 2) = 15$

Step 1: Distribute. Multiply the $3$ by everything inside the parentheses. $3 \cdot x - 3 \cdot 2 = 15$ $3x - 6 = 15$

Step 2: Undo the subtraction. Add $6$ to both sides. $3x - 6 + 6 = 15 + 6$ $3x = 21$

Step 3: Undo the multiplication. Divide both sides by $3$. $\frac{3x}{3} = \frac{21}{3}$ $x = 7$

(Note: For Example 3, you could also start by dividing both sides by 3 to get $x - 2 = 5$, and then add 2 to get $x = 7$. Both methods are correct!)