Multi-Step Equations

Multi-Step Equations

Some equations require you to simplify each side before you can apply the two-step method. This usually means distributing parentheses and/or combining like terms first. The overall strategy is:

1. Simplify each side of the equation separately (distribute, combine like terms).
2. Isolate the variable using inverse operations, just like a two-step equation.

Why Order Matters

You cannot move terms between sides of the equation until each side is fully simplified on its own. Trying to isolate the variable too early, before distributing or combining like terms, is one of the most common sources of errors.

Example

Solve 3(2x - 4) + 5 = 2x + 7.

1. Distribute on the left side: 3(2x - 4) = 6x - 12, so the equation becomes 6x - 12 + 5 = 2x + 7
2. Combine like terms on the left side (-12 + 5 = -7): 6x - 7 = 2x + 7
3. Move the variable terms to one side by subtracting 2x from both sides: 4x - 7 = 7
4. Add 7 to both sides: 4x = 14
5. Divide both sides by 4: x = \frac{14}{4} = \frac{7}{2}

Checking the Solution

Substitute x = \tfrac{7}{2} into the original equation: left side is 3(2 \cdot \tfrac{7}{2} - 4) + 5 = 3(7-4)+5 = 3(3)+5=14; right side is 2 \cdot \tfrac{7}{2} + 7 = 7 + 7 = 14. Both sides match, confirming the solution.

Equations with Distribution on Both Sides

Some multi-step equations require distributing on both sides before you can combine anything. For example, in 2(x+3) = 4(x-1), distribute on the left to get 2x + 6 and on the right to get 4x - 4, giving 2x + 6 = 4x - 4. From there, treat it like an equation with variables on both sides: subtract 2x from both sides to get 6 = 2x - 4, add 4 to both sides to get 10 = 2x, and divide by 2 to get x = 5.

Key Takeaways

• Fully simplify each side (distribute, combine like terms) before isolating the variable.
• Only after both sides are simplified should you start moving terms across the equals sign.
• Fraction answers are perfectly valid solutions — do not round unless the problem asks for it.
• Verify your answer in the original (unsimplified) equation.