Multi-step equations are equations that take more than one inverse operation to solve. They matter because many real situations, such as budgeting, comparing phone plans, and calculating distances, lead to equations with several parts. The goal is to isolate the variable while keeping both sides balanced.
Each step should make the equation simpler without changing its solution.
Key Facts
- Keep equations balanced by doing the same operation to both sides.
- Use the distributive property before combining unlike parts: a(b + c) = ab + ac.
- Combine like terms before moving terms across the equal sign, such as 3x + 5x = 8x.
- To move a variable term, add or subtract the same variable term on both sides.
- A linear equation often simplifies to x = number, such as 4x = 20 gives x = 5.
- Special cases: 0 = 7 means no solution, and 0 = 0 means all solutions.
Vocabulary
- Variable
- A variable is a letter or symbol that represents an unknown number.
- Coefficient
- A coefficient is the number multiplied by a variable, such as 6 in 6x.
- Like terms
- Like terms have the same variable part and can be combined, such as 2x and 5x.
- Distributive property
- The distributive property lets you multiply a number by each term inside parentheses.
- Inverse operation
- An inverse operation undoes another operation, such as subtraction undoing addition.
Common Mistakes to Avoid
- Forgetting to distribute to every term inside parentheses is wrong because 3(x + 4) becomes 3x + 12, not 3x + 4.
- Combining unlike terms is wrong because terms such as 5x and 5 do not represent the same type of quantity.
- Changing only one side of the equation is wrong because it destroys the balance and can create a false solution.
- Dividing by the coefficient too early is wrong when there are still added or subtracted terms attached to the variable term.
Practice Questions
- 1 Solve for x: 3(x + 4) - 5 = 2x + 10.
- 2 Solve for y: 4y - 7 + 2y = 3(y + 5) + 8.
- 3 Explain how you can tell whether an equation has one solution, no solution, or all solutions after simplifying both sides.