Solving Multi-Step Equations
Learn to solve equations that require multiple operations using inverse operations and the properties of equality.
Definition
Try it now
Worked Examples
Solve:
Identify the operations on
is multiplied by , then is added → Two operations to undo
Undo addition first (subtract 7)
→
Undo multiplication (divide by 2)
→
Check the answer
✓ → Solution verified!
Answer:
Common Mistakes
Forgetting to distribute to ALL terms inside parentheses
Why it's wrong: In , both AND must be multiplied by . Many students only multiply the first term.
Correct: , not
Performing operations on only one side of the equation
Why it's wrong: An equation is like a balance scale. Whatever you do to one side must be done to the other to keep it balanced.
Correct: If you subtract from the left side, subtract from the right side too.
Combining unlike terms
Why it's wrong: You can only combine terms with the same variable part. and are unlike terms.
Correct: (combine and , keep separate)
Moving terms incorrectly (sign errors)
Why it's wrong: When moving a term to the other side, its sign changes. Subtracting is the inverse of adding.
Correct: If , then , not
Interactive Visual
Equation Solver
Ready to solve some equations?
Follow along as we solve the equation step by step.
Balance Scale
Solution: x = 4
Interactive Sandbox
Expression Calculator
Try these:
History
No calculations yet
Practice Problems
16 problemsWhat is the first step to solve ?
Why It Matters
- Personal Finance: Calculating how many months to save for a goal when you have starting savings and add monthly contributions
- Science: Converting temperatures, calculating speeds, or balancing chemical equations
- Business: Determining break-even points or profit margins
- Construction: Calculating materials needed when dealing with multiple measurements
Real World Applications
Savings Goal
Calculate how many months you need to save to reach a financial goal.
Example:
You have 50 dollars saved and add 25 dollars each month. The equation tells you how many months () to save for a 200 dollar item.
You want to buy a gaming console for 350 dollars. You already have 80 dollars and can save 30 dollars per month.
How many months until you can buy the console?
Step 1: Write the mathematical expression
Set up the equation:
Mobile Phone Plan
Compare phone plans to find the break-even point.
Example:
Plan A costs 20 euros plus 0.10 euros per minute. Plan B costs 35 euros with unlimited minutes. The equation finds when they cost the same.
Gym A charges a 50 euro sign-up fee plus 20 euros per month. Gym B charges no sign-up fee but 30 euros per month.
After how many months do they cost the same?
Step 1: Write the mathematical expression
Set up the equation:
Key Takeaways
- 1Multi-step equations require two or more operations to solve
- 2Always simplify each side first (distribute and combine like terms)
- 3Use inverse operations in reverse order: undo addition/subtraction before multiplication/division
- 4Move all variables to one side and constants to the other
- 5Whatever you do to one side, do to the other side
- 6Always check your answer by substituting back into the original equation
Frequently Asked Questions
Glossary
- Multi-step equation
- An equation that requires two or more operations to solve
- Inverse operations
- Operations that undo each other: addition/subtraction, multiplication/division
- Distributive property
- The rule for multiplying a number by a sum
- Like terms
- Terms with the same variable and exponent (e.g., and )
- Coefficient
- The number multiplied by a variable (in , the coefficient is )
- Isolate
- To get a variable alone on one side of an equation
Formula Card
Strategy
The four-step process for solving multi-step equations
Distributive Property
Multiply outside term by each term inside parentheses
Inverse Operations
Pairs of operations that undo each other