Solving Multi-Step Equations

Learn to solve equations that require multiple operations using inverse operations and the properties of equality.

Intermediate25 minLesson

Definition

A multi-step equation is an equation that requires two or more operations to solve. To solve these equations, we use inverse operations in the correct order.

Strategy for solving multi-step equations:

1.Simplify each side (distribute and combine like terms)
2.Move variables to one side using addition or subtraction
3.Move constants to the other side
4.Isolate the variable by dividing or multiplying

\text{Example: } 3x + 5 = 2x + 12

3x - 2x = 12 - 5

x = 7

Try it now

What is the first step to solve

2x + 5 = 13

Worked Examples

Solve: $2x + 7 = 15$

Identify the operations on $x$

$x$ is multiplied by $2$ , then $7$ is added → Two operations to undo

Undo addition first (subtract 7)

$2x + 7 - 7 = 15 - 7$ → $2x = 8$

Undo multiplication (divide by 2)

$\frac{2x}{2} = \frac{8}{2}$ → $x = 4$

Check the answer

$2(4) + 7 = 8 + 7 = 15$ ✓ → Solution verified!

Answer: $x = 4$

Common Mistakes

Forgetting to distribute to ALL terms inside parentheses

Why it's wrong: In $3(x + 4)$ , both $x$ AND $4$ must be multiplied by $3$ . Many students only multiply the first term.

Correct: $3(x + 4) = 3x + 12$ , not $3x + 4$

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 $5$ from the left side, subtract $5$ from the right side too.

Combining unlike terms

Why it's wrong: You can only combine terms with the same variable part. $3x$ and $5$ are unlike terms.

Correct: $3x + 5 + 2x = 5x + 5$ (combine $3x$ and $2x$ , keep $5$ 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 $x + 5 = 12$ , then $x = 12 - 5$ , not $x = 12 + 5$

Interactive Visual

Equation Solver

Ready to solve some equations?

Equation type:Multi-step

Follow along as we solve the equation step by step.

Balance Scale

x + 3=7

Apply to both sides:

Solution: x = 4

Interactive Sandbox

Expression Calculator

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Practice Problems

16 problems

Problem 1 of 16

Easy

What is the first step to solve $2x + 5 = 13$ ?

Why It Matters

Multi-step equations are the foundation of algebra and appear everywhere in real life:

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

Mastering these equations gives you the tools to solve complex real-world problems!

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 $50 + 25m = 200$ tells you how many months ( $m$ ) to save for a 200 dollar item.

1Try It Yourself

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: $80 + 30m = 350$

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 $20 + 0.10m = 35$ finds when they cost the same.

2Try It Yourself

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: $50 + 20m = 30m$

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

No! You can collect variables on either side. However, it's often easier to move them to the side that will give a positive coefficient. For example, in

2x = 5x - 9

, moving to the right gives

9 = 3x

(positive), while moving left gives

-3x = -9

(negative).

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 $a(b + c) = ab + ac$ for multiplying a number by a sum
Like terms: Terms with the same variable and exponent (e.g., $3x$ and $5x$ )
Coefficient: The number multiplied by a variable (in $5x$ , the coefficient is $5$ )
Isolate: To get a variable alone on one side of an equation

Formula Card

Strategy

Simplify → Collect variables → Collect constants → Isolate variable

The four-step process for solving multi-step equations

Distributive Property

a(b + c) = ab + ac

Multiply outside term by each term inside parentheses

Inverse Operations

+ \leftrightarrow -

and

\times \leftrightarrow \div

Pairs of operations that undo each other