Solving Two-Step Equations

Learn how to solve equations that require two operations to isolate the variable.

Intermediate30 minLesson
Printable worksheet

Definition

A two-step equation is an algebraic equation that requires exactly two operations to solve for the variable.
The general form is:
ax+b=corxa+b=cax + b = c \quad \text{or} \quad \frac{x}{a} + b = c
To solve, we use inverse operations in reverse order:
  1. 1.Undo addition/subtraction first (deal with the constant)
  2. 2.Undo multiplication/division second (deal with the coefficient)
This follows the reverse order of operations - we undo PEMDAS backwards!

Try it now

What is the first step to solve 2x+7=152x + 7 = 15?

Worked Examples

Solve for xx: 2x+5=132x + 5 = 13
1

Identify the operations applied to x

xx is multiplied by 2, then 5 is added → Two operations: ×2\times 2, then +5+ 5
2

Undo addition first (subtract 5 from both sides)

2x+5−5=13−52x + 5 - 5 = 13 - 5 → 2x=82x = 8
3

Undo multiplication (divide both sides by 2)

2x2=82\frac{2x}{2} = \frac{8}{2} → x=4x = 4
4

Check the solution

2(4)+5=8+5=132(4) + 5 = 8 + 5 = 13 ✓ → Correct!

Common Mistakes

Dividing before subtracting/adding

Why it's wrong: Students divide first but only divide some of the terms. In 2x+5=132x + 5 = 13, dividing 2x2x and 1313 by 2 but not the 55 gives x+5=6.5x + 5 = 6.5, so x=1.5x = 1.5, which is wrong: 2(1.5)+5=82(1.5) + 5 = 8, not 13.

Correct: Undo addition/subtraction FIRST, then multiplication/division: 2x=82x = 8, so x=4x = 4. (Dividing first only works if you divide every term: x+2.5=6.5x + 2.5 = 6.5.) Remember: reverse PEMDAS!

Only performing the operation on one side

Why it's wrong: Forgetting that whatever you do to one side, you must do to the other to keep the equation balanced.

Correct: Think of the equation as a balance scale. Both sides must stay equal!

Sign errors when moving terms

Why it's wrong: Confusing subtraction with addition. In 3x−7=143x - 7 = 14, adding 7 (not subtracting) to both sides gives 3x=213x = 21.

Correct: To undo subtraction, add. To undo addition, subtract. Use the opposite operation!

Forgetting to check the answer

Why it's wrong: Not verifying means errors go unnoticed. Always substitute back into the original equation.

Correct: Substitute your answer back into the original equation to verify it works.

Interactive Visual

Balance Scale

2x + 5=13
2x
5
13
Apply to both sides:

Solution: x = 4

Equation Solver

Solve for xx:

2x+5=132x + 5 = 13
1
2x+5=132x + 5 = 13
Subtract 55 from both sidesRemove the constant term

Tips:

  • • Whatever you do to one side, do to the other
  • • Undo operations in reverse order (PEMDAS backwards)
  • • Check your answer by substituting back

Follow along as we solve the equation step by step.

Interactive Sandbox

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the first step to solve 2x+7=152x + 7 = 15?

Why It Matters

Two-step equations are the foundation for solving more complex algebraic problems:
  • Shopping: You buy 3 identical T-shirts, use a 15 dollar coupon and pay 45 dollars. How much does one T-shirt cost?
  • Cooking: One batch of a recipe makes 4 pancakes, and you already have 2 left over. How many batches do you need for 14 pancakes in total?
  • Sports: A basketball player scores twice as many points as last game, plus 5 points from free throws. If they scored 23 points, how many did they score last game?
  • Savings: You save a fixed amount each week plus a 50 dollar bonus. After 8 weeks you have 130 dollars. How much do you save weekly?
Mastering two-step equations prepares you for multi-step equations, systems of equations, and real-world problem solving!

Real World Applications

Memberships and Fees

Gyms and streaming services often charge a monthly price plus a one-time fee.

Example:

A streaming service costs 8 dollars per month plus a one-time 20 dollar activation fee. If you paid 68 dollars total, how many months did you sign up for?

1Try It Yourself

A gym charges 25 dollars per month plus a 50 dollar signup fee. You have 200 dollars to spend.

How many months of membership can you afford?

Step 1: Write the mathematical expression

From 25m+50=20025m + 50 = 200, find mm (a calculation or its value):

Temperature Conversion

Converting between Celsius and Fahrenheit uses a two-step equation.

Example:

To convert Celsius to Fahrenheit: F=95C+32F = \frac{9}{5}C + 32. If it's 77°F outside, what's the temperature in Celsius?

2Try It Yourself

The weather app shows 86°F. You want to know the temperature in Celsius.

What is 86°F in Celsius?

Step 1: Write the mathematical expression

Use 86=95C+3286 = \frac{9}{5}C + 32 and solve for CC (a calculation or its value):

Earning Money

Many jobs pay a base rate plus commission or tips.

Example:

A salesperson earns 200 dollars base pay plus 15 dollars per item sold. This week they earned 350 dollars. How many items did they sell?

3Try It Yourself

A waiter earns 50 dollars per shift. This week they also got 120 dollars in tips and earned 320 dollars in total.

How many shifts did they work?

Step 1: Write the mathematical expression

Let ss = number of shifts: 50s+120=32050s + 120 = 320. Find ss (a calculation or its value):

Key Takeaways

  • 1Two-step equations require exactly two inverse operations to solve
  • 2Always undo addition/subtraction FIRST, then undo multiplication/division
  • 3Whatever you do to one side of the equation, do the same to the other side
  • 4Always check your answer by substituting back into the original equation
  • 5The solution can be positive, negative, or zero

Frequently Asked Questions

Think of it like getting dressed and undressed. You put on your socks THEN shoes, but you take off your shoes FIRST then socks. In 2x+52x + 5, we multiply by 2 first, then add 5. To undo, we subtract 5 first, then divide by 2.
Think of it like getting dressed and undressed. You put on your socks THEN shoes, but you take off your shoes FIRST then socks. In 2x+52x + 5, we multiply by 2 first, then add 5. To undo, we subtract 5 first, then divide by 2.
Look at what's happening to the variable. Start by undoing whatever is farthest from the variable (the constant term), then undo what's closest (the coefficient).
Yes! For example, 2x+1=42x + 1 = 4 gives x=1.5x = 1.5 or x=32x = \frac{3}{2}. Not all equations have whole number solutions.

Glossary

Two-step equation
An equation that requires exactly two inverse operations to solve for the variable
Inverse operation
An operation that undoes another operation (addition/subtraction, multiplication/division)
Coefficient
The number multiplied by the variable (in 3x3x, the coefficient is 3)
Constant
A number without a variable (in 2x+52x + 5, the constant is 5)
Isolate
To get the variable alone on one side of the equation
Today's challengeOne problem, three tries

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