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Solving Two-Step Equations · How to solve two-step linear equations by undoing operations in reverse order while keeping both sides balanced

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Prerequisite: Complete or review Solving One-Step Equations with a Balance Scale

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Solving Two-Step Equations

with Atlas

Atlas stands beside a glowing balance scale floating in starlight, a small box marked x resting on the left pan among scattered number tiles waiting to be moved.

Identify the two operations applied to the variable in a two-step equation
Undo operations in reverse order using inverse operations
Keep an equation balanced by applying the same step to both sides
Check a solution by substituting it back into the original equation

Key terms

variable: A letter, such as x, that stands for an unknown number you are solving for
coefficient: The number multiplied by the variable, such as the 3 in 3x
inverse operation: An operation that undoes another, like subtraction undoing addition
isolate: To get the variable alone on one side of the equation

Reading a two-step equation

An equation like 3x + 4 = 19 tells a short story about a number. The variable x was first multiplied by 3, and then 4 was added. To find x you will reverse that story. Naming the two operations before you touch the equation keeps you from guessing and helps you plan your two steps.

Undo in reverse order

Because the expression was built by multiplying and then adding, you undo it by subtracting and then dividing. This mirrors the order of operations run backward. If you try to divide first while the 4 is still attached, you would have to divide every term, which is messier and a common source of mistakes. Removing the added number first leaves a clean multiplication to undo.

Keep both sides balanced

An equation is a promise that two sides are equal. Every step must protect that promise, so whatever you subtract or divide on the left you also subtract or divide on the right. If you change only one side, the two sides stop being equal and your answer will be wrong. Doing the same thing to both sides is what keeps the scale level.

Worked examples

Solve 3x + 4 = 19

Subtract 4 from both sides: 3x = 15
Divide both sides by 3: x = 5

Answer: x = 5

Solve x/2 - 5 = 1

Add 5 to both sides: x/2 = 6
Multiply both sides by 2: x = 12

Answer: x = 12

Hi, I'm Atlas. A two-step equation hides your variable behind two operations, and your job is to peel them away one at a time. Picture the equation as a balanced scale: whatever you do to one side you must also do to the other, or the balance breaks and the equation is no longer true. Start by undoing addition or subtraction, then undo multiplication or division. That order is the reverse of how the expression was built, which is why we call it working backward. Each move uses an inverse operation: subtraction undoes addition, and division undoes multiplication. When the variable finally stands alone, you have your answer, and you can always confirm it by putting the number back into the original equation.

Activity

Put the solving steps for 2x + 5 = 13 in the correct order

Practice

Solve 6x + 2 = 20 and check your answer by substitution

6x + 2 = 20

Solve x/3 + 4 = 7 step by step in your notebook

Common mistakes to avoid

Divide by the coefficient before undoing the additionUndo addition or subtraction first, then divide by the coefficient, because you reverse the order of operations.
Change only one side of the equationWhatever you do to one side you must do to the other, or the equation is no longer balanced.

Check your understanding

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

Solve for x: 5x - 3 = 12

Why do you divide last when solving 4x + 6 = 26?

Recap

A two-step equation applies two operations to the variable. To solve it, undo those operations in reverse order using inverse operations, always changing both sides equally, and then check your solution by substituting it back into the original equation.

Reflect

Which step felt trickiest, and how would you explain undoing operations in reverse order to a friend?