Mathematics › 🔄 Grade 7 › Solving Two-Step Equations

Solving Two-Step Equations

Two operations are tangled around x — undo them in reverse order and the balance scale tells you exactly what x must be.

Grade 7Middle school

Solving Two-Step Equations — illustration

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The big idea: A two-step equation has two operations applied to x, like multiplying then adding. To solve it, undo those operations in reverse order — addition or subtraction first, then multiplication or division — while keeping both sides of the equation equal at every step, just like keeping a balance scale level.

🎯 By the end, you'll be able to

Model a two-step equation as a balanced scale
Solve a two-step equation by undoing operations in reverse order
Check a solution by substituting it back into the original equation
Solve two-step equations that involve negative numbers

📎 You should already know

Two operations, tangled together

A one-step equation undoes a single operation. A two-step equation, like 3x + 5 = 20, has two things happening to x: it's being multiplied by 3, and 5 is being added. To solve it, you have to carefully unwind both operations — and the order you unwind them in matters.

🔑 Whatever you do to one side, do to the other

Think of an equation as a balance scale: both sides are equal, like two pans holding the same weight. Add, subtract, multiply, or divide — whatever operation you apply, apply it to both sides, or the scale tips and the equation stops being true.

\[ ax + b = c \]

The general shape of a two-step equation: x is multiplied by a, then b is added, giving c.

Undo in reverse order

The operations were applied to x in the order “multiply, then add.” To isolate x again, undo them in the opposite order: first undo the addition or subtraction (the last thing that happened), then undo the multiplication or division (the first thing that happened).

🎮 Two-Step Balance LIVE

Undo addition then undo multiplication — reverse the order of operations to solve for x.

The two-step routine

Every two-step equation follows the same routine: (1) add or subtract the same number on both sides to isolate the multiplication term, then (2) divide (or multiply) both sides by the same number to isolate x completely. Two clean moves, every time.

📝 Worked example: Solve 3x + 5 = 20.

Undo the addition first: subtract 5 from both sides. 3x + 5 − 5 = 20 − 5, so 3x = 15.
Undo the multiplication: divide both sides by 3. 3x ÷ 3 = 15 ÷ 3.
Check: 3(5) + 5 = 15 + 5 = 20. ✓

✓ x = <strong>5</strong>.

📝 Worked example: Solve −2x − 4 = 10.

Undo the subtraction first: add 4 to both sides. −2x − 4 + 4 = 10 + 4, so −2x = 14.
Undo the multiplication: divide both sides by −2. −2x ÷ (−2) = 14 ÷ (−2).
Check: −2(−7) − 4 = 14 − 4 = 10. ✓

✓ x = <strong>−7</strong>.

⚠️ Watch your signs when dividing

Dividing both sides by a negative number is where most mistakes happen — it's easy to forget that a positive divided by a negative gives a negative result. Always double-check your solution by substituting it back into the original equation.

Check your understanding

1. Solve 2x + 3 = 11.

Subtract 3: 2x = 8. Divide by 2: x = 4.

2. Solve 5x − 7 = 18.

Add 7: 5x = 25. Divide by 5: x = 5.

3. When solving a two-step equation, which operation should you undo first?

Undo the operations in reverse order: addition/subtraction first, then multiplication/division.

4. Solve −3x + 6 = 0.

Subtract 6: −3x = −6. Divide by −3: x = 2.

5. Is x = −4 a solution to 2x + 10 = 2?

2(−4) + 10 = −8 + 10 = 2, which matches the right side, so x = −4 checks out.

✅ Key takeaways

A two-step equation has two operations applied to x, such as multiply then add.
Keep the equation balanced: whatever you do to one side, do to the other.
Undo the operations in reverse order — addition/subtraction first, then multiplication/division.
Always check your solution by substituting it back into the original equation.
Dividing by a negative number is where sign mistakes most often happen — check carefully.