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7th Grade algebra · Ages 12–13

Two Steps Back

Two-step equations, negative numbers and real-world stories — the year algebra starts to look like algebra.

xthe unknown

The one idea

Undo in reverse order: peel off the adding first, then the multiplying — like taking off shoes before socks.

A two-step equation like 3x + 4 = 19 is just two one-step equations in a row. The order matters: undo the + 4 before the × 3, because the + 4 was the last thing done to x. Negative answers are normal now — x = −4 is a perfectly good answer.

📚 The 4 lessons⚖️ Balance scale✏️ Practice

This page is the right one if they can already…

  • Solves one-step equations with all four operations
  • Combines like terms
  • Adds and subtracts negative numbers

Not yet? Start at 6th Grade: Balance and Tip — algebra only works one rung at a time.

The four 7th Grade algebra lessons

In order — each one leans on the last. Every example is broken into single moves and ends by checking itself.

1

Two-step equations

Undo the last thing that was done to x first. Usually that means: deal with the adding, then the multiplying.

xxx
19

Worked example

3x + 4 = 19

  1. 1

    x was multiplied by 3, then 4 was added. Undo the + 4 first.

    3x = 19 − 4

  2. 2

    Work it out.

    3x = 15

  3. 3

    Now undo the × 3.

    x = 15 ÷ 3 = 5

Answer: x = 5

Check: 3(5) + 4 = 15 + 4 = 19 ✓

⚠️ The usual mistake

Divides by 3 first and gets x + 4 = 19 ÷ 3.

They undo in the order they read, left to right.

Fix: Undo in reverse: socks go on before shoes, so shoes come off first. The + 4 went on last, so it comes off first.

💬 Say this

“What was the last thing done to x? Undo that one first.”

🏠 Try it at home

“Think of a number, triple it, add 4. Tell me the answer.” Work out their number in your head. Then swap.

2

Negative answers

If adding something made the total smaller, x must be negative. Negative answers are real answers.

−6−5−4−3−2−10123456

Worked example

x + 7 = 3

  1. 1

    Undo + 7 by subtracting 7 from both sides.

    x = 3 − 7

  2. 2

    Start at 3 on the number line and go back 7.

    3 → 0 is 3 steps, 4 more to go

  3. 3

    Land on the answer.

    x = −4

Answer: x = −4

Check: −4 + 7 = 3 ✓

⚠️ The usual mistake

Writes x = 4.

They calculate 7 − 3 because taking the bigger from the smaller “can’t be done”.

Fix: Use a thermometer: 3° and the temperature drops 7°. Where is it? Below zero.

💬 Say this

“Adding 7 made it 3. So was x bigger or smaller than zero?”

🏠 Try it at home

Use a lift or a thermometer: “We’re on floor 3, we went up 7 to get here. Where did we start?” (Basement 4.)

3

Brackets in equations

Either share out the brackets first, or divide both sides by the number outside. Both roads arrive at the same answer.

Worked example

2(x + 3) = 16

  1. 1

    Share out the brackets.

    2x + 6 = 16

  2. 2

    Undo + 6.

    2x = 10

  3. 3

    Undo × 2.

    x = 5

Answer: x = 5

Check: Shortcut road: divide both sides by 2 → x + 3 = 8 → x = 5. Same answer ✓

⚠️ The usual mistake

Writes 2x + 3 = 16.

The 2 only multiplied the x, not the 3.

Fix: The number outside visits everyone inside. Draw the area rectangle to see two pieces.

💬 Say this

“Who does the 2 have to multiply?”

🏠 Try it at home

“Two friends each buy a ticket and a $3 drink. They spend $16. What is a ticket?” That is 2(t + 3) = 16.

4

Write the equation from a story

Find the thing that changes (multiplied by x) and the thing that stays the same (added on). Then solve.

MilesCost ($)
03
15
27
m2m + 3

Worked example

A taxi charges $3 plus $2 for every mile. The ride cost $15. How many miles?

  1. 1

    Let m be the miles. Each mile adds $2, plus the fixed $3.

    2m + 3 = 15

  2. 2

    Undo + 3.

    2m = 12

  3. 3

    Undo × 2.

    m = 6

Answer: 6 miles

Check: 2(6) + 3 = $15 ✓

⚠️ The usual mistake

Writes 3m + 2 = 15.

They attach the letter to the first number mentioned.

Fix: Ask which number happens once (the $3) and which happens every mile (the $2). The every-time number goes with m.

💬 Say this

“Which cost happens once? Which happens again and again?”

🏠 Try it at home

Find a real price list with a fixed fee — delivery, a gym, a phone plan — and write it as an equation.

⚖️ Keep it fair

Get x on its own using only fair moves. This is exactly how equations are solved at every grade, right up to Grade 10. Try the wrong button too — see what happens.

Puzzle: 2x + 3 = 11

xx
ten

2x + 3 = 11

Tip: clear the loose dots first, then share. Undo the adding before the multiplying.

✏️ Practice that never runs out

Every question is new. A wrong answer gets a hint; a second one shows the worked solution, so no one stays stuck.

Two-step equations

Undo the adding first, then the multiplying. Answers can be negative.

Solve for x.

−3x − 6 = −33

Simplify first

Tidy expressions with negatives in them.

Which is the simplified version?

−2x + 5 − 3x + 8

The words, in plain English

Two-step equation
Needs two undoing moves.
3x + 4 = 19
Negative number
A number less than zero.
−4
Distributive property
a(b + c) = ab + ac.
2(x + 3) = 2x + 6
Constant
A number that doesn’t change.
The $3 taxi fee.
Rate
How much per one of something.
$2 per mile.

🌍 Algebra in an ordinary day

  • Phone plans: $10 a month plus $5 per gigabyte
  • Temperatures dropping below zero
  • “Think of a number” tricks — every one is a two-step equation

🏁 By the end of 7th Grade

  • Solves two-step equations in the right order
  • Accepts and checks negative answers
  • Solves equations with brackets two different ways
  • Writes an equation from a real-world story

Where this fits

One idea, twelve rungs. The hiding spot just gets cleverer.

  1. Pre-K❓The Hiding Game
  2. K🎁Make It Fair
  3. 1□The Same As
  4. 2?Number Machines
  5. 3nLetters Are Boxes
  6. 4nNumber Recipes
  7. 5xUndo It
  8. 6xBalance and Tip
  9. 7xTwo Steps Back
  10. 8y = mx + bLines and Slope
  11. 9xAlgebra 1
  12. 10f(x)Curves and Functions

← One rung down

6th Grade: Balance and Tip

One rung up →

8th Grade: Lines and Slope

All 12 gradesMore maths, grade by grade →

Frequently Asked Questions

What algebra should a 7th Grade child learn?

Undo in reverse order: peel off the adding first, then the multiplying — like taking off shoes before socks. By the end of the year they should be able to: solves two-step equations in the right order; accepts and checks negative answers; solves equations with brackets two different ways; writes an equation from a real-world story.

Isn’t 7th Grade too early for algebra?

No — two steps back is standard 7th Grade maths in most curricula. This page breaks it into single moves so it never feels like a jump.

What is the most common 7th Grade algebra mistake?

Writes x = 4. They calculate 7 − 3 because taking the bigger from the smaller “can’t be done”. The fix: Use a thermometer: 3° and the temperature drops 7°. Where is it? Below zero.

What if my child is stuck?

Go down one rung. Every lesson here builds on the 6th Grade page, and the “ready for this page if” list at the top tells you exactly what they need first. In practice, a wrong answer gives a hint, and a second wrong answer shows the full worked solution.

Is it free?

Yes. All 4 lessons, the pictures, the worked examples, the balance scale and the practice questions are free with no signup. The practice never runs out — every question is generated fresh and checked.

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