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# The balance method

Algebra Foundations · Equations

<!-- Speaker note: Students solve linear equations by doing the same to both sides (the balance method), including x on both sides, and verify every answer in the first equation (AC9M7A03, node 3). About 50 minutes (Predict and Run 10 min · Investigate 12 min · Modify 12 min · Practise 10 min · Make 6 min). -->

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## Today

**We are learning to** do the same to both sides to get x on its own, then check your answer in the first equation.

- I can solve an equation by doing the same thing to both sides, and show each step.
- I can solve an equation with x on both sides.
- I can check my answer by putting it back into the first equation, every time.

<!-- Speaker note: Before class: This is a **hinge** lesson: later modelling work (ratios, rates, money problems) depends on fluent solving *and* the checking habit. A student who skips the check will not notice a wrong answer. Do the "x on both sides" scale yourself. The key move is taking a **bag of x** from both sides, which students rarely think of until they see the scale. Teacher: Leo's mistake (one side only) is the misconception this lesson exists to fix. Ask: what happened to the right side? -->

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## Why it matters

> Engineers, nurses working out doses and anyone splitting a bill rearrange equations this way: keep it balanced, and every step stays true.

<!-- Speaker note: Backtracking stops working when $x$ is on **both** sides, as in $x + 7 = 3x + 1$. The **balance method** works for every linear equation: whatever you do to one side, do to the other, so the scale stays level. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**$2x + 3 = 11$. You take 3 from **both** sides. What equation is left?**

- $2x = 8$
- $2x = 11$
- $2x + 3 = 8$
- $x = 8$

<!-- Speaker note: What happens: 2x = 8. The left loses its 3 blocks and the right loses 3 too: 11 − 3 = 8. The bags of x stay. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: each wrong option is a real slip (the right side left alone, the bags dropped, the 3 taken from the wrong side). Ask students who chose differently to explain. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Falgebra%2Fbalance-method%2Fstep-2&code=widget%3A+balance%0Amode%3A+solve%0Aleft%3A+2x+%2B+3%0Aright%3A+11%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/algebra/balance-method/?full=step-2#step-2)

<!-- Speaker note: Use **Take away** and **Share** on the board. Each step does the same to both sides, so the scale stays level, and the list underneath writes your steps as algebra. -->

<!-- Speaker note: Teacher: the scale never tips, because every button does the same to both sides. The list underneath is the written method: point at it. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 2x = 8. The left loses its 3 blocks and the right loses 3 too: 11 − 3 = 8. The bags of x stay.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## In order

Each line keeps the equation balanced.

$$3x + 5 = 20$$

1) $3x + 5 - 5 = 20 - 5$ — Take 5 from **both** sides.
2) $3x = 15$ — Now three lots of $x$ are 15.
3) $\frac{3x}{3} = \frac{15}{3}$ — Divide **both** sides by 3.
4) $x = 5$ — $x$ is on its own.
5) $3(5) + 5 = 20 \checkmark$ — Check in the **first** equation, not the last line.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 3x + 5 = 20 and say "Start with the equation." Line 1: write 3x + 5 - 5 = 20 - 5 and say "Take 5 from both sides." Line 2: write 3x = 15 and say "Now three lots of x are 15." Line 3: write 3x ÷ 3 = 15 ÷ 3 and say "Divide both sides by 3." Line 4: write x = 5 and say "x is on its own." Line 5: write 3(5) + 5 = 20 ✓ and say "Check in the first equation, not the last line." -->

<!-- Speaker note: Teacher: reveal the worked example one line at a time. Stress the last line: check in the first equation, not the line above. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## In order

**Put the steps for solving $4x + 3 = 23$ in order.**

```maths
4x = 20
x = 5
4x + 3 − 3 = 23 − 3
4x = 26
Check: 4 × 5 + 3 = 23
4x + 3 = 23
```

<!-- Speaker note: Take 3 from both sides (4x = 20), divide both sides by 4 (x = 5), then check in the first equation. Correct order: 4x + 3 = 23 4x + 3 − 3 = 23 − 3 4x = 20 x = 5 Check: 4 × 5 + 3 = 23 -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Leo solves $2x + 3 = 11$, but his check fails. Click the first line that goes wrong.**

```
 1  2x + 3 = 11
 2  2x = 11
 3  x = 5.5
 4  Check: 2 × 5.5 + 3 = 14, not 11
```

* Answer: line 2

<!-- Speaker note: Line 2: Leo took 3 from the left side only. The right side has to lose 3 too: 2x = 8, so x = 4. Check: 2 × 4 + 3 = 11. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Which step first?

**To solve $3x + 5 = 20$, which first step keeps whole numbers?**

- Divide both sides by 3
- Take 5 from both sides

* Answer: Take 5 from both sides

<!-- Speaker note: Both are allowed, but dividing first gives x + 5/3 = 20/3. Taking 5 first gives 3x = 15, then x = 5. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Harder scales



- **Bigger numbers:** Each task loads a new scale. Same moves, bigger numbers.
- **Bags on both sides:** Now there are bags of $x$ on both sides. Backtracking cannot solve this; the balance method can.
- **Bags and blocks on both sides:** Bags and blocks on both sides. Get the bags on one side first.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/algebra/balance-method/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Taking away bags of x from both sides is the same move as taking away blocks, and it handles x on both sides. Hints, in order: Bags of x can be taken away too. / Take 2 bags of x from both sides. / There are bags and blocks on both sides. / Take one bag of x from both sides first. For students who need more support: Use the board for the Practise questions first, then write each button press as a line of algebra. For early finishers: 'I think of a number, double it and add 5 to get 19.' Write the equation, solve it with the balance method, and check. Then invent one where the answer must be a whole number of people, and explain why. When students get stuck: A student changes one side only (2x + 3 = 11 becomes 2x = 11). Ask: What did you do to the left side? Did the right side get the same? Fix: Write the step on both sides: 2x + 3 − 3 = 11 − 3. / A student checks in the last line (x = 4 → 4 = 4) and calls it verified. Ask: Which equation were we asked to solve? Fix: Check in the first equation: 2 × 4 + 3 = 11. / With x on both sides, the student is stuck. Ask: You took blocks from both sides. What else is on both sides? Fix: Take the smaller number of x from both sides first: x + 7 = 3x + 1 becomes 7 = 2x + 1. -->

<!-- Speaker note: Teacher: for x + 7 = 3x + 1, let students struggle briefly, then ask what else they could take away from both sides. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Two steps

**Solve $5x + 4 = 39$.**

* Answer: 7

<!-- Speaker note: 5x + 4 − 4 = 39 − 4, so 5x = 35 and x = 7. Check: 5 × 7 + 4 = 39. -->

<!-- Speaker note: Teacher: students should write each step as a line of algebra, then check in the first equation. Equations with subtraction (adding to both sides) come in a later lesson. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Bags on both sides

**Solve $9x = 4x + 35$.**

* Answer: 7

<!-- Speaker note: Take 4x from both sides: 5x = 35. Divide by 5: x = 7. Check: 9 × 7 = 63 and 4 × 7 + 35 = 63. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Bags and blocks on both sides

**Solve $6x + 2 = 4x + 12$.**

* Answer: 5

<!-- Speaker note: Take 4x from both sides: 2x + 2 = 12. Take 2: 2x = 10. Divide by 2: x = 5. Check: 6 × 5 + 2 = 32 and 4 × 5 + 12 = 32. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: your own balance

Give it to a friend to solve with the balance method, then check their answer in your first equation.

<!-- Speaker note: What this step is for: Writing an equation with x on both sides from a chosen solution shows the balance method is reversible. Hints, in order: Pick two expressions with x, such as 4x + 1 and 2x + 7. / Work out both when x = 3: 13 and 13. They match, so 4x + 1 = 2x + 7 works. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Solve 4x + 3 = 2x + 11. Show each step and check your answer in the first equation.**

* Take 2x from both sides: 2x + 3 = 11. Take 3: 2x = 8. Divide by 2: x = 4. Check: 4 × 4 + 3 = 19 and 2 × 4 + 11 = 19.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next: equations from word problems, then graphing linear relations in Year 8, where the solution is where two lines meet. Where it leads in class: rearranging formulas in science (speed = distance ÷ time), and money problems with a fixed fee plus a rate. -->