---
title: "Solving harder equations"
subtitle: "Algebra Foundations · Equations and inequalities · Lesson 13"
---

# Solving harder equations

**Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}}

> **Key question:** Two phone plans: 7 dollars plus 3 a gigabyte, or 5 a gigabyte minus a 9 dollar credit. When do they cost the same? 3x + 7 = 5x − 9.


:::teacher-note

**In one sentence:** Students solve linear equations with x on both sides, brackets, negative answers and fractions by doing the same to both sides, writing one step per line, and verify every answer by substitution; a graph shows the solution as where the two sides meet (AC9M8A02, node 2).

**Students should be able to say:** I can solve an equation with x on both sides, including negatives. I can solve an equation with brackets by expanding first. I can check my solution by substituting it into the first equation.

**Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet.

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## Predict

Try each option in both sides. Or solve it: what could you take from both sides first?

:::keep-together

**Which value of $x$ makes $3x + 7 = 5x - 9$ true?**

☐ $x = 2$

☐ $x = -8$

☐ $x = 1$

☐ $x = 8$

My prediction, and why:

{{lines-3}}

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:::teacher-note

**What happens:** x = 8. Check: 3 × 8 + 7 = 31 and 5 × 8 − 9 = 31. Both sides are equal.

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## Run

Try it on the board in the tutorial. Was your prediction right? What was different?

{{lines-2}}

## Same to both sides


:::keep-together

### Solve, then check

Here is how to solve with $x$ on both sides.

1. $3x + 7 = 5x - 9$ — $x$ is on both sides.
2. $7 = 2x - 9$ — Take $3x$ from both sides (the smaller $x$ term, so $2x$ stays positive).
3. $16 = 2x$ — Add 9 to both sides: this undoes $- 9$.
4. $x = 8$ — Divide both sides by 2.
5. $3(8) + 7 = 31,\; 5(8) - 9 = 31$ — Check in the first equation: both sides are 31.

Now solve it yourself, one step per line. The checker marks the first line that does not follow.

**Solve $3x + 7 = 5x - 9$. Write one step per line.**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 3x + 7 = 5x - 9 → 7 = 2x - 9 → 16 = 2x → x = 8

Take 3x from both sides: 7 = 2x − 9. Add 9: 16 = 2x. Halve: x = 8.

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:::keep-together

### Find the mistake

**Ben solves $2x + 5 = 4x - 7$. Click the line where his working goes wrong.**

```
 1  2x + 5 = 4x − 7
 2  Take 2x from both sides: 5 = 2x − 7
 3  Take 7 from both sides: −2 = 2x
 4  So x = −1.
```

Line: {{field-15}}

Why is it wrong?

{{lines-2}}

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:::teacher-note

**Answer:** line 3

Line 3: to undo − 7, add 7 to both sides: 12 = 2x, so x = 6. Check: 2 × 6 + 5 = 17 and 4 × 6 − 7 = 17.

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:::keep-together

### Check by substituting

**Check $x = 8$ in $3x + 7 = 5x - 9$: what is the value of each side?**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 31

3 × 8 + 7 = 31 and 5 × 8 − 9 = 31: equal, so x = 8 is right.

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:::typical

## One change at a time


**A bracket.** Each task changes one thing. Check each answer by substituting.

**A negative answer.** 

**A fraction.** 

**Brackets on both sides.** 

**No solution.** 

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

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## Solve and check


:::keep-together

### x on both sides

**Solve $5x - 4 = 2x + 11$. One step per line.**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 5x - 4 = 2x + 11 → 3x - 4 = 11 → 3x = 15 → x = 5

Take 2x: 3x − 4 = 11. Add 4: 3x = 15. x = 5. Check: 21 = 21.

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:::keep-together

### Negative x

**Solve $6 - x = 2x - 9$.**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 5

Add x: 6 = 3x − 9. Add 9: 15 = 3x. x = 5. Check: 1 = 1.

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:::keep-together

### Angles

**Angles on a straight line are $3x$ and $x + 20$ degrees. Find $x$.**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 40

3x + x + 20 = 180, so 4x = 160 and x = 40. The angles are 120° and 60°.

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:::keep-together

### Review: expanding

**Review: expand $4(2x - 3)$.**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 8x - 12

4 × 2x = 8x and 4 × (−3) = −12.

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## Make: build an equation

Write your equation, then solve it on paper to check.

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

## Exit ticket

:::keep-together

**Solve 4(x − 1) = 2x + 6 and check.**

{{lines-3}}

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:::teacher-note

**Answer:** 4x − 4 = 2x + 6, 2x = 10, x = 5. Check: 16 = 16.

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