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# Solving equations with graphs

Linear relations · Solving and modelling with lines

<!-- Speaker note: Students solve linear equations by drawing each side as a line and reading the x where the lines meet, then check by substitution (AC9M8A02). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 15 min · Practise and Make 10 min). -->

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## Today

**We are learning to** draw each side of an equation as a line. Where the lines meet, both sides are equal, so the x there is the solution.

- I can solve an equation by drawing each side as a line and finding where they meet.
- I can explain why the solution is the x of the meeting point, not the y.
- I can check a graph's answer by substitution, and recognise parallel lines as no solution.

<!-- Speaker note: Before class: Do the Run and Modify boards yourself first. This lesson joins the two units: the Equations unit's equations (solved with the balance and backtracking) and this unit's lines. The key idea is which number to read. At the meeting point both sides have the same value (the y), and the solution is the x that gives it. Expect students to read the y: it is the most common mistake, and the Predict and Kai's working are built around it. The Make board uses two sliders and shows the meeting point in green. Any line through (3, 7) other than y = 2x + 1 works. -->

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## Why it matters

> A graph can solve an equation you have no method for yet: draw both sides and look where they meet. Calculators, spreadsheets and engineers do exactly this.

<!-- Speaker note: An equation such as $2x + 3 = 11$ says two things are equal. Each side can be drawn as a line: $y = 2x + 3$, and $y = 11$ (a flat line, because 11 does not change with $x$). In the Equations unit you solved equations with a balance and by backtracking. Now you solve them by looking at a graph. -->

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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

**The lines $y = 2x + 3$ and $y = 11$ are drawn. What is $x$ at the point where they meet?**

- $x = 3$
- $x = 8$
- $x = 11$
- $x = 4$

<!-- Speaker note: What happens: x = 4. Where they meet, both lines have the same y, so 2x + 3 = 11. That happens when x = 4 (2 × 4 + 3 = 11). The point is (4, 11): 11 is its y, not its x. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. 'x = 11' is the common wrong answer (reading the y). Do not correct it: the Run board settles it. -->

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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>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Flinear-relations%2Fsolve-graphs%2Fstep-2&code=widget%3A+graph%0Ax%3A+-1%2C+6%0Ay%3A+-2%2C+14%0Aplot%3A+y+%3D+2x+%2B+3%0Aplot%3A+y+%3D+11%0Apoint+P%3A+1%2C+1%2C+drag%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/solve-graphs/?full=step-2#step-2)

<!-- Speaker note: Drag $P$ until it is on both lines. Read its coordinates under the board. Which number is the solution of $2x + 3 = 11$? -->

<!-- Speaker note: Teacher: when P is on both lines, ask: what is the same about the two lines at this point? (The y.) So which number is the solution? (The x.) -->

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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?

* x = 4. Where they meet, both lines have the same y, so 2x + 3 = 11. That happens when x = 4 (2 × 4 + 3 = 11). The point is (4, 11): 11 is its y, not its x.

<!-- 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>

## Solving with a graph

Here is the method, one line at a time.

$$2x + 3 = 11$$

1) $y = 2x + 3, \quad y = 11$ — Draw each side as a line.
2) $(4, 11)$ — Find where they meet. There, both sides have the same value, 11.
3) $x = 4$ — The solution is the $x$ of the meeting point, not the $y$.
4) $2(4) + 3 = 11 \; \checkmark$ — Check by substitution in the first equation.

<!-- 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 2x + 3 = 11 and say "Solve this equation with a graph." Line 1: write y = 2x + 3, y = 11 and say "Draw each side as a line." Line 2: write (4, 11) and say "Find where they meet. There, both sides have the same value, 11." Line 3: write x = 4 and say "The solution is the x of the meeting point, not the y." Line 4: write 2(4) + 3 = 11 ✓ and say "Check by substitution in the first equation." -->

<!-- Speaker note: Teacher: model the worked example live. Sketch the two lines, mark where they meet, then say 'the solution is the x, because that is the input that makes both sides equal'. Finish with the substitution check. -->

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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>

## Solving with a graph

**On a graph, $y = 3x - 2$ meets $y = 7$ at $(3, 7)$. What is the solution of $3x - 2 = 7$?**

* Answer: 3

<!-- Speaker note: x = 3. Check: 3 × 3 − 2 = 7. -->

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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

**Kai solves an equation with a graph. Click the line where his working goes wrong.**

```
 1  Solve x + 5 = 2x + 1 with a graph.
 2  Draw y = x + 5 and y = 2x + 1.
 3  The lines meet at (4, 9).
 4  So the solution is x = 9.
```

* Answer: line 4

<!-- Speaker note: Line 4: 9 is the y of the meeting point, the value of both sides. The solution is the x: x = 4. Check: 4 + 5 = 9 and 2 × 4 + 1 = 9. -->

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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 equation?

**The lines $y = x + 5$ and $y = 2x + 1$ meet at $(4, 9)$. Which equation has the solution $x = 4$?**

- $x + 5 = 2x + 1$
- $2x + 1 = 4$
- $x + 5 = 4$

* Answer: $x + 5 = 2x + 1$

<!-- Speaker note: The meeting point is where the two sides are equal, so it solves x + 5 = 2x + 1. Check: 4 + 5 = 9 and 2 × 4 + 1 = 9. -->

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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>

## Change the equation



- **Subtraction:** Each task draws both sides of a new equation. Solve $3x - 4 = 8$.
- **A negative answer:** Solve $2x + 3 = 1$.
- **$x$ on both sides:** Solve $x + 5 = 3x + 1$. Now both sides are sloping lines.
- **Between the lines:** Solve $2x + 1 = 6$.
- **Never meeting:** $2x + 1 = 2x + 4$. Look at the two lines. What does the last lesson say about them?

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/solve-graphs/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a subtraction, a negative solution, x on both sides, an answer between the grid lines, and two lines that never meet (no solution). Hints, in order: The lines meet to the left of the y-axis. / Both lines slope now. Follow them until they cross. / P moves in halves on this board. The lines meet between two grid lines. For students who need more support: Give squared paper and have students draw both lines from tables of values (x = 0 to 4) before using the board. For early finishers: Solve 3x − 2 = x + 5 with a graph: the answer is not a whole number. How accurate can a graph be? (x = 3.5) When students get stuck: A student gives the y of the meeting point as the solution. Ask: Put your answer into the equation. Is it true? Fix: At the meeting point, read the x off the bottom axis. The y is the value of both sides there, not the answer. / A student cannot find where the lines meet on the negative task. Ask: Which way do the lines get closer together? Fix: Follow both lines to the left of the y-axis. The solution can be negative. / A student says parallel lines meet somewhere far away. Ask: How far apart are they at x = 0? At x = 10? Fix: Work out both rules at a few values of x: the gap is always 3. They never meet, so there is no solution. / A student moves the Make sliders at random. Ask: Where must your line go through? Fix: Work out y = 2x + 1 at x = 3 (it is 7). Choose m, then find c so that 3m + c = 7. -->

<!-- Speaker note: Teacher: the last task has parallel lines. Ask: what did the last lesson say about lines with the same m? Then: what does that mean for the equation? -->

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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>

## From a graph

**A graph shows $y = 4x - 3$ meeting $y = 9$ at $(3, 9)$. Solve $4x - 3 = 9$.**

* Answer: 3

<!-- Speaker note: x = 3. Check: 4 × 3 − 3 = 9. -->

<!-- Speaker note: Teacher: for each answer, ask a student to say which number is the y and why it is not the answer. -->

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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>

## Both sides sloping

**$y = 2x - 1$ and $y = -x + 5$ meet at $(2, 3)$. What value of $x$ makes $2x - 1 = -x + 5$ true?**

* Answer: 2

<!-- Speaker note: x = 2. Check: 2 × 2 − 1 = 3 and −2 + 5 = 3. -->

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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 equation?

**The lines $y = 5$ and $y = x + 2$ meet. Which equation does the meeting point solve?**

- $x + 2 = 5$
- $x = 5$
- $x + 5 = 2$

* Answer: $x + 2 = 5$

<!-- Speaker note: Set the two rules equal: x + 2 = 5. They meet at (3, 5), so x = 3. -->

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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>

## Review

**Review: where does $y = 3x - 4$ cross the $y$-axis? Give the value of $y$.**

* Answer: -4

<!-- Speaker note: −4: when x = 0, y = 3 × 0 − 4 = −4. -->

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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: an equation with a chosen answer

Your line is $y = mx + c$. The green point shows where it meets $y = 2x + 1$. Move it so they meet at $x = 3$. (Your line cannot be $y = 2x + 1$ itself.)

Stretch: write down your equation, then check $x = 3$ by substitution.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/solve-graphs/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Building a second line through a chosen meeting point reverses the lesson: from the solution to the equation. Hints, in order: At x = 3, the line y = 2x + 1 is at 7. Your line must pass through (3, 7). / Pick m first. Then find c so that 3m + c = 7. -->

<!-- Speaker note: Teacher: collect two different lines that work and compare them: both pass through (3, 7). -->

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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

**The lines y = 2x − 3 and y = 5 meet at (4, 5). Solve 2x − 3 = 5 and check your answer.**

* x = 4. Check: 2 × 4 − 3 = 5.

<!-- 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: modelling with two lines. Two phone plans or two ways to pay become two rules; where their lines cross is where they cost the same. Where it leads in class: break-even points in business, when two runners meet, and solving equations that have no neat algebra method. -->