Solving equations with graphs

Classroom resources

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Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 15 min · Practise and Make 10 min

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.

Running the lesson

1. Predict: take a show of hands. 'x = 11' is the common wrong answer (reading the y). Do not correct it: the Run board settles it.
2. Run: 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.)
3. Investigate: 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.
4. Modify: 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?
5. Practise: for each answer, ask a student to say which number is the y and why it is not the answer.
6. Make: collect two different lines that work and compare them: both pass through (3, 7).

When students get stuck

Answers to the checks

1. where they meet: The lines $y = 2x + 3$ and $y = 11$ are drawn. What is $x$ at the point where they meet?
   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.
2. find the meeting point: Drag $P$ to the point where the two lines meet.
   
3. Solving with a graph: On a graph, $y = 3x - 2$ meets $y = 7$ at $(3, 7)$. What is the solution of $3x - 2 = 7$?
   3
   x = 3. Check: 3 × 3 − 2 = 7.
4. Find the mistake: Kai solves an equation with a graph. Click the line where his working goes wrong.
   line 4
   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.
5. Which equation?: The lines $y = x + 5$ and $y = 2x + 1$ meet at $(4, 9)$. Which equation has the solution $x = 4$?
   $x + 5 = 2x + 1$
   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.
6. Subtraction: Solve $3x - 4 = 8$: drag $P$ to where the lines meet.
   
   They meet at (4, 8), so x = 4. Check: 3 × 4 − 4 = 8.
7. A negative answer: Solve $2x + 3 = 1$: drag $P$ to where the lines meet.
   
   They meet at (−1, 1), so x = −1. Check: 2 × (−1) + 3 = 1.
8. $x$ on both sides: Solve $x + 5 = 3x + 1$: drag $P$ to where the lines meet.
   
   They meet at (2, 7), so x = 2. Check: 2 + 5 = 7 and 3 × 2 + 1 = 7.
9. Between the lines: Solve $2x + 1 = 6$: drag $P$ to where the lines meet.
   
   They meet at (2.5, 6), so x = 2.5. Check: 2 × 2.5 + 1 = 6.
10. Never meeting: Where do $y = 2x + 1$ and $y = 2x + 4$ meet?
    Nowhere: they are parallel, so $2x + 1 = 2x + 4$ has no solution
    Both have m = 2, so they are parallel and never meet: 2x + 1 is always 3 less than 2x + 4. The equation has no solution.
11. From a graph: A graph shows $y = 4x - 3$ meeting $y = 9$ at $(3, 9)$. Solve $4x - 3 = 9$.
    3
    x = 3. Check: 4 × 3 − 3 = 9.
12. Both sides sloping: $y = 2x - 1$ and $y = -x + 5$ meet at $(2, 3)$. What value of $x$ makes $2x - 1 = -x + 5$ true?
    2
    x = 2. Check: 2 × 2 − 1 = 3 and −2 + 5 = 3.
13. Which equation?: The lines $y = 5$ and $y = x + 2$ meet. Which equation does the meeting point solve?
    $x + 2 = 5$
    Set the two rules equal: x + 2 = 5. They meet at (3, 5), so x = 3.
14. Review: Review: where does $y = 3x - 4$ cross the $y$-axis? Give the value of $y$.
    -4
    −4: when x = 0, y = 3 × 0 − 4 = −4.
15. an equation with a chosen answer: Set $m$ and $c$ so that the equation $2x + 1 = mx + c$ has the solution $x = 3$.
    
    Your line meets y = 2x + 1 at (3, 7), so 2x + 1 = mx + c is true when x = 3.

Solutions

One possible solution for each task. Other correct programs are fine.

change the equation › Subtraction

Open task: any program that does what the step asks.

change the equation › A negative answer

Open task: any program that does what the step asks.

change the equation › $x$ on both sides

Open task: any program that does what the step asks.

change the equation › Between the lines

Open task: any program that does what the step asks.

change the equation › Never meeting

Open task: any program that does what the step asks.

an equation with a chosen answer

Open task: any program that does what the step asks.

Stretch and support

Stretch: Solve 3x − 2 = x + 5 with a graph: the answer is not a whole number. How accurate can a graph be? (x = 3.5)

Support: Give squared paper and have students draw both lines from tables of values (x = 0 to 4) before using the board.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Solving equations with graphs".

The idea: Solving linear equations algebraically: I can Students solve linear equations of the form ax + b = cx + d algebraically, including equations requiring expansion, and verify by substitution.

What understanding looks like:
- Solves a linear equation with unknowns on both sides correctly.
- Verifies the solution by substitution.
- Does not leave the unknown on both sides after the first step.

Mistakes students often make here:
- A student gives the y of the meeting point as the solution.
- A student cannot find where the lines meet on the negative task.
- A student says parallel lines meet somewhere far away.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Graphing linear relations including inequalities), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Solving equations with graphs".

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.

What each step is for

• where they meet: Committing to which number the meeting point gives (x or y) is the whole lesson: students who read y here will read y on every graph.
• find the meeting point: Dragging P onto both lines at once makes 'where they meet' physical: there is exactly one such point, and its x makes both sides equal.
• reading the solution: The solution is the x of the meeting point, never its y; and a graph's answer is still checked by substitution.
• change the equation: 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).
• read the graph: Reading the solution as the x of a meeting point, and naming the equation it solves, in mixed forms; the review keeps c fresh.
• an equation with a chosen answer: Building a second line through a chosen meeting point reverses the lesson: from the solution to the equation.

Where it 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.