Solving harder equations

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

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• Worksheet Download Copy address for Prepare Print the student copy; the Teacher edition has the answers.

Students will be able to

• 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.

Timing

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

Before class

Do the working check yourself first. Students need The balance method (Year 7), Expanding brackets, and adding and multiplying negatives.

This is a hinge node: inequalities (next) use the same moves, and sign slips carry over. The common mistake is undoing a term with its own sign (taking 7 away to undo − 7); it is planted in Ben's working and again in Practise.

The working check marks each line against the one before, so students see the first step that went wrong; the last line must be x = …

Running the lesson

1. Predict: take a show of hands, then ask how students chose: trying values is fine, and leads to checking.
2. Run: ask what P's y-coordinate means (the value of both sides).
3. Investigate: model the worked example, saying why the smaller x term moves (so x stays positive).
4. Modify: for the no-solution task, ask what the graph would look like.
5. Make: collect students' equations and solve one together.

When students get stuck

Answers to the checks

1. when do they cost the same?: Which value of $x$ makes $3x + 7 = 5x - 9$ true?
   x = 8. Check: 3 × 8 + 7 = 31 and 5 × 8 − 9 = 31. Both sides are equal.
2. where both sides are equal: The two lines are $y = 3x + 7$ and $y = 5x - 9$. Drag $P$ to where they meet. Was your prediction right?
   
3. Solve, then check: Solve $3x + 7 = 5x - 9$. Write one step per line.
   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.
4. Find the mistake: Ben solves $2x + 5 = 4x - 7$. Click the line where his working goes wrong.
   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.
5. Check by substituting: Check $x = 8$ in $3x + 7 = 5x - 9$: what is the value of each side?
   31
   3 × 8 + 7 = 31 and 5 × 8 − 9 = 31: equal, so x = 8 is right.
6. A bracket: A bracket: solve $2(x + 3) = x + 10$. Expand first. One step per line.
   2(x + 3) = x + 10 → 2x + 6 = x + 10 → x + 6 = 10 → x = 4
   Expand: 2x + 6 = x + 10. Take x: x + 6 = 10. Take 6: x = 4. Check: 2(4 + 3) = 14 and 4 + 10 = 14.
7. A negative answer: A negative answer: solve $4x - 3 = 7x + 12$.
   -5
   Take 4x: −3 = 3x + 12. Take 12: −15 = 3x. So x = −5. Check: 4(−5) − 3 = −23 and 7(−5) + 12 = −23.
8. A fraction: A fraction: solve $\dfrac{x}{3} + 2 = 6$.
   12
   Take 2: x/3 = 4. Multiply by 3: x = 12. Check: 12 ÷ 3 + 2 = 6.
9. Brackets on both sides: Brackets on both sides: solve $3(x - 2) = 2(x + 1)$. One step per line.
   3(x - 2) = 2(x + 1) → 3x - 6 = 2x + 2 → x - 6 = 2 → x = 8
   Expand both: 3x − 6 = 2x + 2. Take 2x: x − 6 = 2. Add 6: x = 8. Check: 3(6) = 18 and 2(9) = 18.
10. No solution: Edge case: solve $2x + 3 = 2x + 5$. What happens?
    No value works: taking 2x away leaves 3 = 5, which is never true
    Taking 2x from both sides leaves 3 = 5: false for every x, so there is no solution. On a graph, the two lines are parallel and never meet.
11. x on both sides: Solve $5x - 4 = 2x + 11$. One step per line.
    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.
12. Negative x: Solve $6 - x = 2x - 9$.
    5
    Add x: 6 = 3x − 9. Add 9: 15 = 3x. x = 5. Check: 1 = 1.
13. Angles: Angles on a straight line are $3x$ and $x + 20$ degrees. Find $x$.
    40
    3x + x + 20 = 180, so 4x = 160 and x = 40. The angles are 120° and 60°.
14. Review: expanding: Review: expand $4(2x - 3)$.
    8x - 12
    4 × 2x = 8x and 4 × (−3) = −12.
15. build an equation: Set $m$ and $c$ so that $2x + 1 = mx + c$ has the solution $x = 3$, with $m$ more than 2.
    
    For example m = 4, c = −5: 2x + 1 = 4x − 5 gives x = 3. Any line through (3, 7) works.

Solutions

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

build an equation

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

Stretch and support

Stretch: Solve (x + 4)/2 = (x − 1)/3. (Multiply both sides by 6: 3x + 12 = 2x − 2, x = −14.)

Support: Use the Run graph to check answers: drag P to the meeting point.

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 harder equations".

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 takes 7 away to undo − 7 (Ben's mistake).
- A student leaves x on both sides after the first step.
- A student expands only the first term of a bracket.

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 harder equations".

Exit ticket

Solve 4(x − 1) = 2x + 6 and check.

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

What each step is for

• when do they cost the same?: Committing to a solution of 3x + 7 = 5x − 9 before solving shows whether they can reason about both sides; the graph then shows the solution as where the two sides are equal.
• where both sides are equal: Each side of the equation is a line; the solution is where they meet, because there both sides have the same value.
• same to both sides: Move the smaller x term (so the x term stays positive), then the numbers, then divide; substitute to check. Ben's mistake takes 7 away to undo − 7, instead of adding it.
• one change at a time: One change at a time: a bracket, a negative answer, a fraction, brackets on both sides, then the edge case of an equation with no solution.
• solve and check: Equations with x on both sides, negatives and a geometry context, mixed; the inverse-sign slip is planted again; the review keeps expanding fresh.
• build an equation: Choosing a second line that meets y = 2x + 1 at x = 3 builds an equation with that solution, reversing the lesson: from the answer to the equation.

Where it leads

Next: Inequalities (node 3), which solve the same way except when multiplying or dividing by a negative.

Where it leads: simultaneous equations (Year 9 and 10), and rearranging formulas with the subject in two places.