032i - Solving Two-Step Equations

Lesson 32 — Solving Two-Step Equations

Strand: Algebra | Descriptor: AC9M7A03 | Duration: 45 minutes

Learning Intentions

• To solve linear equations requiring two inverse operations.
• To undo operations in the correct order.

Success Criteria

I can:

1. Identify the two operations applied to the variable, and their order.
2. Undo them in reverse order.
3. Solve equations of the form and .
4. Check my solution by substitution.

Warmup

(6 minutes — the function machine, mini whiteboards)

Draw a machine: input output.

1. If the input is , what is the output?
2. If the input is , what is the output?
3. If the output is , what was the input? How did you work it out?
4. Describe, in order, how you reversed the machine.

Answers: ; ; input . To reverse: subtract first, then divide by .

The key insight to name: reversing a machine means undoing the operations in reverse order — last on, first off.

Activities

Activity 1 — Explicit Instruction: Undo in Reverse Order (12 min)

The two-step protocol:

1. Identify what has been done to the variable, in order.
2. Undo those operations in reverse order.
3. Do the same to both sides at every step.
4. Check by substituting back.

I do. Solve .

What was done to ? Multiplied by , then was added. So undo the first.

Check: ✓

Why not divide first? Show the wrong route explicitly: dividing by gives — correct, but far messier. Removing the added term first keeps the numbers whole.

I do — subtraction version. Solve .

Check: ✓

I do — division version. Solve .

Check: ✓

We do: ; ; ; .

Activity 2 — Independent Practice (10 min)

You do: Solve, showing every step and a check.

(Answers: ; ; ; ; ; ; ; ; ; .)

Activity 3 — Inquiry: Variables on both Sides (10 min)

Extends the Lesson 31 balance puzzle into symbolic form.

I do. Solve .

Strategy: first gather the variable terms on one side, then proceed as a one- or two-step equation.

Check: left ; right ✓

Investigation task — pairs:

Solve each, then look for a pattern in your method.

Socratic scaffolding for Q3:

Prompt	Purpose
Understand: what is different from before?	The variable appears on both sides.
What is the goal?	Get all the terms on one side, all the numbers on the other.
Which side should the variables go to?	The one with more of them — here the left, avoiding negatives.
First move?	Subtract from both sides: .
What kind of equation is that now?	A two-step one, which you can already solve.
Carry it out	, so .
Check	Left: . Right: ✓
Generalise	Collecting the variables first reduces the problem to one already solved.

Answers: 1. ; 2. ; 3. ; 4. .

Checks for Understanding

(5 minutes — exit ticket)

1. Solve, showing working: (a) (b) (c) .
2. Solve .
3. Reasoning. To solve , which operation should you undo first, and why?
4. Check whether solves .
5. A student solves by dividing everything by first, getting , so . Find the error.

Answers: 1. (a) (b) (c) ; 2. , so ; 3. Undo the first — operations are reversed in the opposite order to how they were applied, and it keeps the numbers whole; 4. ✓ yes; 5. They divided only the and the , not the . Dividing correctly gives , so .

Common Misconceptions

Misconception	How to pre-empt it
Undoing in the wrong order (dividing before subtracting).	Anchor to the function machine: last on, first off. Say the order aloud before writing.
Dividing only some terms, as in exit ticket Q5.	Every term on both sides must be divided. Show the full line.
Sign errors with .	Model “add to both sides” explicitly rather than jumping to the answer.
Collecting variables onto the side with fewer of them, creating negatives.	Advise choosing the side with the larger coefficient — though either works.
Believing is not a valid answer.	Include deliberately and check it.
Skipping the check.	Keep the check compulsory. It catches nearly every slip.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). If , what is the value of ?

Answer

, so and .

E2 (AMC Junior style). Solve for .

Answer

E3 (Challenge). A number is multiplied by , then is subtracted, giving . What is the number?

Answer

E4 (Challenge). If , find the value of .

Answer

, so and .

E5 (Challenge). Two equations share the same solution: and . Find .

Answer

From the first, , so . Substituting: , giving .

Homework

1. Solve, showing working and a check: (a) (b) (c) (d) (e) (f) .
2. Solve: (a) (b) (c) (d) .
3. Solve (variables on both sides): (a) (b) (c) (d) .
4. Check whether the given value is the solution: (a) for (b) for (c) for .
5. Reasoning. Explain, using a function machine, why you undo addition before division when solving .
6. A number is multiplied by and then is added, giving . Write an equation and solve it.
7. Challenge. If , find and then find .
8. Challenge. Two equations have the same solution: and . Find .

Answers: Q1 — (a) (b) (c) (d) (e) (f) . Q2 — (a) (b) (c) (d) . Q3 — (a) (b) (c) (d) . Q4 — (a) yes, (b) no, (c) yes, . Q6 — , so . Q7 — , so and . Q8 — , so ; then and .