034i - Verifying Solutions by Substitution

Lesson 34 — Verifying Solutions by Substitution

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

Learning Intentions

• To verify the solution of a linear equation by substitution.
• To use verification to locate and correct errors in algebraic working.

Success Criteria

I can:

1. Substitute a proposed solution into both sides of an equation and compare.
2. State clearly whether a value is or is not the solution.
3. Find the step at which an error was made in someone’s working.
4. Explain why verification is a reliable check.

Warmup

(6 minutes — true or false, mini whiteboards)

Decide whether each statement is true, showing your substitution.

1. solves
2. solves
3. solves
4. solves

Answers: 1. True — . 2. True — . 3. True — . 4. False — .

Activities

Activity 1 — Explicit Instruction: the Verification Protocol (10 min)

Why verify? Solving is a chain of steps; a single slip anywhere gives a wrong answer that looks finished. Substitution tests the answer against the original problem, independently of how you got there.

The protocol — evaluate each side separately, then compare.

I do. Is the solution to ?

So is the solution.

Setting out to insist on. Label LHS and RHS. Evaluate each on its own line. Never write a chain that assumes the answer is correct — that is circular.

I do — a value that fails. Is the solution to ?

So is not the solution. (Solving properly: gives , so — not a natural number. A useful reminder that not every equation has a whole-number solution.)

We do: Verify each.

1. Is the solution to ?
2. Is the solution to ?
3. Is the solution to ?

(Answers: 1. Yes, . 2. No, . 3. Yes, .)

Activity 2 — Error Hunting (14 min)

The core activity. Students find not just that an answer is wrong, but where the error occurred.

I do. A student’s working:

Verify: LHS ; RHS . Not equal — so something is wrong.

Locate the error: step (1) to (2) should subtract , giving , not add it. The correct solution is .

Pairs task — find the error and give the correct solution:

Student A

Student B

Student C

Student D

Socratic scaffolding for Student D:

Prompt	Purpose
First, verify their answer.	LHS ; RHS . Not equal.
So an error exists. Where do you look first?	The first step that changes the equation.
What did they do from line 1 to line 2?	They added to the left instead of subtracting it from both sides.
What should line 2 be?	, i.e. .
Solve correctly.	.
Verify the correction.	LHS ; RHS ✓
Looking back	The error was a sign error in collecting terms — the most common kind.

Answers: A — failed to expand fully; should be , giving . B — subtracted instead of adding; should be , giving . C — divided only two of the three terms; should be , giving . D — added instead of subtracting; correct answer .

Activity 3 — Inquiry: Which Equation? (8 min)

Pairs. Reverses the usual direction.

Here are four equations:

1. Without solving fully, use substitution to decide which of these have as a solution.
2. Which have ? Which do not, and what is their solution?
3. Write two more equations with solution : one with brackets, one with variables on both sides.

Socratic scaffolding for Q3:

Prompt	Purpose
Understand: what are you building?	An equation you already know the answer to.
Start from the answer. What is true?	.
Now do the same thing to both sides.	Multiply by : .
Again.	Add : .
Check	✓
For a bracket version, work backwards from .	, then : .
For both sides	Start with ; add to each side: . Check: ✓
Looking back	Building an equation is just solving in reverse.

Answers: 1–2. (i) ✓; (ii) ✓; (iii) ✓; (iv) ✓ — all four have . 3. Answers vary; check by substitution.

Teacher note: all four working is deliberate — it lets students see that many different equations can share a solution, which is exactly why an equation must be checked rather than guessed from its appearance.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Verify whether solves . Show LHS and RHS separately.
2. Verify whether solves .
3. Find the error in this working and give the correct solution:

4. Reasoning. Explain why substituting into the original equation is a better check than re-reading your working.
5. Write an equation with solution that requires two steps to solve.

Answers: 1. LHS , RHS ✓ yes; 2. LHS , RHS ✓ yes; 3. They subtracted instead of adding; , so ; 4. Re-reading tends to repeat the same mistake, because you follow the same reasoning again. Substitution tests the answer independently, against the original problem; 5. Answers vary, e.g. or .

Common Misconceptions

Misconception	How to pre-empt it
Substituting into the simplified equation, so an expansion error survives.	State the rule: always check against the original.
Writing a circular check that assumes the answer is right.	Require LHS and RHS to be evaluated separately and only then compared.
Concluding “wrong answer” without locating the error.	Every error-hunt answer must name the step and the type of error.
Believing verification proves the method was correct.	Verification confirms the answer, not the reasoning — two errors can cancel out. Discuss briefly.
Assuming that if the check fails, the equation has no solution.	The equation is fine; the arithmetic was not. Re-solve.
Treating a non-whole-number solution as automatically an error.	It may be correct — Year 7 equations usually give whole numbers, but not always.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Which of these has as its solution: , , ?

Answer

First: ✓. Second: ✗. Third: ✓. So the first and third.

E2 (AMC Junior style). For what value of does solve ?

Answer

, so and .

E3 (Challenge). The equations and have the same solution. Find .

Answer

From the second: , so . Then , giving .

E4 (Challenge). A student claims solves . Verify the claim, and if it is wrong, find the correct solution.

Answer

LHS ; RHS ✓ — the claim is correct.

E5 (Challenge). Build three different equations, each with solution : one one-step, one two-step, and one with brackets. Verify each.

Answer

For example: ; ; . Each checks out at .

Homework

1. Verify whether each value is the solution, showing LHS and RHS separately: (a) for (b) for (c) for (d) for
2. Find the error in each and give the correct solution: (a) ; ; (b) ; ; (c) ; ; (d) ; ;
3. For what value of does solve ?
4. Write an equation with solution that (a) is one-step (b) is two-step (c) contains brackets.
5. The equations and share a solution. Find .
6. Reasoning. Two students both get for the same equation, but one made two errors that cancelled out. Does verification detect this? Explain what verification does and does not confirm.
7. Challenge. For what value of do the equations and have the same solution?

Answers: Q1 — (a) yes, (b) no, (c) yes, (d) yes, . Q2 — (a) added instead of subtracting; , (b) failed to expand fully; , (c) added instead of subtracting; , (d) divided only two terms; , . Q3 — . Q5 — so ; then and . Q6 — verification confirms the answer is correct, not that the method was. Marking usually requires valid working as well. Q7 — so ; then and .