Equations — Solve and Verify by Substitution

Before you begin

Key Words

Tap each card to flip it and read the full definition with examples. These six words are the building blocks of this topic.

Substitution

Replacing a letter with a number

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Definition

Replacing a pronumeral with a given numerical value to evaluate an expression or check an equation.

Example

If x = 3, substitute into 2x + 1:
2(3) + 1 = 7

The key step

Write brackets around the number when replacing: 2(3) + 1

Used for

Verifying solutions and evaluating formulas

Remember

Always follow order of operations after substituting

Verify

Prove that a solution is correct

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Definition

To confirm a solution is correct by substituting it back into the original equation and checking both sides are equal.

How to verify

Substitute → simplify LHS → simplify RHS → check LHS = RHS

Also called

"Checking the solution" or "proof by substitution"

If LHS = RHS

✅ The solution IS correct

If LHS ≠ RHS

❌ The solution is NOT correct — check your working

Formula

A rule written using pronumerals

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Definition

A mathematical rule that shows the relationship between quantities, written using pronumerals.

Examples

A = lw (area)
d = st (distance)
P = 2(l+w)

Not a formula

3x + 5 (expression)
3x + 5 = 11 (equation)

Key feature

Has a subject (pronumeral by itself) on the left

Used in

Science, measurement, finance, geometry

LHS & RHS

Left-hand side & right-hand side

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Definition

The expression on the left of the = sign (LHS) and the expression on the right (RHS).

Example

In 3x − 1 = 8:
LHS = 3x − 1
RHS = 8

To verify

Evaluate LHS and RHS separately, then check they're equal

Verified when

LHS = RHS ✅

Tip

Always write "LHS =" and "RHS =" in your working

Evaluate

Find the numerical value

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Definition

To calculate the numerical value of an expression after substituting values for the pronumerals.

Example

Evaluate 4x − 3 when x = 5:
4(5) − 3 = 17

Steps

1. Substitute
2. Apply BODMAS
3. Simplify

Common error

Forgetting BODMAS — always multiply/divide before add/subtract

Also means

"Work out the value of"

Subject of a Formula

The pronumeral by itself on one side

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Definition

The pronumeral that is expressed in terms of all other quantities — it appears alone on one side of the formula.

Example

In A = lw,
A is the subject

Example

In d = st,
d is the subject

Why it matters

When the subject is known, substitute all other values to evaluate

Note

Sometimes we need to solve for a different pronumeral

✅ Key idea: Substitution is the bridge between algebra and arithmetic. Once you replace every pronumeral with a number, you're left with a calculation you can solve using BODMAS.

Core concept

Solving vs Verifying

These two skills go hand in hand — solving finds the answer, verifying proves it's right. Make sure you understand both.

🔢 Solving

Starting with an equation, use inverse operations to find the value of the unknown.

EXAMPLE

2x + 3 = 11

→ subtract 3: 2x = 8

→ divide by 2:

x = 4

✅ Verifying

Starting with a proposed answer, substitute it in and check LHS = RHS.

EXAMPLE — verify x = 4

LHS = 2(4) + 3 = 11

RHS = 11

LHS = RHS ✅

How to verify a solution — 4 steps

Substitute

Replace the pronumeral with the given value on both sides

Evaluate LHS

Simplify the left-hand side using BODMAS

Evaluate RHS

Simplify the right-hand side

Compare

LHS = RHS? ✅ Verified. LHS ≠ RHS? ❌ Not a solution.

Common formulas you'll use

These formulas appear frequently in Stage 4. Learn to recognise them and what each pronumeral represents.

Formula	Name	What the pronumerals mean
A = lw	Area of a rectangle	A = area, l = length, w = width
P = 2(l + w)	Perimeter of a rectangle	P = perimeter, l = length, w = width
A = 12bh	Area of a triangle	A = area, b = base, h = perpendicular height
d = st	Distance formula	d = distance, s = speed, t = time
C = 2πr	Circumference of a circle	C = circumference, r = radius, π ≈ 3.14

⚠️ BODMAS reminder: After substituting, always apply operations in the correct order — Brackets, Orders (powers), Division, Multiplication, Addition, Subtraction. A very common mistake is adding before multiplying.

Interactive

Try It Live

Type a value for the pronumeral and watch the substitution happen step by step. Try different values — including incorrect ones — to see what happens.

2x + 3 = 11

Let x =

🔁 The Substitution Process

Step 1 Write brackets around the value replacing the pronumeral. This keeps your working clear.

Step 2 Simplify both sides separately using BODMAS.

Step 3 Compare LHS and RHS. If they're equal, the value is a solution.

Step 4 For formulas: substitute all known values, then evaluate the subject.

I Do

Worked Examples

Work through each example step by step. Click Show Next Step to reveal one step at a time.

Example 1 — Verify a correct solution

Foundation

Verify that x = 5 is the solution of 3x − 4 = 11

→

The equation and proposed solution:

3x − 4 = 11, x = 5

We will substitute x = 5 into the left-hand side and check if it equals 11.

Evaluate the LHS — substitute x = 5

LHS = 3(5) − 4

Replace x with 5. Write brackets around the 5.

Apply BODMAS — multiply first

LHS = 15 − 4 = 11

State the RHS

RHS = 11

✓

Compare

LHS = RHS = 11 ✅

Since LHS = RHS, x = 5 is verified as the solution.

Example 2 — Show a value is NOT a solution

Foundation

Show that x = 3 is NOT a solution of 4x + 1 = 17

→

Substitute x = 3 into the LHS

LHS = 4(3) + 1

Even though we suspect it's wrong, we still follow the full process.

Multiply first (BODMAS)

LHS = 12 + 1 = 13

State the RHS

RHS = 17

✗

Compare

LHS = 13 ≠ RHS = 17 ❌

Since LHS ≠ RHS, x = 3 is NOT a solution. (The correct solution is x = 4.)

Example 3 — Substitute into a formula

Core

A = lw Find A when l = 8 and w = 5

→

Write the formula

A = lw

We know l = 8 and w = 5. We need to find A.

Substitute the known values

A = (8)(5)

Replace l with 8 and w with 5. Use brackets.

Evaluate

A = 40

✓

Write the answer with units

The area is 40 square units.

Always include units in your final answer for a formula problem.

Example 4 — Use a formula to find an unknown

Core

d = st Find t when d = 120 and s = 40

→

Write the formula

d = st

d is the subject, but we need to find t. We'll substitute what we know and then solve.

Substitute d = 120 and s = 40

120 = 40t

Solve for t — divide both sides by 40

t = 120 ÷ 40

Evaluate

t = 3

✓

Verify: substitute t = 3, s = 40 back into d = st

d = 40 × 3 = 120 ✅

The journey takes 3 hours.

Example 5 — Formula with a fraction

Extension

A = 12bh Find A when b = 9 and h = 6

→

Write the formula

A = 12bh

This is the area of a triangle. b = base, h = perpendicular height.

Substitute b = 9 and h = 6

A = 12 × (9) × (6)

Multiply — work left to right

A = 12 × 54

9 × 6 = 54. Now halve it.

Evaluate

A = 27

The area of the triangle is 27 square units.

Spot the mistake

Error Detective

Each solution below has exactly ONE mistake. Find which line is wrong, then reveal the explanation.

🔍

Detective Case 1 — Verify x = 4 in the equation 3x + 2 = 14

Line 1: LHS = 3x + 2, substitute x = 4

Line 2: LHS = 3 + 2 × 4

Line 3: LHS = 3 + 8 = 11

Line 4: RHS = 14

Line 5: LHS ≠ RHS, so x = 4 is not a solution ✗

I think the error is on line:

🔍 The error is on Line 2.

When substituting x = 4 into 3x + 2, the x must be placed inside brackets: 3(4) + 2. Writing it as 3 + 2 × 4 treats 3 as a separate constant — but 3x means 3 multiplied by x.

Correct Line 2: LHS = 3(4) + 2 = 12 + 2 = 14

LHS = RHS = 14 ✅ → x = 4 IS a solution.

Common mistake: separating the coefficient from the pronumeral when substituting. Always write the coefficient immediately before the bracket: 3(4).

🔍

Detective Case 2 — Evaluate P = 2(l + w) when l = 7 and w = 3

Line 1: P = 2(l + w), l = 7, w = 3

Line 2: P = 2(7 + 3)

Line 3: P = 2 × 7 + 3

Line 4: P = 14 + 3 = 17

I think the error is on line:

🔍 The error is on Line 3.

After substituting, we have P = 2(7 + 3). BODMAS says we must evaluate the bracket first: 7 + 3 = 10. Then multiply by 2. Line 3 expanded the bracket incorrectly — it only multiplied the 7, not the (7 + 3).

Correct: P = 2(7 + 3) = 2(10) = 20

Common mistake: distributing the multiplication before evaluating what's inside the bracket. Always evaluate brackets first.

Applying formulas

Formula Problems

Use the 5-step method: identify the formula → write it → substitute → evaluate → write a sentence answer.

Problem 1 — Distance, speed and time

A car travels at a speed of 60 km/h for 2.5 hours. Use the formula d = st to find the distance travelled.

1 Identify the formula and the known values.

Formula: d = st Known: s = , t =

2 Substitute the values into the formula.

d = ×

3 Evaluate and write a sentence answer.

The car travelled km.

✅ Full Solution

Formula: d = st

Known values: s = 60, t = 2.5

Substitute: d = 60 × 2.5

Evaluate: d = 150

Answer: The car travelled 150 km.

Problem 2 — Verify a formula result

A student claims the area of a rectangle with length 12 cm and width 4 cm is 48 cm². Use A = lw to verify whether this is correct.

1 Write the formula and substitute.

A = lw = () × ()

2 Evaluate. Does it match the student's claim?

A = The student's claim of 48 cm² is .

✅ Full Solution

Formula: A = lw

Substitute: A = (12)(4)

Evaluate: A = 48

Answer: A = 48 cm². The student's claim is correct ✅

Problem 3 — Find an unknown from a formula

The perimeter of a rectangle is 36 cm and the length is 11 cm. Use P = 2(l + w) to find the width.

1 Substitute what you know.

36 = 2( + w)

2 Solve the equation for w. (Two steps needed.)

w = cm

3 Verify by substituting l and w back into P = 2(l + w).

P = 2( + ) = ✓

✅ Full Solution

Substitute: 36 = 2(11 + w)

Divide both sides by 2: 18 = 11 + w

Subtract 11: w = 7

Verify: P = 2(11 + 7) = 2(18) = 36 ✅

Answer: The width is 7 cm.

You Do

Practice Questions

Choose your level and work through the questions. Read the feedback carefully — it explains the reasoning, not just the answer.

Foundation: Substituting values and verifying solutions. Focus on the LHS / RHS method.

Q1. Substitute x = 3 into 2x + 4. What is the result?

💡 Need a hint?

Replace x with 3: 2(3) + 4. Then use BODMAS — multiply first.

Q2. Is x = 6 a solution of x + 5 = 11?

💡 Need a hint?

Substitute x = 6 into the LHS: 6 + 5 = ? Compare to RHS = 11.

Q3. Is x = 2 a solution of 5x − 3 = 8?

💡 Need a hint?

LHS = 5(2) − 3. Use BODMAS: multiply first → 10 − 3 = 7. Compare to RHS = 8.

Q4. Use A = lw to find A when l = 6 and w = 4.

💡 Need a hint?

lw means l multiplied by w. Substitute: A = (6)(4).

Core: Verifying two-step equations and substituting into real formulas.

Q1. Verify whether x = 4 is the solution of 3x − 2 = 10.

💡 Need a hint?

LHS = 3(4) − 2. BODMAS: multiply first → 12 − 2 = 10. Then compare to RHS.

Q2. Using d = st, find t when d = 200 and s = 50.

💡 Need a hint?

Substitute d = 200, s = 50 into d = st: 200 = 50t. Then divide both sides by 50.

Q3. The formula for the area of a triangle is A = 12bh. Find A when b = 10 and h = 8.

💡 Need a hint?

Substitute: A = ½ × (10) × (8). Multiply b × h first, then multiply by ½ (or divide by 2).

Q4. The perimeter of a rectangle is 30 cm. The length is 9 cm. Use P = 2(l + w) to find the width.

💡 Need a hint?

Substitute P = 30, l = 9: 30 = 2(9 + w). Divide by 2: 15 = 9 + w. Now solve for w.

Extension: Multi-step formula problems, verifying more complex equations, and reasoning about solutions.

Q1. Which value of x satisfies 4x − 7 = 2x + 3? Verify by substituting your answer.

💡 Need a hint?

Collect x terms on one side first: subtract 2x from both sides. Then solve the resulting equation and verify by substituting back.

Q2. The circumference of a circle is C = 2πr. A circle has radius 7 cm. Using π ≈ 3.14, which value is closest to C?

💡 Need a hint?

C = 2 × π × r. Substitute π ≈ 3.14 and r = 7. Multiply step by step: 2 × 3.14 first, then × 7.

Q3. A student says x = −2 is a solution of 3x + 10 = 4. Are they correct?

💡 Need a hint?

Substitute x = −2: LHS = 3(−2) + 10. Remember: 3 × (−2) = −6. Then −6 + 10 = ?

Q4. Using A = 12bh, a triangle has area 36 cm² and height 9 cm. What is the base?

💡 Need a hint?

Substitute A = 36, h = 9: 36 = ½ × b × 9 → 36 = 4.5b. Divide both sides by 4.5.

Solve & Verify by Substitution

Key Words

Solving vs Verifying

🔢 Solving

✅ Verifying

How to verify a solution — 4 steps

Common formulas you'll use

Try It Live

🔁 The Substitution Process

Worked Examples

Example 1 — Verify a correct solution

Example 2 — Show a value is NOT a solution

Example 3 — Substitute into a formula

Example 4 — Use a formula to find an unknown

Example 5 — Formula with a fraction

Error Detective

Detective Case 1 — Verify x = 4 in the equation 3x + 2 = 14

Detective Case 2 — Evaluate P = 2(l + w) when l = 7 and w = 3

Formula Problems

Problem 1 — Distance, speed and time

Problem 2 — Verify a formula result

Problem 3 — Find an unknown from a formula

Practice Questions