LESSON 13 / 100 | 13% COMPLETE | STAGE I — FOUNDATIONS

Stage I — Foundations

Substitution

The moment a variable receives a value, algebra becomes arithmetic — and the answer becomes real.

Section 01

What Is Substitution?

Substitution is the process of replacing a variable with a specific numerical value and then evaluating the resulting arithmetic expression. It is how algebraic formulas are put to work — every engineering calculation, every physics equation, every financial formula operates through substitution.

The Three-Step Process

Write the original expression

Replace each variable with its given value, placing the value in brackets

Evaluate the resulting arithmetic using BODMAS

Given: x = 3, y = −2

Expression: 2x² − 3y + 1

After swap: 2(3)² − 3(−2) + 1

Evaluate: 2(9) + 6 + 1 = 18 + 6 + 1 = 25

A formula is a machine. Substitution is the act of feeding values into it. The algebraic expression is the blueprint; substitution is the moment the blueprint becomes a built object with measurable dimensions.

Section 02

The Bracket Rule — Why It Matters

Always place substituted values in brackets before evaluating. This is not optional — it is the mechanism that prevents sign errors and order-of-operations mistakes, especially with negative values and powers.

Without Brackets — Dangerous

Substitute x = −3 into x²:

x² = −3² = −9 ✗ Wrong!

Without brackets, BODMAS applies the exponent before the negative sign.

With Brackets — Safe

Substitute x = −3 into x²:

x² = (−3)² = 9 ✓ Correct

Brackets force the negative into the base before the exponent applies.

The Golden Habit

Every time you substitute, write the value in round brackets first — even for positive integers. The habit takes one second and eliminates an entire class of errors that costs marks at every level of mathematics.

Section 03

Single-Variable Substitution

Worked Example A — Evaluate 4x² − 7x + 3 when x = −2

Write with brackets: 4(−2)² − 7(−2) + 3

Orders: (−2)² = 4 → 4(4) − 7(−2) + 3

Multiplication: 16 − (−14) + 3 = 16 + 14 + 3

Result: 33

Worked Example B — Evaluate (3x − 1)/(x + 2) when x = 5

Substitute: (3(5) − 1) / ((5) + 2)

Numerator: 15 − 1 = 14

Denominator: 5 + 2 = 7

Result: 14/7 = 2

Section 04

Multi-Variable Substitution

When expressions contain more than one variable, substitute all values simultaneously — write every replacement before evaluating any of it.

Worked Example C — Evaluate 3a² − 2ab + b² when a = 4, b = −1

Substitute all: 3(4)² − 2(4)(−1) + (−1)²

Orders: (4)² = 16, (−1)² = 1 → 3(16) − 2(4)(−1) + 1

Multiplication: 48 − (−8) + 1 = 48 + 8 + 1

Result: 57

Worked Example D — Evaluate √(p² + q²) when p = 3, q = 4

Substitute: √((3)² + (4)²)

Inside the root: 9 + 16 = 25

Result: √25 = 5 (Pythagorean triple!)

Section 05

Substitution into Real-World Formulas

Science and engineering express their laws as algebraic formulas. Substitution is the action that turns a formula into a specific result.

Area of a triangle	A = ½bh	b = base, h = height
Kinetic energy	KE = ½mv²	m = mass, v = velocity
Distance-speed-time	d = vt	v = speed, t = time
Celsius to Fahrenheit	F = (9/5)C + 32	C = degrees Celsius
Simple interest	I = PRT/100	P = principal, R = rate, T = time

Worked Example E — Convert 37°C to Fahrenheit

Formula: F = (9/5)C + 32

Substitute: F = (9/5)(37) + 32

Evaluate: (9 × 37)/5 + 32 = 333/5 + 32 = 66.6 + 32

Result: 98.6°F

Worked Example F — Find simple interest on R5 000 at 7% p.a. for 3 years

Formula: I = PRT/100

Substitute: I = (5000)(7)(3)/100

105 000 / 100 = R1 050

Section 06

Common Substitution Errors

Expression	Wrong	Correct	Error
x²	−9	9	Missing brackets on negative base
2x²	−18	18	Applying exponent to −3 without brackets: 2(−3)² = 2(9) = 18
−x	−3	3	−(−3) = +3, not −3
x + 2	32	−1	Writing x = −3 as 3, losing the negative sign entirely

The Double-Negative Trap

When the expression has a minus sign and the substituted value is negative, the two negatives combine to give positive. This surprises students every time:

Expression: −x | x = −5

Result: −(−5) = +5

The expression “the negative of x” becomes “the negative of negative five” — which is positive five.

Exercises

Practice Set

1. Evaluate 3x − 7 when x = 4

3(4) − 7 = 12 − 7 = 5

2. Evaluate x² + 2x − 5 when x = −3

(−3)² + 2(−3) − 5 = 9 − 6 − 5 = −2

3. Evaluate 2a − 3b + ab when a = 5, b = −2

2(5) − 3(−2) + (5)(−2) = 10 + 6 − 10 = 6

4. Evaluate (2p + 3)/(p − 1) when p = 4

Numerator: 2(4) + 3 = 11. Denominator: 4 − 1 = 3. Answer: 11/3 (or 3⅔)

5. The formula for the area of a trapezium is A = ½(a + b)h. Find A when a = 5, b = 9, h = 6.

A = ½(5 + 9)(6) = ½(14)(6) = ½(84) = 42 square units

6. Evaluate −x² + 3x − 1 when x = −2. Watch the signs carefully.

−(−2)² + 3(−2) − 1. Note: −(−2)² = −(4) = −4. So: −4 + (−6) − 1 = −4 − 6 − 1 = −11

7. A ball is thrown upward. Its height (in metres) after t seconds is given by h = 20t − 5t². Find the height at t = 3. Is the ball still rising or falling?

h = 20(3) − 5(3)² = 60 − 5(9) = 60 − 45 = 15 metres. At t = 2: h = 40 − 20 = 20 m. At t = 3: h = 15 m. Since height decreased from t=2 to t=3, the ball is falling.

8. Challenge: Find the value of k such that the expression 3x² + kx − 2 equals 8 when x = 2.

Substitute x = 2: 3(4) + k(2) − 2 = 8. 12 + 2k − 2 = 8. 10 + 2k = 8. 2k = −2. k = −1. Verify: 3(4) + (−1)(2) − 2 = 12 − 2 − 2 = 8 ✓

Summary

Lesson Checklist

You Can Now

Replace variables with given values using the bracket-first habit
Evaluate the resulting expression correctly using BODMAS
Handle negative substituted values, especially in powers and products
Substitute into expressions with multiple variables simultaneously
Apply substitution to real-world formulas from science and finance
Identify double-negative situations and resolve them correctly
Work backwards: find an unknown constant given a target output value

Up Next → Lesson 14

Writing Expressions from Words

Translate real-world situations and word problems into algebraic expressions — the reverse direction of substitution, and the gateway to solving equations.

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Lesson 14 - Writing Expressions from Words