Lesson 3 ~30 min Unit 2 · Patterns & Algebra +90 XP

Substitution

What is $3x + 5$ actually worth? It depends on $x$! Learn to swap letters for numbers and calculate the value of any expression, one step at a time.

Today's hook: If $x = 4$, what is $2x + 3$? Try it in your head, then check if your method matches the one in this lesson.

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Think First

warm-up

If $x = 3$, calculate $2x + 7$. Show every step of your working.

Record your answer in your workbook.

The Big Idea

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Substitution means replacing a variable with its numerical value, then calculating the result. It's like opening a mystery box and finding out what number was hiding inside.

When we substitute $x = 4$ into $2x + 3$, we replace every $x$ with 4, then use order of operations (BODMAS) to calculate. The answer is called the value of the expression.

$2x + 3 \text{ when } x = 4 \rightarrow 2(4) + 3 = 11$

Replace every variable

If $x = 3$, then $2x + x = 2(3) + 3 = 9$. Every $x$ must be replaced.

Use brackets

Always write $2(4)$ not $24$. Brackets prevent confusion between digits.

Follow BODMAS

After substituting, use order of operations: Brackets, Orders, Divide, Multiply, Add, Subtract.

What You'll Master

objectives

Know

What substitution means
Why we use brackets when substituting
How BODMAS applies after substitution

Understand

That an expression has different values for different inputs
Why negative values need extra care
That substitution preserves the structure of the expression

Can Do

Substitute positive and negative values
Evaluate expressions with multiple variables
Handle powers and fractions when substituting

Words You Need

vocabulary

SubstituteTo replace a variable with a number. "Substitute $x = 3$" means write 3 wherever $x$ appears.

EvaluateTo calculate the numerical value of an expression after substitution.

ExpressionA maths phrase with variables and numbers. e.g. $2x + 3$. It has a value only after substitution.

ValueThe numerical result after evaluating an expression. e.g. When $x = 4$, the value of $2x + 3$ is 11.

BODMASOrder of operations: Brackets, Orders (powers), Division, Multiplication, Addition, Subtraction.

InputThe number you put into an expression. Also called the "substituted value".

Spot the Trap

heads-up

Wrong: If $x = 3$, then $2x = 23$

Right: $2x = 2(3) = 6$. Always use brackets: $2(3)$, never $23$.

Wrong: If $x = -2$, then $x^2 = -4$

Right: $x^2 = (-2)^2 = (-2) \times (-2) = 4$. A negative squared is positive!

The Substitution Method

+5 XP

Follow this three-step method every time. It prevents the most common mistakes and guarantees full working marks.

Step 1: Write the expression. Copy it exactly.
Step 2: Substitute. Replace variables with values in brackets.
Step 3: Evaluate. Use BODMAS to calculate.

$\text{Write} \rightarrow \text{Substitute} \rightarrow \text{Evaluate}$

Try It Now: Evaluate $4x + 1$ when $x = 3$. Use the three-step method. Answer: Step 1: $4x + 1$ → Step 2: $= 4(3) + 1$ → Step 3: $= 12 + 1 = 13$

Substituting Negative Numbers

+5 XP

When the value you substitute is negative, brackets are essential. $-x$ when $x = -3$ becomes $-(-3)$, which equals $+3$.

Always wrap negatives in brackets. $x^2$ when $x = -2$ is $(-2)^2 = 4$, NOT $-2^2 = -4$. The brackets tell the calculator (and your brain) to square the whole number including its sign.

$(-2)^2 = 4 \quad \text{but} \quad -2^2 = -4$

Expression	Substitute $x = -3$	Working	Answer
$2x + 1$	$2(-3) + 1$	$= -6 + 1$	$-5$
$x^2$	$(-3)^2$	$= 9$	$9$
$-x$	$-(-3)$	$= +3$	$3$
$5 - x$	$5 - (-3)$	$= 5 + 3$	$8$

Brackets are mandatory

$(-3)^2$ not $-3^2$. The brackets protect the negative sign.

$-(-n) = +n$

Two negatives make a positive. $-(-3) = +3$.

$5 - (-3) = 8$

Subtracting a negative = adding. $5 - (-3) = 5 + 3 = 8$.

Multiple Variables

+5 XP

Some expressions have more than one variable. You'll be given a value for each one. Substitute them all, then evaluate.

When evaluating $2a + 3b$ with $a = 4$ and $b = 2$, replace every $a$ with 4 and every $b$ with 2. Then follow BODMAS: $2(4) + 3(2) = 8 + 6 = 14$.

$2a + 3b \text{ when } a=4, b=2 \rightarrow 2(4) + 3(2) = 14$

Powers and Fractions

+5 XP

When expressions contain powers or fractions, the same rules apply, just be extra careful with the order of operations.

For $x^2$, the power is evaluated after substitution but before other operations. For fractions like $\frac{x}{2}$, the fraction bar acts as brackets, evaluate the numerator and denominator first.

$x^2 = (x)^2 \quad \frac{x+1}{2} = (x+1) \div 2$

$x^2$ needs brackets

$(-3)^2 = 9$ but $-3^2 = -9$. Always bracket the substituted value.

Fraction bar = brackets

$\frac{x+1}{2}$ means $(x+1) \div 2$. Calculate the top first.

Orders before multiply

In $2x^2$, calculate $x^2$ first, then multiply by 2. Not $(2x)^2$!

Watch Me Solve It · Worked example

Watch Me Solve It · Multi-step substitution

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PROBLEM

Evaluate $3x^2 - 2x + 4$ when $x = -2$.

1

Write the expression

$3x^2 - 2x + 4$

Always start by copying the expression exactly. This earns you a method mark.
2

Substitute $x = -2$ with brackets

$= 3(-2)^2 - 2(-2) + 4$

Every $x$ becomes $(-2)$. Notice $x^2$ becomes $(-2)^2$, not $-2^2$.
3

Evaluate powers first (Orders)

$= 3(4) - 2(-2) + 4$

$(-2)^2 = (-2) \times (-2) = 4$. A negative squared is positive!
4

Multiply next

$= 12 - (-4) + 4$

$3(4) = 12$ and $-2(-2) = +4$, so $-2(-2) = -(-4)$. Two negatives make positive!
5

Add and subtract left to right

$= 12 + 4 + 4 = 20$

$12 - (-4) = 12 + 4 = 16$, then $16 + 4 = 20$.

Answer $20$

Common Pitfalls

heads-up

Writing $-3^2$ instead of $(-3)^2$

$-3^2 = -9$ because the square only applies to 3. $(-3)^2 = 9$ because the square applies to the whole $-3$.

Fix: always write $(-3)^2$ when substituting negative values into squared terms.

Forgetting BODMAS after substitution

$2 + 3(4)$ is NOT $5(4) = 20$. You must multiply before adding.

Fix: after substituting, take a breath and identify the operations in BODMAS order.

$5 - (-3)$ becoming $5 - 3 = 2$

Subtracting a negative means adding. $5 - (-3) = 5 + 3 = 8$.

Fix: two negatives = positive. $-(-n) = +n$. Write out the double negative step.

Copy Into Your Books

The Substitution Method

Step 1: Write the expression
Step 2: Substitute (use brackets)
Step 3: Evaluate using BODMAS
Always show all working!

Negative Numbers

$(-n)^2 = n^2$ (always positive)
$-(-n) = +n$
$a - (-b) = a + b$
Always bracket negatives!

Multiple Variables

Replace every variable
Use different brackets if needed
$2a + 3b$ with $a=4, b=2$:
$= 2(4) + 3(2) = 8 + 6 = 14$

Powers & Fractions

$x^2$ with $x = -3$: $(-3)^2 = 9$
Fraction bar = brackets
$\frac{x+1}{2}$: do top first
Orders before multiply!

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Mixed

4 problems

Four problems to test your substitution skills. Work each one, then reveal the answer.

1 Evaluate $5x - 3$ when $x = 4$.

$= 5(4) - 3 = 20 - 3 =$ $17$17
2 Evaluate $x^2 + 2x$ when $x = -3$.

$= (-3)^2 + 2(-3) = 9 + (-6) = 9 - 6 =$ $3$. Remember: $(-3)^2 = 9$!3
3 Evaluate $2a + 3b$ when $a = 5$ and $b = -2$.

$= 2(5) + 3(-2) = 10 + (-6) = 10 - 6 =$ $4$4
4 Evaluate $\frac{x + 4}{2}$ when $x = 6$.

$= \frac{6 + 4}{2} = \frac{10}{2} =$ $5$. Do the top first!5

Complete in your workbook.

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Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 2 MARKS

Q6. Evaluate $4n + 7$ when $n = 3$.

Answer in your workbook.

Show model answer

$= 4(3) + 7$ (1 mark) $= 12 + 7 =$ $19$ (1 mark).

Apply Medium 3 MARKS

Q7. Evaluate $x^2 - 3x$ when $x = -2$.

Answer in your workbook.

Show model answer

$= (-2)^2 - 3(-2)$ (1 mark) $= 4 - (-6)$ (1 mark) $= 4 + 6 =$ $10$ (1 mark).

Apply Medium 4 MARKS

Q8. The formula for the area of a trapezium is $A = \frac{h(a + b)}{2}$. Find the area when $h = 6$, $a = 5$, and $b = 9$.

Answer in your workbook.

Show model answer

$A = \frac{6(5 + 9)}{2}$ (1 mark) $= \frac{6(14)}{2}$ (1 mark) $= \frac{84}{2}$ (1 mark) $=$ $42$ units$^2$ (1 mark).

Comprehensive Answers

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Extension Problems

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Formulas

Key formulas and rules.

Watch Out

Common mistakes to avoid.

Check

Always verify your answers.

Practice

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Interactive: Algebra Machine

Substitute numbers into algebraic expressions and see them evaluate step by step.

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Unit overview · Maths subject page · Unit quiz

Substitution

Printable Worksheets

Know

Understand

Can Do

The Substitution Method

Negative Numbers

Multiple Variables

Powers & Fractions

Quick Check

Show Your Working Model Answers

Extension Problems

Key Concept

Formulas

Watch Out

Check

Practice

Next

Interactive: Algebra Machine

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