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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.
1
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.

$2x+3$ expression x = 4 $2(4)+3$ substituted BODMAS: multiply first $8+3=11$ VALUE = 11
$2x + 3 \text{ when } x = 4 \rightarrow 2(4) + 3 = 11$
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.
2
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
3
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".
4
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!

5
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.

STEP 1: WRITE $2x + 3$ STEP 2: SUBSTITUTE $= 2(4) + 3$ STEP 3: EVALUATE $= 8 + 3 = 11$
$\text{Write} \rightarrow \text{Substitute} \rightarrow \text{Evaluate}$
EXAMPLE: Evaluate $3n - 2$ when $n = 5$ $n = 5$ STEP 1: WRITE $3n - 2$ STEP 2: SUBSTITUTE $= 3(5) - 2$ STEP 3: EVALUATE $= 15 - 2$ $= 13$ ANSWER = 13
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: $= 12 + 1 = 13$
6
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.

$-2^2$ NO BRACKETS $-4$ ✗ $(-2)^2$ WITH BRACKETS $4$ ✓ Brackets change everything! ALWAYS: $(-2)^2$
$(-2)^2 = 4 \quad \text{but} \quad -2^2 = -4$
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. $5 - (-3) = 5 + 3 = 8$.
7
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: $2(4) + 3(2) = 8 + 6 = 14$.

$2a + 3b$ a = 4, b = 2 $2(4)$ $+3(2)$ $= 8 + 6$ $= 14$ ANSWER = 14
$2a + 3b \text{ when } a=4, b=2 \rightarrow 2(4) + 3(2) = 14$
Evaluate $a - b$ when $a = 7, b = 3$ $= 7 - 3$ $= 4$ Evaluate $3x + 2y$ when $x = 2, y = 5$ $= 3(2) + 2(5)$ $= 16$ Evaluate $mn + n$ when $m = 3, n = 4$ $= (3)(4) + 4 = 12 + 4$ $= 16$
8
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.

$x^2 + 3$ when $x = 3$ $(3)^2 + 3$ $= 9 + 3$ $= 12$ $\frac{x+1}{2}$ when $x = 5$
$x^2 = (x)^2 \quad \frac{x+1}{2} = (x+1) \div 2$
$x^2$ 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
+15 XP per step
Q
PROBLEM
Evaluate $3x^2 - 2x + 4$ when $x = -2$.
  1. 1
    Write the expression
    $3x^2 - 2x + 4$
    Always start by copying the expression exactly. This earns you a method mark.
  2. 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. 3
    Evaluate powers first (Orders)
    $= 3(4) - 2(-2) + 4$
    $(-2)^2 = (-2) \times (-2) = 4$. A negative squared is positive!
  4. 4
    Multiply next
    $= 12 - (-4) + 4$
    $3(4) = 12$ and $-2(-2) = +4$, so $-2(-2) = -(-4)$. Two negatives make positive!
  5. 5
    Add and subtract left to right
    $= 12 + 4 + 4 = 20$
    $12 - (-4) = 12 + 4 = 16$, then $16 + 4 = 20$.
Answer $20$
9
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

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Brain Trainer · Mixed
4 problems

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

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

    $= 5(4) - 3 = 20 - 3 =$ $17$17

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

    $= (-3)^2 + 2(-3) = 9 + (-6) = 9 - 6 =$ $3$. Remember: $(-3)^2 = 9$!3

  3. 3 Evaluate $2a + 3b$ when $a = 5$ and $b = -2$.

    $= 2(5) + 3(-2) = 10 + (-6) = 10 - 6 =$ $4$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).

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Extension Problems

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Key formulas and rules.

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