Mathematics · Topic 1.2
Algebra
Algebra uses letters (variables) to represent unknown numbers. In this topic you will write expressions, solve simple equations, and explore patterns in sequences.
What You'll Learn
- Understand the difference between variables, expressions, equations, and formulas
- Simplify algebraic expressions by collecting like terms
- Solve one-step and two-step equations using inverse operations
- Recognise and continue number sequences
- Find the nth term rule for a linear sequence
- Substitute values into expressions and formulas
IB Assessment Focus
Criterion A: Select and apply the correct inverse operations to solve equations.
Criterion B: Investigate number patterns and describe them as general rules.
Criterion C: Use correct algebraic notation and explain your reasoning.
Criterion D: Apply algebra to solve real-world problems.
Key Vocabulary
| Term | Definition | Example |
|---|---|---|
| Variable | A letter that represents an unknown value | x, y, n |
| Expression | Numbers and variables combined with operations (no equals sign) | 3x + 5 |
| Equation | A statement that two expressions are equal | 3x + 5 = 14 |
| Coefficient | The number multiplied by a variable | In 3x, the coefficient is 3 |
| Constant | A fixed number with no variable attached | In 3x + 5, the constant is 5 |
| Term | Each part of an expression separated by + or − | 3x + 5 has two terms: 3x and 5 |
| Like terms | Terms with the same variable and power | 3x and 7x are like terms |
| Inverse operation | The operation that undoes another | + undoes −; × undoes ÷ |
| Sequence | An ordered list of numbers following a rule | 2, 5, 8, 11… |
Mathematics · Topic 1.2
Variables & Expressions
An expression is like a mathematical phrase. Learning to write, read, and simplify expressions is the first step in algebra.
Writing Algebraic Expressions
In algebra, we use shorthand conventions to write expressions clearly:
- Multiplication: We write 3 × x as 3x (no × sign needed)
- Division: We write x ÷ 4 as x/4 (fraction form)
- 1 × x: We just write x (not 1x)
- x × x: We write x² (x squared)
Translating Words to Algebra
| Words | Algebraic Expression |
|---|---|
| 5 more than a number | x + 5 |
| 3 less than a number | x − 3 |
| Double a number | 2x |
| A number divided by 4 | x/4 |
| Triple a number, then add 7 | 3x + 7 |
| The square of a number | x² |
Collecting Like Terms
To simplify an expression, combine terms that have the same variable (like terms). You cannot combine unlike terms.
Like terms — can combine:
- 3x + 5x = 8x
- 7y − 2y = 5y
- 4a + 3b + 2a + b = 6a + 4b
Unlike terms — cannot combine:
- 3x + 5y (different variables — leave as is)
- 2x + 3x² (different powers — leave as is)
Common Mistake: Students sometimes write 3x + 5 = 8x. This is wrong because 5 is a constant, not a term with x. You cannot combine 3x and 5. The expression 3x + 5 is already fully simplified.
Mathematics · Topic 1.2
Solving Equations
Solving an equation means finding the value of the unknown variable that makes the equation true. We use inverse operations to "undo" what has been done to the variable.
The Balance Rule
Golden Rule
Whatever you do to one side of an equation, you MUST do to the other side.
Think of an equation like a balance scale. Both sides must stay equal, so any operation you apply must be done to both sides.
One-Step Equations
Example 1: Solve x + 7 = 12
The variable has 7 added to it inverse of + is −
Subtract 7 from both sides: x = 12 − 7 = 5
Check: 5 + 7 = 12 ✓
Example 2: Solve 4x = 20
The variable is multiplied by 4 inverse of × is ÷
Divide both sides by 4: x = 20 ÷ 4 = 5
Check: 4 × 5 = 20 ✓
Two-Step Equations
For two-step equations, always undo addition/subtraction first, then undo multiplication/division.
Example: Solve 2x + 3 = 11
Undo addition: Subtract 3 from both sides → 2x = 8
Undo multiplication: Divide both sides by 2 → x = 4
Verify: 2(4) + 3 = 8 + 3 = 11 ✓
Example: Solve 5x − 7 = 18
Undo subtraction: Add 7 to both sides → 5x = 25
Undo multiplication: Divide both sides by 5 → x = 5
Verify: 5(5) − 7 = 25 − 7 = 18 ✓
Critical Rule: Always verify your answer by substituting it back into the original equation. If both sides are equal, your answer is correct.
Mathematics · Topic 1.2
Sequences & the nth Term
A sequence is an ordered list of numbers following a rule. Finding the nth term rule lets you predict any term in the sequence without listing every number.
Types of Sequences
| Type | Description | Example |
|---|---|---|
| Linear | Increases (or decreases) by the same amount each time | 3, 7, 11, 15… (add 4) |
| Square numbers | 1², 2², 3², 4²… | 1, 4, 9, 16, 25… |
| Triangular numbers | Sum of consecutive integers | 1, 3, 6, 10, 15… |
| Fibonacci-type | Each term = sum of the two before it | 1, 1, 2, 3, 5, 8… |
Finding the Common Difference
For a linear sequence, the common difference (d) is the amount each term increases by.
Example: 5, 9, 13, 17…
9 − 5 = 4, 13 − 9 = 4, 17 − 13 = 4
Common difference d = 4
Finding the nth Term Rule
- Find the common difference (d)
- The nth term starts with dn (common difference × n)
- Work out the adjustment: first term − d
- Rule: nth term = dn + (first term − d)
Example: Find the nth term of 3, 7, 11, 15…
Common difference d = 4, so start with 4n
Adjustment: 3 − 4 = −1
nth term = 4n − 1
Check: n=1 → 4(1) − 1 = 3 ✓ n=2 → 4(2) − 1 = 7 ✓
Example: Find the nth term of 5, 9, 13, 17…
Common difference d = 4, so start with 4n
Adjustment: 5 − 4 = 1
nth term = 4n + 1
Check: n=1 → 4(1) + 1 = 5 ✓ n=2 → 4(2) + 1 = 9 ✓
Always check your nth term rule by substituting n = 1, n = 2, and n = 3 to make sure it gives the correct terms.
Mathematics · Topic 1.2
Substitution
Substitution means replacing variables with given numbers and calculating the result. It is used to evaluate expressions and verify equation solutions.
How to Substitute
- Write out the expression
- Replace each variable with the given value
- Use brackets around negative numbers
- Calculate, following the order of operations (BIDMAS/BODMAS)
Examples
Example 1: Evaluate 3x + 7 when x = 4.
= 3(4) + 7 = 12 + 7 = 19
Example 2: Evaluate 2a² − 5 when a = 3.
= 2(3)² − 5 = 2(9) − 5 = 18 − 5 = 13
Example 3: Evaluate 4x − 2y when x = 5 and y = 3.
= 4(5) − 2(3) = 20 − 6 = 14
Order of Operations (BIDMAS)
| Letter | Operation | Priority |
|---|---|---|
| B | Brackets | First |
| I | Indices (powers) | Second |
| D | Division | Third (left to right) |
| M | Multiplication | Third (left to right) |
| A | Addition | Fourth (left to right) |
| S | Subtraction | Fourth (left to right) |
Common Mistake: When substituting a = 3 into a², the answer is 3² = 9, NOT 3 × 2 = 6. The ² means "squared" (multiply by itself), not "times 2."
Mathematics · Topic 1.2
Worked Examples
These examples show the clear, step-by-step working expected in assessments.
EXAMPLE 1Simplify 4x + 3y + 2x − y.
+
Full Solution
Collect like terms: group the x terms and the y terms.
4x + 2x = 6x and 3y − y = 2y
Answer: 6x + 2y
4x + 2x = 6x and 3y − y = 2y
Answer: 6x + 2y
EXAMPLE 2Solve 3x + 4 = 19. Show your working.
+
Full Solution
Step 1: Subtract 4 from both sides: 3x = 15.
Step 2: Divide both sides by 3: x = 5.
Verify: 3(5) + 4 = 15 + 4 = 19 ✓
Step 2: Divide both sides by 3: x = 5.
Verify: 3(5) + 4 = 15 + 4 = 19 ✓
EXAMPLE 3Find the nth term of the sequence 6, 11, 16, 21…
+
Full Solution
Step 1: Common difference d = 11 − 6 = 5.
Step 2: Start with 5n.
Step 3: Adjustment: 6 − 5 = 1.
Step 4: nth term = 5n + 1.
Check: n=1: 5(1)+1 = 6 ✓ n=2: 5(2)+1 = 11 ✓ n=3: 5(3)+1 = 16 ✓
Step 2: Start with 5n.
Step 3: Adjustment: 6 − 5 = 1.
Step 4: nth term = 5n + 1.
Check: n=1: 5(1)+1 = 6 ✓ n=2: 5(2)+1 = 11 ✓ n=3: 5(3)+1 = 16 ✓
EXAMPLE 4Evaluate 2x² + 3x − 1 when x = 4.
+
Full Solution
Replace x with 4:
= 2(4)² + 3(4) − 1
= 2(16) + 12 − 1
= 32 + 12 − 1
= 43
= 2(4)² + 3(4) − 1
= 2(16) + 12 − 1
= 32 + 12 − 1
= 43
EXAMPLE 5A taxi charges $3 fixed fee plus $2 per kilometre. Write an expression for the cost of a journey of k kilometres. Find the cost of a 7 km ride.
+
Full Solution
Expression: Cost = 2k + 3 (dollars).
For k = 7: Cost = 2(7) + 3 = 14 + 3 = $17.
For k = 7: Cost = 2(7) + 3 = 14 + 3 = $17.
EXAMPLE 6The 20th term of a sequence with nth term = 3n − 2. What is it?
+
Full Solution
Substitute n = 20 into the rule:
3(20) − 2 = 60 − 2 = 58
3(20) − 2 = 60 − 2 = 58
Mathematics · Topic 1.2
Practice Q&A
Attempt each question before revealing the model answer.
SIMPLIFYSimplify 7a + 3b − 2a + 5b.
+
Model Answer
Collect like terms: 7a − 2a = 5a and 3b + 5b = 8b. Answer: 5a + 8b.
SOLVESolve the equation 4x − 1 = 11.
+
Model Answer
Add 1 to both sides: 4x = 12. Divide both sides by 4: x = 3. Check: 4(3) − 1 = 12 − 1 = 11 ✓
SOLVESolve the equation x/3 + 5 = 9.
+
Model Answer
Subtract 5 from both sides: x/3 = 4. Multiply both sides by 3: x = 12. Check: 12/3 + 5 = 4 + 5 = 9 ✓
DESCRIBEA sequence begins 5, 9, 13, 17… Describe the pattern and find the 10th term.
+
Model Answer
The common difference is 4 — each term increases by 4. The nth term = 4n + 1. The 10th term = 4(10) + 1 = 41.
EVALUATEIf a = 3 and b = −2, evaluate 2a − 3b.
+
Model Answer
= 2(3) − 3(−2) = 6 − (−6) = 6 + 6 = 12.
WRITEWrite an expression for: "I think of a number, multiply it by 5, then subtract 3."
+
Model Answer
Let the number be n. The expression is 5n − 3.
FINDThe nth term of a sequence is 2n + 5. Find the first four terms.
+
Model Answer
n=1: 2(1)+5 = 7. n=2: 2(2)+5 = 9. n=3: 2(3)+5 = 11. n=4: 2(4)+5 = 13.
First four terms: 7, 9, 11, 13.
First four terms: 7, 9, 11, 13.
SOLVESolve 6x + 2 = 3x + 14.
+
Model Answer
Subtract 3x from both sides: 3x + 2 = 14. Subtract 2: 3x = 12. Divide by 3: x = 4. Check: 6(4)+2 = 26 and 3(4)+14 = 26 ✓
Mathematics · Review
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.
What is a variable?
A letter that represents an unknown value, e.g., x, y, or n.
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What is the difference between an expression and an equation?
An expression has no equals sign (e.g., 3x + 5). An equation states two things are equal (e.g., 3x + 5 = 14).
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In the expression 7x − 3, what is the coefficient and the constant?
Coefficient = 7 (the number before x). Constant = −3.
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What are like terms?
Terms with the same variable and power. Example: 3x and 5x are like terms; 3x and 3y are not.
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What is the inverse of addition?
Subtraction. And the inverse of multiplication is division.
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The golden rule of equations?
Whatever you do to one side, you MUST do to the other side. This keeps the equation balanced.
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Solve x + 9 = 15.
x = 15 − 9 = 6.
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Solve 3x = 21.
x = 21 ÷ 3 = 7.
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How do you find the common difference in a sequence?
Subtract any term from the next term. E.g., 3, 7, 11 → 7 − 3 = 4.
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Find the nth term of 2, 5, 8, 11…
d = 3. nth term = 3n − 1. Check: 3(1) − 1 = 2 ✓
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What does BIDMAS stand for?
Brackets, Indices, Division, Multiplication, Addition, Subtraction — the order of operations.
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Evaluate 5x − 2 when x = 3.
5(3) − 2 = 15 − 2 = 13.
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Simplify 3x + 2y + 5x − y.
8x + y (combine like terms: 3x+5x = 8x, 2y−y = y).
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What is substitution?
Replacing a variable with a given number and calculating the result.
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Write "triple a number then add 4" in algebra.
3n + 4 (where n is the number).
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Mathematics · Practice Test
Practice Test — 20 Questions
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