Algebra begins with recognising and describing patterns. We use letters to represent numbers, helping us generalise and write rules for number relationships.
Instead of specific numbers, we can use letters such as x or n to describe patterns. Example: If the pattern adds 3 each time, the nth term is 3n.
Write the rule for the pattern: 5, 8, 11, 14, ...
Starts at 5, adds 3 each time → nᵗʰ term = 3n + 2
If x = 6, evaluate 2x + 5.
2(6) + 5 = 17
We simplify algebraic expressions by combining like terms, and expand brackets using distributive properties.
Combine like terms: 3x + 5x = 8x
Expand brackets: 2(x + 4) = 2x + 8
Simplify: 7a + 3a − 5.
10a − 5
Expand: 3(x + 2).
3x + 6
Equations show that two expressions are equal. To solve for a variable, we perform the same operation on both sides until the variable is isolated.
To solve 2x + 3 = 11: Subtract 3 → 2x = 8 → divide by 2 → x = 4.
Solve for x: 3x + 5 = 17.
3x = 12 → x = 4
Solve for y: 4y − 2 = 10.
4y = 12 → y = 3
Linear equations form straight lines when graphed. They have the form y = mx + c, where m is the slope and c is the y-intercept.
The line y = 2x + 1 crosses the y-axis at 1 and rises 2 units for every 1 unit across.
For y = 3x + 2, find y when x = 4.
y = 3(4) + 2 = 14
The graph passes through (0, 5) and (2, 9). Find the slope.
m = (9 − 5)/(2 − 0) = 2
Indices (powers) represent repeated multiplication. Substitution lets us evaluate algebraic expressions by replacing variables with given values.
a² × a³ = a⁵
(a³)² = a⁶
a⁰ = 1
Evaluate: a³ × a² when a = 2.
a⁵ = 2⁵ = 32
If x = 5, find the value of 2x² + 3x.
2(25) + 15 = 65