Algebra · Ordinary Level
Algebra Basics
Ordinary Level · Class 1 of 8 · Tap NEXT to begin
Section 1 of 2
Letters & Evaluating
Algebra is using letters in place of numbers. It lets us write a rule with letters, then swap real numbers in when we want a real answer.
What algebra is
Algebra = using letters instead of numbers. In $\text{Pay} = 200000 + 700000m$, the $200000$ is the constant (fixed), $700000$ is the coefficient, and $m$ is the variable.
Evaluating
Evaluation = put numbers in instead of the letters, then work it out with BOMDAS (Brackets, Orders, Multiply & Divide, then Add & Subtract).
Worked example — substitute the values
Given $x = 2$ and $y = 3$, find $x + y$, $3x + 5y$ and $x^2 + y^2$.
$x + y = 2 + 3 = 5$
$3x + 5y = 3(2) + 5(3) = 6 + 15 = 21$
$x^2 + y^2 = 2^2 + 3^2 = 4 + 9 = 13$
$5,\ 21,\ 13$
Worked example — bigger numbers
Given $a = 12$ and $b = 15$, find $3a + 5b$ and $5a + 2b$.
$3a + 5b = 3(12) + 5(15) = 36 + 75 = 111$
$5a + 2b = 5(12) + 2(15) = 60 + 30 = 90$
$111$ and $90$
Negatives need brackets
When the value you put in is negative, slip brackets around it first, then BOMDAS. Most marks are lost when the minus sign goes missing.
You try
Given $x = -2$ and $y = 4$, find $x + y$, $x^2 - y$ and $3x - 2y$.
Brackets around $-2$ every time. Remember $(-2)^2 = +4$.
$x + y = (-2) + 4 = 2$
$x^2 - y = (-2)^2 - 4 = 4 - 4 = 0$
$3x - 2y = 3(-2) - 2(4) = -6 - 8 = -14$
$2,\ 0,\ -14$
Words you’ll meet in $2x^3 + 3x^2 + 7x + 9$: a term is a piece separated by $+$ or $-$ (this has $4$ terms); the coefficient is the number in front ($2$ in $2x^3$); the variable is the letter; the power (index) is the raised number ($3$ in $x^3$); the constant is the term with no letter ($9$).
Section 2 of 2
Simplifying Like Terms
Simplifying just means tidying: collect the terms that match (same letter and power) and add or subtract their coefficients. Terms that don’t match are left alone.
The rule
Only add and subtract like variables. $x$ and $x^2$ are not like terms — the power must match too. And $x$ means $1x$ (the $1$ is silent), so $x + 3x = 4x$.
Worked example — one variable
Simplify $2x + 3 + 5x + 7$ and $7x + 2 + 9x + 3$.
$2x + 5x = 7x$, $3 + 7 = 10$ → $7x + 10$
$7x + 9x = 16x$, $2 + 3 = 5$ → $16x + 5$
$7x + 10$ and $16x + 5$
Worked example — two variables
Simplify $5g + 3p + 2g + 9p$ and $5x + 2y + 3x + 4y$.
$5g + 2g = 7g$, $3p + 9p = 12p$ → $7g + 12p$
$5x + 3x = 8x$, $2y + 4y = 6y$ → $8x + 6y$
$7g + 12p$ and $8x + 6y$
You try
Simplify $7a + 8b - 2a - 12b$ and $x^2 + 8x - 12x - 9$.
Collect the matching terms. The sign travels with the term: $+$ I have, $-$ I owe.
$7a - 2a = 5a$, $8b - 12b = -4b$ → $5a - 4b$
$8x - 12x = -4x$ → $x^2 - 4x - 9$
$5a - 4b$ and $x^2 - 4x - 9$
Worked example — with powers
Simplify $5x \cdot x + 2x$ and $x^2 - 3x - 2x + 9$.
$5x \cdot x = 5x^2$ → $5x^2 + 2x$
$-3x - 2x = -5x$ → $x^2 - 5x + 9$
$5x^2 + 2x$ and $x^2 - 5x + 9$
Some expressions are already as tidy as they get — if the variables don’t match, leave them: $3x + 2y$ (different variables), $3ab + 2c$ (different terms), $3xy + 4xz$ (only one letter matches). Don’t force them together.
That’s Class 1.
Letters for numbers, evaluating with BOMDAS, and simplifying like terms. Class 2: multiplying out brackets.