Why This Matters for Grade 10#
Basic algebra is the engine of every Grade 10 topic:
- Algebraic Expressions: Expanding $3(2x + 1)$ and collecting like terms is the first thing you do
- Equations: Solving equations is just algebra — isolate $x$ using inverse operations
- Functions: Substituting $x = 2$ into $f(x) = 3x^2 - 1$ to find $f(2)$
- Exponents: Simplifying $\frac{2x^3 \cdot 3x^2}{6x^4}$ needs algebraic skills
If you can’t do these things fluently, every Grade 10 topic will feel harder than it should.
1. Like Terms & Unlike Terms#
Like terms have the same variable(s) raised to the same power(s). You can only add or subtract like terms.
| Like terms (CAN combine) | Unlike terms (CANNOT combine) |
|---|---|
| $3x + 5x = 8x$ | $3x + 5x^2$ (different powers) |
| $2ab - 7ab = -5ab$ | $2ab + 3a$ (different variables) |
| $4x^2y + x^2y = 5x^2y$ | $4x^2y + 4xy^2$ (different arrangement) |
Key: Constants (plain numbers) are like terms with each other: $3 + 7 = 10$.
2. The Distributive Law#
$$ a(b + c) = ab + ac $$This is the most important law in algebra. Expanding brackets = applying the distributive law.
Examples#
$3(x + 4) = 3x + 12$
$-2(3a - 5) = -6a + 10$ (watch the signs!)
$x(x + 3) = x^2 + 3x$
Common mistake: $-2(3a - 5) \neq -6a - 10$. The negative multiplies BOTH terms. $(-2)(-5) = +10$.
3. Substitution#
Replace the variable with the given value, then calculate using BODMAS.
Examples#
If $x = 3$: $\quad 2x^2 - 5x + 1 = 2(3)^2 - 5(3) + 1 = 18 - 15 + 1 = 4$
If $a = -2$: $\quad a^3 + 3a = (-2)^3 + 3(-2) = -8 + (-6) = -14$
Always use brackets when substituting negative numbers: $a^2 = (-2)^2 = 4$, NOT $-2^2 = -4$.
4. Solving Simple Equations#
An equation has an equals sign. Your job: get $x$ alone on one side using inverse operations.
The Inverse Operations#
| Operation | Inverse |
|---|---|
| $+ 5$ | $- 5$ |
| $- 3$ | $+ 3$ |
| $\times 4$ | $\div 4$ |
| $\div 2$ | $\times 2$ |
Worked Example#
Solve $3x - 7 = 14$
$$3x - 7 = 14$$$$3x = 14 + 7 = 21$$$$x = \frac{21}{3} = 7$$Check: $3(7) - 7 = 21 - 7 = 14$ ✓
With Brackets#
Solve $2(x + 3) = 16$
$$2x + 6 = 16$$$$2x = 10$$$$x = 5$$Variables on Both Sides#
Solve $5x - 3 = 2x + 9$
$$5x - 2x = 9 + 3$$$$3x = 12$$$$x = 4$$Golden rule: Whatever you do to one side, you must do to the other side.
5. Formulae & Changing the Subject#
A formula like $A = \frac{1}{2}bh$ has $A$ as the subject. You can rearrange to make any variable the subject.
Example: Make $h$ the subject#
$$A = \frac{1}{2}bh$$$$2A = bh$$$$h = \frac{2A}{b}$$This skill is used heavily in Grade 10 literal equations and finance (rearranging $A = P(1+in)$ for $i$ or $n$).
6. Algebraic Notation You Must Know#
| Notation | Meaning |
|---|---|
| $2x$ | $2 \times x$ |
| $x^2$ | $x \times x$ |
| $3x^2$ | $3 \times x \times x$ |
| $(3x)^2$ | $(3x)(3x) = 9x^2$ |
| $\frac{x}{3}$ | $x \div 3$ |
| $-x$ | $(-1) \times x$ |
🚨 Common Mistakes#
- $2x + 3x = 5x^2$: WRONG. $2x + 3x = 5x$. You add the coefficients, not the powers.
- $-2(x - 3) = -2x - 6$: WRONG. $(-2)(-3) = +6$, so it’s $-2x + 6$.
- Forgetting brackets when substituting: $x = -3$, then $x^2 = (-3)^2 = 9$, NOT $-3^2 = -9$.
- Moving terms across the equals sign without changing sign: $x - 5 = 3$ gives $x = 8$, not $x = -2$.
- $(x + y)^2 \neq x^2 + y^2$: $(x + y)^2 = x^2 + 2xy + y^2$. This is the Grade 10 special product!
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