The Fundamental Idea#
An equation is a balance scale. Whatever you do to the left side, you MUST do to the right side. Your goal is to isolate $x$ by “undoing” operations in reverse order (undo the last thing first).
1. Linear Equations#
The Strategy#
Undo operations step by step. Addition undoes subtraction, multiplication undoes division.
Worked Example 1: Basic#
$3x + 7 = 22$
$3x = 15$ ← Subtract 7 from both sides
$x = 5$ ← Divide both sides by 3
Worked Example 2: Variables on Both Sides#
$5x - 3 = 2x + 12$
$3x = 15$ ← Move $x$ terms to left, numbers to right
$x = 5$
Worked Example 3: With Brackets#
$3(2x - 1) = 4(x + 3)$
$6x - 3 = 4x + 12$ ← Expand both sides first
$2x = 15$
$x = 7.5$
Worked Example 4: With Fractions#
$\frac{x}{3} + \frac{x}{4} = 7$
Multiply everything by the LCD (12):
$4x + 3x = 84$
$7x = 84$
$x = 12$
Strategy for fractions: Multiply every term by the LCD to eliminate all denominators in one step.
2. Literal Equations (Changing the Subject)#
A literal equation has multiple letters. You need to isolate one specific variable.
The Logic#
Treat every letter that ISN’T the one you’re solving for as if it were a number. Then solve normally.
Worked Example 1#
Make $r$ the subject of $A = \pi r^2$
$\frac{A}{\pi} = r^2$
$r = \sqrt{\frac{A}{\pi}}$ (take $r > 0$ since it’s a radius)
Worked Example 2#
Make $x$ the subject of $y = \frac{2x + 1}{3}$
$3y = 2x + 1$
$3y - 1 = 2x$
$x = \frac{3y - 1}{2}$
Worked Example 3#
Make $b$ the subject of $a^2 + b^2 = c^2$
$b^2 = c^2 - a^2$
$b = \sqrt{c^2 - a^2}$
3. Simultaneous Equations (Two Unknowns)#
Two unknowns need two equations. There are two methods:
Method 1: Substitution#
- Make one variable the subject in one equation.
- Substitute into the other equation.
- Solve for the remaining variable.
- Substitute back to find the first variable.
Worked Example#
$2x + y = 7$ … (1)
$x - y = 2$ … (2)
From (2): $x = y + 2$ … (3)
Substitute (3) into (1): $2(y + 2) + y = 7$
$2y + 4 + y = 7$
$3y = 3$
$y = 1$
From (3): $x = 1 + 2 = 3$
Answer: $x = 3$, $y = 1$
Method 2: Elimination#
Add or subtract the equations to eliminate one variable.
$2x + y = 7$ … (1)
$x - y = 2$ … (2)
Add (1) + (2): $3x = 9 \Rightarrow x = 3$
Substitute back: $3 - y = 2 \Rightarrow y = 1$
Tip: Use elimination when the coefficients are easy to match. Use substitution when one equation is already solved for a variable.
4. Linear Inequalities#
Inequalities work exactly like equations with ONE critical difference:
⚠️ When you multiply or divide by a NEGATIVE number, FLIP the inequality sign.
Worked Example 1#
$3x - 5 > 7$
$3x > 12$
$x > 4$
Worked Example 2 (Sign Flip)#
$-2x + 3 \leq 9$
$-2x \leq 6$
$x \geq -3$ ← Sign flipped because we divided by $-2$!
Worked Example 3 (Compound Inequality)#
$-1 < 2x + 3 \leq 9$
Subtract 3 from all parts: $-4 < 2x \leq 6$
Divide by 2: $-2 < x \leq 3$
Graphing on a Number Line#
- Open circle ($\circ$): The value is NOT included ($<$ or $>$)
- Closed circle ($\bullet$): The value IS included ($\leq$ or $\geq$)
For $-2 < x \leq 3$: Open circle at $-2$, closed circle at $3$, shade between.
5. Word Problems → Equations#
The hardest part of word problems is translating English into algebra.
Key Translations#
| English | Algebra |
|---|---|
| “3 more than $x$” | $x + 3$ |
| “twice a number” | $2x$ |
| “5 less than $x$” | $x - 5$ |
| “the product of $x$ and $y$” | $xy$ |
| “the sum of two consecutive numbers” | $x + (x+1)$ |
Worked Example#
The sum of three consecutive numbers is 42. Find the numbers.
Let the numbers be $x$, $x+1$, $x+2$.
$x + (x+1) + (x+2) = 42$
$3x + 3 = 42$
$3x = 39$
$x = 13$
The numbers are 13, 14, 15.
🚨 Common Mistakes#
- Forgetting to apply the operation to BOTH sides: If you subtract 5 from the left, you must subtract 5 from the right too.
- Not expanding brackets before solving: $2(x + 3) = 10$ → expand to $2x + 6 = 10$ first.
- Inequality sign flip: ONLY flip when multiplying or dividing by a negative. Adding or subtracting a negative does NOT cause a flip.
- Simultaneous equations — substituting into the wrong equation: After finding one variable, substitute into the SIMPLER equation to find the other.
- Literal equations — treating the target variable as a number: Remember, $r$ is not a number. You isolate it the same way, but the answer will have other letters in it.
💡 Pro Tip: Always Check Your Answer#
Substitute your answer back into the original equation. If both sides are equal, you’re correct. This takes 10 seconds and catches most errors.
$3x + 7 = 22$ with $x = 5$: $3(5) + 7 = 15 + 7 = 22$ ✓
🔗 Related Grade 10 topics:
- Factorization — you need factoring to solve quadratic equations
- Exponent Laws — exponential equations use the same “make bases equal” logic
- Sketching Graphs — solving $f(x) = 0$ gives you the x-intercepts of a graph
- Analytical Geometry — simultaneous equations find intersection points of lines
📌 Where this leads in Grade 11:
- Quadratic Equations & Discriminant — quadratic formula, nature of roots, simultaneous equations with a quadratic