Solving Equations & Inequalities
Master linear equations, literal equations, simultaneous equations, and inequalities — with full worked examples and the logic behind every step.
Equations and Inequalities
Table of Contents
Equations & Inequalities: Solving for the Unknown#
An equation says two things are equal. Your job is to find the value of $x$ that makes it true. In Grade 10, you solve four types of equations — each builds on the one before.
The Golden Rule#
Whatever you do to one side, you MUST do to the other. This keeps the equation balanced.
| Operation on $x$ | Undo it by… |
|---|---|
| $+ 5$ | Subtract 5 from both sides |
| $\times 3$ | Divide both sides by 3 |
| $\div 2$ | Multiply both sides by 2 |
| $x^2$ | Square root both sides |
The Four Equation Types#
1. Linear Equations (one variable)#
$$3x - 7 = 2x + 5 \Rightarrow x = 12$$Move all $x$-terms to one side, all numbers to the other, then solve.
2. Literal Equations (change the subject)#
Make a specific variable the subject of a formula:
$$A = \pi r^2 \Rightarrow r^2 = \frac{A}{\pi} \Rightarrow r = \sqrt{\frac{A}{\pi}}$$💡 Treat every other letter as if it’s a number — the rules are identical to solving a normal equation.
3. Simultaneous Equations (two unknowns)#
Two equations, two unknowns. Two methods:
| Method | Strategy | Best when… |
|---|---|---|
| Substitution | Solve one equation for $y$, substitute into the other | One equation is already solved for a variable |
| Elimination | Add/subtract equations to eliminate one variable | Coefficients line up neatly |
4. Linear Inequalities#
Solve exactly like an equation, with one critical rule:
$$-2x > 6 \Rightarrow x < -3$$⚠️ When you multiply or divide by a negative number, the inequality sign FLIPS.
Represent the solution on a number line (open circle for $<$/$>$, closed circle for $\leq$/$\geq$).
Deep Dive#
- Solving Equations, Literal Equations & Inequalities — full worked examples for all four types, word problems, and common traps
🚨 Common Mistakes#
- Dropping a sign when moving terms: $3x - 7 = 5$ → $3x = 12$, NOT $3x = -2$.
- Forgetting to flip the inequality: $-x > 3$ becomes $x < -3$. The sign FLIPS.
- Literal equations — treating letters as special: $V = \frac{1}{3}\pi r^2 h$ — solving for $h$ is the same algebra as solving $12 = 3x$.
- Simultaneous — not substituting back: After finding $x$, substitute into the EASIER original equation to find $y$.
- Fractions: Multiply every term by the LCD to clear all fractions before solving.
🔗 Related Grade 10 topics:
- Algebra: Factorisation — factorising is the main tool for solving equations
- Exponents — exponential equations use exponent laws
- Functions — solving $f(x) = 0$ gives x-intercepts
📌 Where this leads in Grade 11: Quadratic Equations & Inequalities
⏮️ Number Patterns | 🏠 Back to Grade 10 | ⏭️ Functions