Why This Matters for Grade 11#
Grade 11 equation solving is an extension of Grade 10 methods:
- Quadratic equations: You still isolate, factorise, and solve — but with $x^2$ terms
- Surd equations: You isolate the surd, then square both sides — but you must check for extraneous solutions
- Trig equations: You solve $\sin\theta = \frac{1}{2}$ using the same inverse-operation logic
- Simultaneous equations: Grade 11 mixes a linear equation with a quadratic (substitution method)
If the Grade 10 methods aren’t automatic, Grade 11 equations will overwhelm you.
1. Linear Equations#
Isolate $x$ using inverse operations.
Worked Example#
Solve $\frac{2x - 3}{4} = 5$
$$2x - 3 = 20 \quad \text{(multiply both sides by 4)}$$$$2x = 23$$$$x = \frac{23}{2} = 11.5$$Always check: $\frac{2(11.5) - 3}{4} = \frac{20}{4} = 5$ ✓
2. Literal Equations (Changing the Subject)#
Make a specific variable the subject of the formula.
Worked Example#
Make $r$ the subject of $A = \pi r^2$
$$r^2 = \frac{A}{\pi}$$$$r = \sqrt{\frac{A}{\pi}} \quad (r > 0)$$Common Formulae You Should Be Able to Rearrange#
| Formula | Make this the subject |
|---|---|
| $v = u + at$ | $t = \frac{v - u}{a}$ |
| $A = P(1 + in)$ | $i = \frac{A - P}{Pn}$ |
| $y = mx + c$ | $x = \frac{y - c}{m}$ |
Grade 11 connection: When you solve $y = a(x-p)^2 + q$ for $x$, you’re doing a literal equation with the quadratic function.
3. Simultaneous Equations#
Two equations, two unknowns. Two methods:
Method 1: Substitution#
- Make one variable the subject in one equation
- Substitute into the other equation
- Solve, then substitute back
Solve: $y = 2x - 1$ and $3x + y = 9$
Substitute: $3x + (2x - 1) = 9$
$5x = 10$, so $x = 2$
$y = 2(2) - 1 = 3$
Solution: $x = 2, y = 3$
Method 2: Elimination#
- Multiply equations so one variable has the same coefficient
- Add or subtract to eliminate that variable
Solve: $2x + 3y = 12$ and $4x - 3y = 6$
Add: $6x = 18$, so $x = 3$
$2(3) + 3y = 12$, so $3y = 6$, $y = 2$
Grade 11 extension: In Grade 11, one equation is linear and one is quadratic (e.g., $y = x + 1$ and $x^2 + y^2 = 25$). You MUST use substitution — elimination won’t work.
4. Equations with Fractions#
Multiply every term by the LCD to clear the fractions.
Worked Example#
Solve $\frac{x}{2} + \frac{x}{3} = 10$
LCD = 6. Multiply every term by 6:
$3x + 2x = 60$
$5x = 60$
$x = 12$
Critical rule: When the variable is IN the denominator (like $\frac{3}{x} = 5$), you must check that your answer doesn’t make the denominator zero. This is called checking for restrictions.
5. Inequalities — The Sign-Flip Rule#
Solve like an equation, BUT: when you multiply or divide by a negative number, FLIP the inequality sign.
Worked Example#
Solve $-3x + 6 > 12$
$-3x > 6$
$x < -2$ (flip the sign because you divided by $-3$)
On a number line: Open circle at $-2$, shade to the LEFT.
Grade 11 extension: Quadratic inequalities like $x^2 - 5x + 6 < 0$ require factorising, finding roots, and using a sign diagram or parabola sketch.
🚨 Common Mistakes#
- Forgetting to apply operations to ALL terms: In $\frac{x}{2} + 3 = 7$, if you multiply by 2, you get $x + 6 = 14$, NOT $x + 3 = 14$.
- Sign errors when moving terms: $x - 5 = 3$ gives $x = 8$, not $x = -2$. Moving $-5$ across the equals sign makes it $+5$.
- Not flipping the inequality: $-2x > 6$ gives $x < -3$, not $x > -3$.
- Literal equations — wrong variable isolated: Read the question carefully. “Make $h$ the subject” means get $h = ...$.
- Not checking simultaneous solutions: Substitute BOTH values back into BOTH original equations.
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