Forming And Solving Equations

Forming Equations from Words

• Use **x** to represent an unknown value; translate phrases like '2 less than something' → **x − 2**, 'double something' → **2x**, '5 lots of something' → **5x**, '3 more than something' → **x + 3**, 'half of something' → **x/2**.
• Key words: **sum/total/more than** (+), **difference/less than** (−), **product/lots of/times as many** (×), **shared/split/grouped** (÷).
• Use brackets to keep order correct: 'something add 1, then multiplied by 3' → **3(x + 1)**; compare with 'something multiplied by 3, then add 1' → **3x + 1**.
• Choose the unknown wisely: if Adam is 10 years younger than Barry, let Barry's $age =$ **x**, $Adam's =$ **x − 10**; if Adam's age is half Barry's, let $Barry's =$ **2x**, $Adam's =$ **x**.
• An **equation** has an equals sign; insert 'is equal to' to place it. E.g., 'Lisa's age is double Aisha's and sum is 27' → **2x $+ x =$ 27**.
• Always give the answer **in context** (e.g., 'Lisa is 18 years old').

Forming Equations from Shapes

• Use all given information and **shape properties**: triangles (equilateral, isosceles, scalene, right-angled), quadrilaterals (square, rectangle, kite, rhombus, parallelogram, trapezium), polygons (sum of interior angles = **180(n − 2)**), parallel lines (alternate, corresponding, co-interior).
• For **perimeter**, add all side lengths; for **area**, use appropriate formula (e.g., rectangle: length × width, triangle: ½ × base $\times height)$.
• For **3D shapes**, volume = cross‑section area × length (prisms); surface area $= sum$ of face areas.
• Sketch a diagram if none given; split irregular shapes into sum/difference of common shapes.
• Put **brackets** around algebraic expressions when substituting into formulas (e.g., perimeter $= 2(3x + 1) + 2(2x - 5))$.
• Read carefully: does the question ask for an angle, perimeter, area, or something else?

Problem Solving with Equations

• Problem solving involves forming and solving equations from a **real‑life or constructed situation**; equations can be **linear** $(e.g., 2(x + 4) = 3x)$ or **quadratic** $(e.g., x^{2} - 7x + 12 = 0)$.
• Answers must be given **in context** with correct units (e.g., 'The population density is 225 people per square km').
• To solve quadratic equations, bring all terms to one side to get '= 0'; choose method (factorising, quadratic formula, completing square).
• If two solutions arise, **justify** which is correct (e.g., length cannot be negative).
• Use algebra in other settings: percentages $(P\% = \frac{P}{100})$, ratios (e.g., x : $(x + 2) = 5$ : 8 → $x/(x+2) = \frac{5}{8})$, or unfamiliar equations $(e.g., \frac{12}{x} = 7 - x$ → multiply by x to get quadratic).
• If part (a) asks to prove an equation and part (b) uses it, you can still do part (b) without having done part (a).

Example: Forming Linear Equations from Words

• A flowerbed has red, yellow, and purple flowers. Yellow $= 3 \times$ red; purple = yellow + 5; difference $(purple - red) = 29$.
• Let $red =$ **x** → yellow = **3x**, purple = **3x + 5**; equation: $(3x + 5) - x = 29$ → $2x + 5 = 29$ → $x = 12$.
• Answer: yellow flowers $= 3 \times 12 =$ **36**.

Example: Forming Equations from Shapes

• Rectangle length $= (3x + 1)$ cm, width $= (2x - 5)$ cm, perimeter $= 22$ cm.
• Perimeter $= 2(3x + 1) + 2(2x - 5) = 6x + 2 + 4x - 10 = 10x -$ 8; set equal to 22 → $10x - 8 = 22$ → $x = 3$.
• Area = length × width $= (3\times 3 + 1) \times (2\times 3 - 5) = 10 \times 1 =$ **10 cm²**.

Example: Problem Solving with a Cube Net

• Cube net has 14 edges; perimeter = **14x** cm. Net has 6 faces; area = **6x²** $cm^{2}$.
• Volume of cube = **x³** $cm^{3}$. Difference (volume − surface $area) = 8 \times$ perimeter → $x^{3} - 6x^{2} = 112x$ → $x^{3} - 6x^{2} - 112x = 0$.
• Factor out $x (x \ne 0): x(x^{2} - 6x - 112) = 0$ → $x^{2} - 6x - 112 = 0$. Solve: $x = 14$ or $x = -8 (reject)$. Volume $= 14^{3} =$ **2744 cm³**.

Common Pitfalls & Tips

• Always **read the question** to see what is asked (angle, perimeter, area, etc.).
• For surface area/volume, check the formula sheet provided in the exam.
• When forming equations, **double‑check** that the algebraic expression matches the wording (e.g., '5 less than x' is x − 5, not 5 − x).
• If you get a quadratic, ensure you have $'= 0'$ before solving; discard extraneous solutions based on context.

Translating Words to Expressions

Forming Equation from Rectangle Perimeter

Cube Net and Equations

Triangle with Equal Base Angles