Forming & Solving EquationsCambridge IGCSE Maths: Revision notes

Section 1

How do you construct expressions and equations from context?

Read the worded problem carefully, define a letter for the unknown quantity, then translate each phrase into algebra step by step.

For example: "write an expression for a number 2 more than $n$ " gives $n + 2$ . To form an equation, use a further piece of information that lets you set two expressions equal, e.g. "a number 2 more than $n$ , doubled, is 30" gives $2(n+2) = 30$ .

Key termsconstruct

Exam tip

Always state your defined letter clearly, e.g. 'let $n$ = the number', before writing the equation — this is expected working and helps avoid misreading the question.

Section 2

How do you form and solve linear equations from geometric or real-life situations?

Geometric problems often give expressions for sides, angles or perimeters that must be set equal using a known geometric fact (e.g. angles on a straight line sum to 180°, or two sides of an isosceles triangle are equal).

Example: A rectangle has length $(x+3)$ cm and width $x$ cm. Its perimeter is 26 cm.

Form the equation: $2(x+3) + 2x = 26$
Expand and simplify: $4x + 6 = 26$
Solve: $x = 5$

Example

Three angles on a straight line are $x$ , $2x$ and $3x-10$ degrees. Since angles on a straight line sum to 180°: $x + 2x + (3x-10) = 180 \Rightarrow 6x = 190 \Rightarrow x = 31.\overline{6}$ .

Section 3

How do you form and solve quadratic equations from context? (Extended)

Some real-life or geometric contexts (typically involving area) lead to a quadratic equation.

Example: A rectangular garden has length $(x+4)$ m and width $x$ m, and area $60\text{ m}^2$ .

Form the equation: $x(x+4) = 60$
Expand: $x^2 + 4x - 60 = 0$
Factorise: $(x+10)(x-6) = 0$
Solve: $x = -10$ or $x = 6$
Reject the negative solution (length cannot be negative), so $x = 6$

Key termsquadratic equation

Common mistake

Always check both solutions of a quadratic against the context — a length, time or quantity cannot be negative, so reject any solution that doesn't make physical sense.

Section 4

How do you form and solve simultaneous equations from contextual problems?

When a problem gives two unknowns and two separate pieces of information, define two letters and write two equations, then solve them simultaneously.

Example: Two numbers have a sum of 20 and a difference of 4.

Let the numbers be $x$ and $y$ : $x + y = 20$ and $x - y = 4$
Add the equations: $2x = 24 \Rightarrow x = 12$
Substitute back: $y = 8$

Key termssimultaneous equations

Exam tip

Always substitute your final values back into the original context (not just the equations) to check they make sense, e.g. that the sum and difference conditions are both satisfied.

Must Know

Define a letter for the unknown before writing any expression or equation
Translate worded phrases into algebra one piece of information at a time
Use known geometric facts (angle sums, equal sides) to form equations from shapes
Area/context problems can lead to quadratic equations — solve then reject any answer that doesn't fit the context (e.g. negative lengths)
Two unknowns with two pieces of information means forming and solving simultaneous equations

That's the notes covered.

Carry on to the next subtopic.