Forming and Solving Equations

Forming and Solving Equations

Many problems in geometry, number and real-world contexts cannot be solved by arithmetic alone — they require first translating the situation into an algebraic equation, solving it, and then interpreting the solution in context. This is one of the most important mathematical skills: moving fluently between a problem described in words or diagrams and its algebraic representation.

General Strategy

  1. Define the unknown: choose a letter and state clearly what it represents (including units if applicable).
  2. Form the equation: translate the given conditions into algebra. Each independent condition gives one equation.
  3. Solve: apply appropriate methods (linear, quadratic, simultaneous, etc.).
  4. Interpret: check the solution is valid in context (e.g. lengths cannot be negative) and answer the question as asked.

Common Contexts

  • Geometry: angles in polygons, perimeter, area. Use known geometric facts to form the equation.
  • Number: consecutive integers, age problems, number puzzles.
  • Ratio and proportion: share quantities, rate problems.
  • Algebra from graphs: set two expressions equal to find an intersection.

Worked Examples

A rectangle has length \( (2x+3) \) cm and width \( (x-1) \) cm. Its perimeter is 36 cm. Find \( x \) and the area.
\[ 2(2x+3) + 2(x-1) = 36 \implies 4x+6+2x-2 = 36 \implies 6x+4 = 36 \implies x = \frac{32}{6} \]

Hmm — try perimeter formula directly: \( 2[(2x+3)+(x-1)] = 36 \implies 2(3x+2)=36 \implies 3x+2=18 \implies x=\frac{16}{3} \). Area \( = (2x+3)(x-1) = \frac{35}{3} \times \frac{13}{3} = \frac{455}{9} \approx 50.6 \) cm².

The sum of three consecutive integers is 51. Find the integers.

Let the integers be \( n \), \( n+1 \), \( n+2 \).

\[ n + (n+1) + (n+2) = 51 \implies 3n+3=51 \implies n=16 \]

The integers are 16, 17 and 18.

Sam is 3 times as old as Tom. In 4 years' time, Sam will be twice Tom's age. Find their current ages.

Let Tom's current age = \( t \). Sam's current age = \( 3t \).

\[ 3t+4 = 2(t+4) \implies 3t+4=2t+8 \implies t=4 \]

Tom is 4, Sam is 12.

A right-angled triangle has legs \( x \) cm and \( (x+3) \) cm. Its hypotenuse is \( (x+6) \) cm. Find \( x \).
\[ x^2 + (x+3)^2 = (x+6)^2 \] \[ x^2 + x^2+6x+9 = x^2+12x+36 \] \[ x^2 - 6x - 27 = 0 \implies (x-9)(x+3)=0 \]

\( x = 9 \) (since \( x > 0 \); \( x = -3 \) is rejected as a length cannot be negative).

The triangle has legs 9 cm and 12 cm and hypotenuse 15 cm — a 3-4-5 Pythagorean triple scaled by 3.

 Key Takeaways

  • Always define the unknown with a clear statement (e.g. "let \(x\) = the width in cm").
  • Form the equation by translating the constraint — perimeter, angle sum, area, difference, etc. — into algebra.
  • Solve using the appropriate method: linear, quadratic (factorise or formula), or simultaneous.
  • Reject solutions that are invalid in context (negative lengths, non-integer ages when integers are required).
  • Always answer the question asked — if the question asks for the area, find and state the area, not just \(x\).