8 min readConcept sheetStructured question, often a word problem worth 5 to 10 marks

Simple and simultaneous equations

Do the same thing to both sides to solve an equation. Two unknowns need two equations; solve them by substitution or elimination.

Linear equationHighest power of x is 1

SimultaneousTwo equations, two unknowns

MethodsSubstitution, elimination, matrices

Last stepCheck in the original words

From a word problem to an answer

Methods and rules

Balance ruleAdd, subtract, multiply or divide both sides by the same thing

FractionsMultiply every term by the LCM of the denominators

SubstitutionMake one letter the subject, put it into the other equation

EliminationMake one letter's coefficients equal, then add or subtract

No solutionParallel lines, e.g. x + y = 5 and 2x + 2y = 12

Infinitely manySame line twice, e.g. x + y = 5 and 2x + 2y = 10

Simple equations

Problem

Solve (a) 5(x − 3) = 2x + 9 (b) x/3 + x/4 = 14.

Working

(a) Open the bracket: 5x − 15 = 2x + 9.
Subtract 2x from both sides: 3x − 15 = 9. Add 15: 3x = 24. So x = 8.
Check: 5(8 − 3) = 25 and 2(8) + 9 = 25. ✓
(b) LCM of 3 and 4 is 12. Multiply every term by 12: 4x + 3x = 168.
7x = 168, so x = 24. Check: 24/3 + 24/4 = 8 + 6 = 14. ✓

Answer

(a) x = 8 (b) x = 24

A wage problem

Problem

A factory worker earns Rs. 1,500 a day plus Rs. 400 for each overtime hour. In a 22-day month she earned Rs. 39,400. How many overtime hours did she work?

Working

Let h = overtime hours.
Basic pay = 22 × 1,500 = Rs. 33,000.
Equation: 33,000 + 400h = 39,400.
400h = 39,400 − 33,000 = 6,400, so h = 6,400 ÷ 400 = 16.
Check: 33,000 + 400 × 16 = 33,000 + 6,400 = 39,400. ✓

Answer

16 overtime hours.

Simultaneous equations by elimination

Problem

Solve 3x + 2y = 18 and 5x − 2y = 14.

Working

The y terms are +2y and −2y, so add the equations.
(3x + 5x) + (2y − 2y) = 18 + 14, so 8x = 32 and x = 4.
Substitute into the first: 3(4) + 2y = 18, so 2y = 6 and y = 3.
Check in the second: 5(4) − 2(3) = 20 − 6 = 14. ✓

Answer

x = 4, y = 3

Simultaneous equations by substitution: a sales mix

Problem

A Kandy tea shop sells plain tea at Rs. 60 and milk tea at Rs. 100. One day it sold 150 cups for Rs. 11,400. How many of each?

Working

Let x = plain teas and y = milk teas.
Cups: x + y = 150. Money: 60x + 100y = 11,400.
From the first, x = 150 − y. Substitute: 60(150 − y) + 100y = 11,400.
9,000 − 60y + 100y = 11,400, so 40y = 2,400 and y = 60.
x = 150 − 60 = 90.
Check: 90 × 60 + 60 × 100 = 5,400 + 6,000 = 11,400. ✓

Answer

90 plain teas and 60 milk teas.

In Sri Lankan business

A garment exporter like MAS, planning how many of two styles to sew with fixed machine time, or a tea shop checking its till against cups sold, is solving simultaneous equations.

Exam angle

Short answer: “Solve 4x − 7 = 2x + 5.” 2x = 12, so x = 6.

Structured question: “Solve 3x + 2y = 18 and 5x − 2y = 14.” Add: 8x = 32, x = 4; then 2y = 6, y = 3. Check in the second equation.

Essay: “A shop sold 150 cups of tea, plain at Rs. 60 and milk at Rs. 100, for Rs. 11,400. Form and solve equations.” x + y = 150, 60x + 100y = 11,400; substitution gives y = 60, x = 90; check the total.

Remember it

Let (name the unknowns), Link (write equations), Solve, Check. A check line in the original words earns a mark and catches slips.