9 min readPathway sheet3-mark or 5-mark question; often a 4-mark case-based question

Word problems on linear equations

Name the two unknowns, turn each condition into one equation, solve by any method, and answer in words with units.

Two unknownsNeed two equations

Ages'n years ago': subtract n from both ages; 'after n years': add n

Two-digit numberTens digit x, units digit y: number = 10x + y; reversed = 10y + x

Fixed charge + rateTotal = fixed charge + (number of days × rate per day)

Method for any word problem

1
Read the question twice. Write 'Let ... = x and ... = y', with units.
2★
Turn the first condition into an equation.
3★
Turn the second condition into an equation.
4
Solve by substitution or elimination.
5
Check the answer against the words of the question.
6
Write the answer as a sentence with units.

★ key step

Common types and how to write the equations

Type	Let	The equations look like
Cost	1 pen = ₹ x, 1 notebook = ₹ y	3x + 2y = total cost
Ages	present ages x and y	x + 5 = 2(y + 5) for 'after 5 years'
Fraction	fraction = x/y	(x + 1)/(y + 1) = given value; clear the fractions
Two-digit number	tens digit x, units digit y	10x + y; reversed number 10y + x
Fixed charge + rate	fixed charge ₹ x, rate per day ₹ y	x + 5y = total for 5 days
Rectangle	length x, breadth y	2(x + y) = perimeter; x − y = given difference

Example 1: fixed charge and daily rate

Problem

A cycle rental shop in Mysuru charges a fixed amount plus a charge for each day (example prices). Meena paid ₹ 160 for 4 days and Arif paid ₹ 220 for 7 days. Find the fixed charge and the charge per day.

Working

Let the fixed charge be ₹ x and the charge per day be ₹ y.
Meena: x + 4y = 160 ... (1)
Arif: x + 7y = 220 ... (2)
Subtract (1) from (2): 3y = 60, so y = 20.
From (1): x = 160 − 80 = 80.
Check: 80 + 7 × 20 = 80 + 140 = 220.

Answer

The fixed charge is ₹ 80 and the charge per day is ₹ 20.

Example 2: a two-digit number

Problem

The sum of the digits of a two-digit number is 11. When the digits are reversed, the new number is 27 more than the original number. Find the number.

Working

Let the tens digit be x and the units digit be y. The number is 10x + y.
Sum of digits: x + y = 11 ... (1)
Reversed number: (10y + x) − (10x + y) = 27, so 9y − 9x = 27 and y − x = 3 ... (2)
Add (1) and (2): 2y = 14, so y = 7. Then x = 4.
The number is 10(4) + 7 = 47.
Check: 4 + 7 = 11 and 74 − 47 = 27.

Answer

The number is 47.

Example 3: ages

Problem

Five years ago, a mother was 4 times as old as her daughter. Fifteen years from now, she will be twice as old as her daughter. Find their present ages.

Working

Let the mother's present age be x years and the daughter's be y years.
Five years ago: x − 5 = 4(y − 5), so x − 4y = −15 ... (1)
Fifteen years from now: x + 15 = 2(y + 15), so x − 2y = 15 ... (2)
Subtract (1) from (2): 2y = 30, so y = 15.
From (2): x = 15 + 30 = 45.
Check: five years ago 40 = 4 × 10; fifteen years from now 60 = 2 × 30.

Answer

The mother is 45 years old and the daughter is 15 years old.

In real life

Auto and taxi fares in many Indian cities have a fixed charge plus a rate per km. Two trips give two equations that reveal both numbers. Splitting a grocery bill between two items works the same way.

Exam angle

3 marks: “The perimeter of a rectangular plot is 64 m, and its length is 8 m more than its breadth. Find its length and breadth.” 2(x + y) = 64 gives x + y = 32, and x − y = 8. Adding: x = 20, y = 12. Length 20 m, breadth 12 m.

5 marks: “A two-digit number is 4 times the sum of its digits. If 18 is added to the number, its digits are reversed. Find the number.” 10x + y = 4(x + y) gives y = 2x. 10x + y + 18 = 10y + x gives x − y = −2. So x = 2, y = 4. The number is 24 (24 + 18 = 42).

Remember it

Let, link, solve, check, say: name the unknowns, link them in equations, solve, check, and state the answer with units.