Word Problems: Linear Equations in Two Variables

1. The perimeter of a rectangular garden is 20 m. If the length is 4 m more than the breadth, find the length and breadth of the garden.

Solution: Let the length of the garden be x m. Therefore, breadth of garden = (x - 4) m.

Since, perimeter is 20 m, so

2[x + (x - 4)] = 20

2(2x - 4) = 20

2x - 4 = 10

2x = 10 + 4 = 14

x = 7

Hence, length = 7 m and breadth = 7 - 4 = 3 m.

Alternatively, you can solve the problem using two variables.

Let the length of garden = x m

Width of garden = y m

Therefore, x = y + 4 ... (1)

Also, perimeter is 20 m, therefore

2(x + y) = 20

x + y = 10 ... (2)

Solving (1) and (2), we get x = 7, y = 3

Hence, length = 7 m and breadth = 3 m

2. Asha is five years older than Robert. Five years ago, Asha was twice as old as Robert was then. Find their present ages.

Solution: Let present age of Asha be x years. Present age of Robert be y years.

Therefore, 

x = y + 5

x - y = 5 ... (1)

5 years ago, Asha was (x - 5) years and Robert was (y - 5) years old. Therefore,

x - 5 = 2(y - 5)

x - 2y = - 5 ... (2)

Solving (1) and (2), we get y = 10 and x = 15

Hence, present age of Asha = 15 years and present age of Robert = 10 years.

3. Two places A and B are 100 km apart. One car starts from A and another from B at the same time. If they travel in the same direction, they meet after 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars. Assume that the speed of car at A is more than the speed of car at B.

Solution: Let speed of the car starting from A be x km/h and speed of the car starting from B be y km/h.

Therefore, the distance travelled by car at A in 5 hours = 5x km and the distance travelled by car at B in 5 hours = 5y km.

Since they meet after 5 hours when they travel in the same direction, the car at A has travelled 100 km more than the car at B. Therefore,

5x - 5y = 100

x - y = 20 ... (1)

When they travel towards each other, they meet after 1 hour. It means, total distance travelled by car at A and car at B in 1 hour is 100 km. Therefore,

x + y = 100 ... (2)

Solving (1) and (2), we get x = 60 and y = 40

Therefore, the speed of car at A = 60 km/h and the speed of car at B = 40 km/h.