## Word Problems on Simultaneous Linear Equations

How to Calculate

Solved Problems :

1. Jacob is six years older than John. In four years' time the sum of their ages will be 30 years. What are their ages now ?

Solution :

Let Jacob's age be x years and that of John be y years.

According to the given conditions :

x - y = 6

x + 4 + y +4 = 30 ( ∵ after 4 years their ages will be (x + 4) years and (y + 4) years )

.i.e. the equations are :

x - y = 6 ....(i)

x + y = 22 ....(ii)

Adding equation (i) and (ii) , we get

2x = 28 ⇒ x = 14

Substituting x = 14 in equation (ii), we get

14 + y = 22 ⇒ y = 8.

∴ Jacob's age is 14 years and John's age is 8 years .

2. The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and the breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area is increased by 67 square units. Find the length and breadth of the rectangle ?

Solution:

Let the length and breadth of the rectangle be x units and y units respectively.

Then area = xy sq. units.

Given : If the length is reduced by 5 units and breadth increased by 3 units, the area is reduced by 9 square units

i.e., xy - 9 = (x - 5)(y + 3)

xy - 9 = xy - 5y + 3x - 15 3x - 5y = 6.

Given : When the length is increased by 3 units and breadth by 2 units, the area is increased by 67 square units

i.e., xy + 67 = (x + 3)(y + 2) xy + 67 = xy + 3y + 2x + 6 2x + 3y = 61

We have to solve the equations :

3x - 5y = 6 ......(1)

2x + 3y = 61 ...(2)

6x - 10y = 12 ...........Multiply (1) by 2 ........(3)

6x + 9y = 183 ..........Multiply (2) by 3 ........(4)

Subtracting equation (3) by (4 ), We get

- 19 y = -171

y = -171 / -19 = 9

Putting y = 9 in (1), we have 3x - 45 = 6

3x = 51 x =17.

The length and breadth of the rectangle are 17 units and 9 units respectively.

3. A boat goes 36km downstream in 4 hours and 30km upstream in 5 hours. Find

(i) The speed of boat in still water.

(ii) Speed of the current.

Solution:

Speed downstream = Distance / Time = 36km / 4hrs. = 9km/hr,

Speed upstream = Distance / Time = 30km / 5hrs. = 6 km/hr.

Let the speed of the boat in still water be x km/hr and speed of the current be y km/hr.

then speed downstream = ( x + y ) km/hr and speed upstream = ( x - y ) km/hr

According to the question.

x + y = 9 .........(i)

x - y = 6 .........(ii)

Adding eq. (i) and (ii) , we get 2x = 15 x = 7.5

Substituting x = 7.5 in (i) , we have 7.5 + y = 9 y = 9 - 7.5 y = 1.5

The speed of the boat in still water is 7.5 km/hr and the speed of the current is 1.5 km/hr.