Chapter 6: Linear Equations in Two Variables (Exercise 6.2)

Chapter 6: Linear Equations in Two Variables (Exercise 6.2)

Question 1:
Give the geometric representations of y = 3 as an equation:
1. In one variable
2. In two variables

Solution:
1. In one variable:
The equation y = 3 represents a single point on a number line at a distance of 3 units to the right of 0.

2. In two variables:
Written as 0x + 1y = 3.
This represents a straight line parallel to the X-axis passing through the point (0, 3) on the Y-axis.

Question 2:
Give the geometric representations of 2x + 9 = 0 as an equation:
1. In one variable
2. In two variables

Solution:
Solving for x:
2x = -9 ⇒ x = -4.5

1. In one variable:
It represents a single point at -4.5 on the horizontal number line.

2. In two variables:
Written as 1x + 0y = -4.5.
This represents a straight line parallel to the Y-axis passing through the point (-4.5, 0) on the X-axis.

Question 3:
The taxi fare in a city is as follows: For the first kilometre, the fare is ₹8 and for the subsequent distance it is ₹5 per km. Taking the distance covered as x km and total fare as ₹y, write a linear equation and draw its graph.

Solution:
* Total distance covered = x km
* Total fare = ₹y
* Fare for the 1^st km = ₹8
* Remaining distance = (x - 1) km
* Fare for remaining distance = ₹5 × (x - 1)

Formulating the equation:
y = 8 + 5(x - 1)
y = 8 + 5x - 5
y = 5x + 3 (or 5x - y + 3 = 0)

Points for graph:
* If x = 0 ⇒ y = 3 ⇒ (0, 3)
* If x = 1 ⇒ y = 8 ⇒ (1, 8)
* If x = 2 ⇒ y = 13 ⇒ (2, 13)

Question 4:
If the work done by a body on application of a constant force is directly proportional to the distance travelled by it, express this in the form of an equation in two variables and draw the graph of the same by taking the constant force as 5 units. Also read from the graph the work done when the distance travelled is:
1. 2 units
2. 0 units

Solution:
* Let the distance travelled be x units.
* Let the work done be y units.
* Constant force = 5 units.

Formula: Work done = Force × Distance
y = 5x

Finding values from the equation:
1. When distance x = 2 units:
y = 5 × 2 = 10 units of Work.

2. When distance x = 0 units:
y = 5 × 0 = 0 units of Work.