Solved Questions on Linear Equations
1. Harry purchases a Jeep and a car from a dealer. The cost of the car is seven thousand less than four times the cost of the jeep. Which linear equation in two variables represents this statement?
a) x + 4y + 7000 = 0
b) x + 4y − 7000 = 0
c) x − 4y + 7000 = 0
d) x − 4y − 7000 = 0
Answer: c) x − 4y + 7000 = 0
Explanation: Let the cost of a car be $x and a jeep be $y.
The cost of the car is seven thousand less than four times the cost of the jeep.
Cost of a car = 4 × cost of a pencil − 7000
⇒ x = 4 × y − 7000
⇒ x = 4y − 7000
∴ x − 4y + 7000 = 0 is a linear equation in two variables to represent this statement.
2. Which one of the following options is true for the following linear equation?
y = 2x − 3
a) No solution
b) One solution
c) Two solution
d) infinite solutions
Answer: d) infinite solutions
Explanation: When we put different values of ‘x’ in the given equation y = 2x − 3, we get different values of ‘y’ as well.
For x = 1:
y = 2x − 3
= 2 × 1 − 3
= 2 − 3
= −1
For x = 2:
y = 2x − 3
= 2 × 2 − 3
= 4 − 3
= 1
For x = 3:
y = 2x − 3
= 2 × 3 − 3
= 6 − 3
= 3 and so on …….
Here, in this equation x can have infinite values and for all the infinite values of x, there are infinite values of y as well.
∴ y = 2x − 3 has infinite solutions.
3. Selena boards a taxi for her office which is 2 kilometres from her home. The fare meter shows $12 after one kilometre and for the upcoming kilometres, it drops to $7. What will be the linear equation for the above-mentioned incident and what is the total fare if she travelled for 12.5 kilometres?
a) y = 7x + 5; $90.5
b) y = 7x + 5; $92.5
c) y = 7x + 12; $90.5
d) y = 7x + 12; $92.5
Answer: b) y = 7x + 5; $92.5
Explanation: Let total distance covered = x and total fare = $y
Fare for the first kilometre = $12
Fare after first kilometre = $7
Total distance left after covering first kilometre = (x − 1)
Fare after covering first kilometre = 7(x − 1)
Total fare = fare of first kilometre + fare after the first kilometre
y = 12 + 7(x −1)
y = 12 + 7x − 7
y = 7x + 5
∴ y = 7x + 5 will be the linear equation for the above-mentioned incident.
If x = 12.5 km, then
Total fare (y) = 7x + 5
= 7 × 12.5 + 5
= $92.5
4. The graph of the linear equation 7x − 3y = 21 cuts the X-axis at the point?
a) (2, 0)
b) (3, 0)
c) (4, 0)
d) (5, 0)
Answer: b) (3, 0)
Explanation: The given line 7x − 3y = 21 cuts the X-axis and at the X-axis coordinate of y = 0.
Putting y = 0 in the above equation 7x − 3y = 21, we get:
7x − 3y = 21
7x − 0 × y = 21
7x − 0 = 21
∴ 7x = 21
∴ x = 21/7
∴ x = 3
So, the point where the line 7x − 3y = 21 cuts the X-axis will be (3, 0).
5. What are the coordinates of the points where the line represented by the linear equation y = 3x − 6 intersects the X-axis and Y-axis?
a) (2, 0), (0, −4)
b) (4, 0), (0, −2)
c) (2, 0), (0, −6)
d) (6, 0), (0, −2)
Answer: c) (2, 0), (0, −6)
Explanation: Given: y = 3x − 6
This line intersects both the X-axis and Y-axis. Thus, to get the pair values of (x, y).
The point on the X-axis which intersects the line is (x, 0) and the point on the Y-axis which intersects the line is (0, y).
At x = 0,
y = 3x − 6
y = 3 × 0 − 6
y = 0 − 6
y = − 6
The point on the Y-axis which intersects the line = (0, −6)
Similarly, at y = 0,
y = 3x − 6
0 = 3 × x − 6
3x = 6
x = 2
The point on the X-axis which intersects the line = (2, 0)
∴ This line will intersect the X-axis at (2, 0) and will intersect the Y-axis at (0, −6).
