Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion : The linear equations x - 2y - 3 = 0 and 3x + 4y - 20 = 0 = have exactly one solution.Reason : The linear equations 2x + 3y - 0 = and 4x + 6y - 18 = 0 = have a unique solution.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Let us first consider the assertion. It says that the linear equations x − 2y − 3 = 0 and 3x + 4y − 20 = 0 have exactly one solution.

Let x − 2y − 3 = 0--- (1) And, 3x + 4y − 20 = 0---(2)

In order to solve these equations, let us multiply the first equation by 3.

3(x − 2y − 3) = 3 × 0

⟹ 3x − 6y − 9 = 0 ----(3)

Subtracting equation 3 from equation 2, we get,

3x + 4y − 20 − (3x − 6y − 9) = 0

Removing the brackets, we get,

3x + 4y − 20 − 3x + 6y + 9 = 0

⟹ 10y − 11 = 0

⟹ y = 11 / 10

Now, in order to find the value of x, substituting the value of y in equation 1, we get,

Thus, the pair of linear equations given possess exactly one solution (unique solution).

Hence, the assertion is correct.

Now, let us consider the reason. It says that the linear equations 2x + 3y − 9 = 0 and 4x + 6y − 8 = 0 have a unique solution.

Let 2x + 3y − 9 = 0---(1)

And, 4x + 6y − 18 = 0---(2)

In order to solve these equations, let us multiply the first equation by 2.

2(2x + 3y − 9) = 2 × 0

⟹4x + 6y − 18=0---(3)

As, equation 2 and 3 are same thus, thus the two linear equations given to us are coincident possessing infinitely many solutions.

Thus, the reason is not correct.

Thus, Assertion is correct but the Reason is incorrect.

Answer

Assertion: The linear equations x - 2y - 3 = 0 and 3x + 4y - 20 = 0 have exactly one solution.
Reason: The linear equations 2x - 3y = 0 and 4x + 6y - 18 = 0 have a unique solution.

Explanation:
To determine if the given Assertion and Reason are true or false, let's analyze each statement separately.

Statement 1: The linear equations x - 2y - 3 = 0 and 3x + 4y - 20 = 0 have exactly one solution.
To find the solution of a system of linear equations, we need to check if the given equations intersect at a single point, which indicates a unique solution.

For the given system of equations:
x - 2y - 3 = 0 ...(1)
3x + 4y - 20 = 0 ...(2)

We can solve these equations simultaneously to find their solution.

Multiplying equation (1) by 3 and equation (2) by 1, we get:
3(x - 2y - 3) = 3(0)
3x - 6y - 9 = 0 ...(3)
3x + 4y - 20 = 0 ...(4)

Now, subtracting equation (3) from equation (4), we eliminate the 'x' term:
(3x + 4y - 20) - (3x - 6y - 9) = 0
3x - 3x + 4y + 6y - 20 + 9 = 0
10y - 11 = 0
10y = 11
y = 11/10

Substituting the value of 'y' into equation (1), we can find the value of 'x':
x - 2(11/10) - 3 = 0
x - 22/10 - 3 = 0
x - 22/10 - 30/10 = 0
x - 52/10 = 0
x - 26/5 = 0
x = 26/5

Therefore, the solution to the given system of equations is x = 26/5 and y = 11/10. This is a unique solution, which means the Assertion is true.

Statement 2: The linear equations 2x - 3y = 0 and 4x + 6y - 18 = 0 have a unique solution.
Similarly, we can solve the given system of equations to determine its solution.

For the system of equations:
2x - 3y = 0 ...(5)
4x + 6y - 18 = 0 ...(6)

Multiplying equation (5) by 2 and equation (6) by 1, we get:
2(2x - 3y) = 2(0)
4x - 6y = 0 ...(7)
4x + 6y - 18 = 0 ...(8)

Now, subtracting equation (7) from equation (8), we eliminate the 'x' term:
(4x

Top Courses for Class 10View all

Science Class 10

Social Studies (SST) Class 10

Mathematics (Maths) Class 10

Past Year Papers for Class 10

English Class 10

Top Courses for Class 10

Top Courses for Class 10

Related Content

About Class 10

Class 10 Study Material

Class 10 All Subjects

NCERT Textbook Solutions

Other Classes

Important Links

Company