Simultaneous Linear Equations — Assertion-Reason Type Questions

Assertion(A): If 8x + 7y = 37 and 7x + 8y = 38, then x = -2, y = 3.

Reason(R): ax + by = c and bx + ay = d is not simultaneous linear equations in two variables.

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Given,

Equations: 8x + 7y = 37 and 7x + 8y = 38

⇒ 8x + 7y = 37

⇒ 8x = 37 - 7y

⇒ x = $\dfrac{37 - 7y}{8}$     .....(1)

Substituting value of x from equation (1) in 7x + 8y = 38, we get :

$\Rightarrow 7\Big(\dfrac{37 - 7y}{8}\Big) + 8y = 38 \\[1em] \Rightarrow \dfrac{259 - 49y}{8} + 8y = 38 \\[1em] \Rightarrow \dfrac{259 - 49y + 64y}{8} = 38\\[1em] \Rightarrow 259 + 15y = 38 \times 8 \\[1em] \Rightarrow 259 + 15y = 304 \\[1em] \Rightarrow 15y = 304 - 259 \\[1em] \Rightarrow 15y = 45\\[1em] \Rightarrow y = \dfrac{45}{15} = 3.$

Substituting value of y in equation (1), we get :

$\Rightarrow x = \dfrac{37 - 7y}{8} \\[1em] \Rightarrow x = \dfrac{37 - 7 \times 3}{8} \\[1em] \Rightarrow x = \dfrac{37 - 21}{8} \\[1em] \Rightarrow x = \dfrac{16}{8} = 2.$

x = 2 and y = 3.

∴ Assertion (A) is false.

⇒ ax + by = c and bx + ay = d are simultaneous linear equations in two variables x and y.

∴ Reason (R) is false.

Hence, option 4 is the correct option.

Assertion(A): $\dfrac{2}{m} + \dfrac{3}{m} = 0 \text{ and } \dfrac{2}{3m} + \dfrac{2}{n} = \dfrac{1}{6}$ is a pair of simultaneous linear equations.

Reason(R): An equation of the form ax + by + c = 0, a ≠ 0, b ≠ 0 is called linear equations in two variables.

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Given,

Equations: $\dfrac{2}{m} + \dfrac{3}{m} = 0 \text{ and } \dfrac{2}{3m} + \dfrac{2}{n} = \dfrac{1}{6}$

$\Rightarrow \dfrac{2}{m} + \dfrac{3}{m} = 0 \\[1em] \Rightarrow \dfrac{5}{m} = 0 \\[1em] \Rightarrow 5 = 0 \times m \\[1em] \Rightarrow 5 \ne 0 .$

Since 5 is not equal to 0, this equation has no solution for m. We cannot find values for m and n that satisfy both equations simultaneously.

$\Rightarrow \dfrac{2}{3m} + \dfrac{2}{n} = \dfrac{1}{6}$ Again, this is not linear in variables m and n, because the variables are in denominators.

Assertion (A) is false.

An equation of the form ax + by + c = 0, a ≠ 0, b ≠ 0, is called a linear equation in two variables.

Reason(R) is true.

A is false, R is true

Hence, option 2 is the correct option.

Assertion(A): A pair of linear equations in two variables cannot have more than one solution.

Reason(R): If we solve a pair of linear equations in two variables, first by elimination method and then by cross multiplication method, then in some cases the two solutions so obtained may be different.

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

A pair of linear equations in two variables cannot have more than one solution. When the pair is consistent and independent, it has exactly one unique solution.

∴ Assertion (A) is true.

If we solve a pair of linear equations in two variables, first by elimination method and then by cross multiplication method, then in some cases the two solutions so obtained may be different.

This is false statement because both methods give same final answers.

∴ Reason(R) is false.

A is true, R is false.

Hence, option 1 is the correct option.