Simultaneous Linear Equations — Multiple Choice Questions

If x = 3, y = k is a solution of the equation 3x - 4y + 7 = 0, then the value of k is

1. 16

2. -16

3. 4

4. -4

Answer

Substituting x = 3 and y = k in 3x - 4y + 7 = 0 we get,

⇒ 3(3) - 4(k) + 7 = 0

⇒ 9 - 4k + 7 = 0

⇒ 4k = 16

⇒ k = 4.

Hence, Option 3 is the correct option.

The solution of the pair of linear equations 2x - y = 5 and 5x - y = 11 is

1. x = -1, y = 2

2. x = 2, y = -1

3. x = 0, y = -5

4. x = $\dfrac{5}{2}$, y = 0

Answer

Given,

2x - y = 5 ......(i)

5x - y = 11 .....(ii)

Subtracting eq. (i) from (ii) we get,

⇒ 5x - y - (2x - y) = 11 - 5

⇒ 5x - y - 2x + y = 6

⇒ 3x = 6

⇒ x = 2.

Substituting value of x in eq. (i),

⇒ 2(2) - y = 5

⇒ 4 - y = 5

⇒ y = 4 - 5

⇒ y = -1.

Hence, Option 2 is the correct option.

If x = a, y = b is the solution of the equations x - y = 2 and x + y = 4, then the values of a and b are respectively,

1. 3 and 5

2. 5 and 3

3. 3 and 1

4. -1 and -3

Answer

Given,

x - y = 2 ......(i)

x + y = 4 ......(ii)

Adding both the equations we get,

⇒ x - y + x + y = 2 + 4

⇒ 2x = 6

⇒ x = 3.

Substituting value of x in (i),

⇒ 3 - y = 2

⇒ y = 3 - 2

⇒ y = 1.

Hence, Option 3 is the correct option.

The solution of the system of equations $\dfrac{4}{x} + 5y = 7 \text{ and } \dfrac{3}{x} + 4y = 5$ is

1. $x = \dfrac{1}{3}, y = -1$

2. $x = -\dfrac{1}{3}, y = 1$

3. x = 3, y = -1

4. x = -3, y = 1

Answer

Given,

$\dfrac{4}{x} + 5y = 7$ .......(i)

$\dfrac{3}{x} + 4y = 5$ .......(ii)

Multiplying eq. (i) by 4 and eq. (ii) by 5 we get,

$\dfrac{16}{x} + 20y = 28$ .......(iii)

$\dfrac{15}{x} + 20y = 25$ .......(iv)

Subtracting eq. (iv) from (iii) we get,

$\Rightarrow \dfrac{16}{x} + 20y - \Big(\dfrac{15}{x} + 20y\Big) = 28 - 25 \\[1em] \Rightarrow \dfrac{16}{x} - \dfrac{15}{x} = 3 \\[1em] \Rightarrow \dfrac{1}{x} = 3 \\[1em] \Rightarrow x = \dfrac{1}{3}.$

Substituting value of x in (ii),

$\Rightarrow \dfrac{3}{\dfrac{1}{3}} + 4y = 5 \\[1em] \Rightarrow 9 + 4y = 5 \\[1em] \Rightarrow 4y = 5 - 9 \\[1em] \Rightarrow 4y = -4 \\[1em] \Rightarrow y = -1.$

Hence, Option 1 is the correct option.

A pair of linear equations which has a unique solution x = 2, y = -3 is

1. x + y = -1
   2x - 3y = -5

2. 2x + 5y = -11
   4x + 10y = -22

3. 2x - y = 1
   3x + 2y = 0

4. x - 4y - 14 = 0
   5x - y - 13 = 0

Answer

Substituting x = 2, y = -3 in x - 4y - 14 = 0 we get,

= 2 - 4(-3) - 14

= 2 + 12 - 14

= 0.

Substituting x = 2, y = -3 in 5x - y - 13 = 0 we get,

= 5(2) - (-3) - 13

= 10 + 3 - 13

= 13 - 13

= 0.

Hence, Option 4 is the correct option.

Consider the following two statements.

Statement 1: A solution to linear equation 5x - 2y = 1 is x = 3, y = 7.

Statement 2: The linear equation 5x - 2y = 1 has a unique solution.

Which of the following is valid?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and Statement 2 is false.

4. Statement 1 is false, and Statement 2 is true.

Answer

If we substitute x = 3 and y = 7 into the equation 5x - 2y = 1, we get

Taking L.H.S.

⇒ 5.(3) - 2.(7)

⇒ 15 - 14

⇒ 1.

Since, L.H.S. = R.H.S.

This confirms that (3, 7) is indeed a solution.

∴ Statement 1 is true.

A single linear equation with two variables (like x and y) represents a line in a coordinate plane.

A line has infinitely many points on it, and each point corresponds to a solution of the equation.

Therefore, a single linear equation has infinitely many solutions, not just one.

∴ Statement 2 is false.

∴ Statement 1 is true, and Statement 2 is false.

Hence, option 3 is the correct option.