Simple Equations — Assertion-Reason Questions

Assertion (A): The equation $3x − 4 = 2 + x$ can also be written as $3x − x = 2 + 4$.

Reason (R): Any term of an equation can be shifted to the other side with change in its sign.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

3. Assertion (A) is true but Reason (R) is false.

4. Assertion (A) is false but Reason (R) is true.

Answer

In the equation $3x − 4 = 2 + x$, on transposing −4 to R.H.S. it becomes +4 and on transposing $x$ to L.H.S. it becomes −$x$.

$\Rightarrow 3x - 4 = 2 + x \\[1em] \Rightarrow 3x - x = 2 + 4.$

So, Assertion (A) is true.

Any term of an equation can indeed be shifted to the other side with change in its sign. This is the rule of transposition, which is exactly the rule used above. So, Reason (R) is true and it is the correct explanation of Assertion (A).

Hence, option 1 is the correct option.

Assertion (A): $x$ = 2 is the solution of the equation $\dfrac{6x}{2} + 3 = 3x$

Reason (R): The solution of the equation satisfies the equation.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

3. Assertion (A) is true but Reason (R) is false.

4. Assertion (A) is false but Reason (R) is true.

Answer

Putting $x$ = 2 in the given equation,

$\text{L.H.S.} = \dfrac{6 \times 2}{2} + 3 = 6 + 3 = 9 \\[1em] \text{R.H.S.} = 3 \times 2 = 6.$

Since L.H.S. ≠ R.H.S., $x$ = 2 is not the solution of the given equation.

In fact, on simplifying the equation we get $3x + 3 = 3x$, i.e., 3 = 0, which is not possible. So, the equation has no solution. Assertion (A) is false.

The solution (root) of an equation is the value of the variable which makes L.H.S. = R.H.S., i.e., it satisfies the equation. So, Reason (R) is true.

Hence, option 4 is the correct option.