Simple Linear Equations — Assertion Reasons Type Questions

Assertion (A): Solution of $\dfrac{0\cdot5x+2}{1\cdot4x+4} = \dfrac{5}{7} \text{ is } -1\dfrac{5}{7}$.

Reason (R): $ax + b$ = 0, $a \neq 0$, $a, b$ ∈ R is a linear equation involving one variable.

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

Solving,

$\Rightarrow \dfrac{0.5x + 2}{1.4x + 4} = \dfrac{5}{7} \\[1em] \Rightarrow 7(0.5x + 2) = 5(1.4x + 4) \\[1em] \Rightarrow 3.5x + 14 = 7x + 20 \\[1em] \Rightarrow 14 - 20 = 7x - 3.5x \\[1em] \Rightarrow -6 = 3.5x \\[1em] \Rightarrow x = \dfrac{-6}{3.5} = -\dfrac{12}{7} = -1\dfrac{5}{7}.$

So, Assertion (A) is true.

$ax + b = 0$, where $a \neq 0$ and $a, b$ ∈ R, is indeed a linear equation in one variable. So, Reason (R) is true.

Hence, option 3 is the correct option.

Assertion (A): The equation $4(2x - 5)(x - 3) = (4x - 3)(2x + 1)$ is not a linear equation.

Reason (R): An equation in which the greatest exponent of the variable after simplification is one, is called a linear equation in one unknown.

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

Solving,

$\Rightarrow 4(2x - 5)(x - 3) = (4x - 3)(2x + 1) \\[1em] \Rightarrow 4(2x^{2} - 6x - 5x + 15) = 8x^{2} + 4x - 6x - 3 \\[1em] \Rightarrow 8x^{2} - 44x + 60 = 8x^{2} - 2x - 3.$

On simplifying, the terms $8x^2$ cancel out and we get $−44x + 60 = −2x − 3$, in which the greatest exponent of $x$ is one.

So, the given equation is a linear equation and Assertion (A) is false.

An equation in which the greatest exponent of the variable after simplification is one, is called a linear equation in one unknown. So, Reason (R) is true.

Hence, option 2 is the correct option.

Assertion (A): Thrice a number when subtracted from 73, gives 7. The number is 68.

Reason (R): For solving a word problem involving one unknown, we first write the statement in equation form in terms of one unknown and then solve the equation to find the value of the unknown.

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

Let the number be $x$.

Solving,

$\Rightarrow 73 - 3x = 7 \\[1em] \Rightarrow 73 - 7 = 3x \\[1em] \Rightarrow 66 = 3x \\[1em] \Rightarrow x = 22.$

Since the number is 22 and not 68, Assertion (A) is false.

The method described for solving a word problem involving one unknown is correct. So, Reason (R) is true.

Hence, option 2 is the correct option.

Assertion (A): Given $7x - 3 = 3x + 9$

⇒ $7x - 3x$ = 9 + 3 [Transposing -3 to RHS and transposing $3x$ to LHS]

Reason (R): Transposition is a process of taking any term of an equation from one side to the other side with its sign changed.

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

On transposing −3 to R.H.S. it becomes +3, and on transposing $3x$ to L.H.S. it becomes −$3x$.

Solving,

$\Rightarrow 7x - 3 = 3x + 9 \\[1em] \Rightarrow 7x - 3x = 9 + 3.$

So, Assertion (A) is true.

Transposition is indeed the process of taking a term of an equation from one side to the other side with its sign changed. So, Reason (R) is true.

Hence, option 3 is the correct option.