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Mathematics

Write true (T) or false (F) :

(i) The smaller of the two numbers whose sum is 26 and the difference is 6 is 10.

(ii) The solutions of an equation in variable z are the values of z for which L.H.S. = R.H.S.

(iii) Sum of two consecutive even natural numbers is 24. This statement can be represented by the equation x + (2x + 2) = 24.

(iv) 3x + 4 > 12 is a linear equation in one variable.

(v) x = 4 is a solution of the equation 5x2−1=4x−7\dfrac{5x}{2} - 1 = 4x - 7.

Linear Eqns One Variable

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Answer

(i) True
Reason —

Let the numbers be x and y.

Sum = x + y = 26

Difference = x - y = 6.

Adding the equations:

x + y + x - y = 26 + 6

⇒ 2x = 32

⇒ x = 322\dfrac{32}{2}

⇒ x = 16.

Substituting the value of x in (x + y = 26) we get:

x + y = 26

⇒ 16 + y = 26

⇒ y = 26 - 16 \quad [Transposing +16 to RHS]

⇒ y = 10.

The smaller number is 10.

(ii) True
Reason — The statement is the fundamental definition of a solution. A value is only a "root" or "solution" if it makes the Left-Hand Side (L.H.S.) equal to the Right-Hand Side (R.H.S.).

(iii) False
Reason —

Consecutive even numbers are represented as x and x + 2. The equation should be:

x + (x + 2) = 24.

The expression 2x + 2 represents twice the first number plus two, which is not the definition of the next consecutive even number.

(iv) False
Reason — The symbol > makes this an inequality, not an equation. A linear equation must have an equal sign (=).

(v) True
Reason —

Given equation:

5x2−1=4x−7\dfrac{5x}{2} - 1 = 4x - 7

Substitute x = 4 into both sides:

∴5x2−1=4x−7⇒5(4)2−1=4(4)−7⇒202−1=16−7⇒20−22=9⇒182=9⇒9=9\therefore \dfrac{5x}{2} - 1 = 4x - 7 \\[1em] \Rightarrow \dfrac{5(4)}{2} - 1 = 4(4) - 7 \\[1em] \Rightarrow \dfrac{20}{2} - 1 = 16 - 7 \\[1em] \Rightarrow \dfrac{20 - 2}{2} = 9 \\[1em] \Rightarrow \dfrac{18}{2} = 9 \\[1em] \Rightarrow 9 = 9

Since, L.H.S. = R.H.S., x = 4 is the correct solution.

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