Which of the following represents a linear equation in two variables?

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$3 p^{\frac{3}{2}} + 7 q \sqrt[]{p} = 8 \sqrt[]{p}$

$\frac{3 x^{2}}{4 y} = \frac{7 x^{2}}{5}$

$\frac{9 a}{2 b} = \frac{11 b}{13 a}$

15x + 17 = 0

answer is A.

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Detailed Solution

Consider alternative 1).

$3 p^{\frac{3}{2}} + 7 q \sqrt[]{p} = 8 \sqrt[]{p}$

$\Rightarrow 3 p \sqrt[]{p} + 7 q \sqrt[]{p} = 8 \sqrt[]{p}$

$\Rightarrow (3 p + 7 q) \sqrt[]{p} = 8 \sqrt[]{p}$

$\Rightarrow 3 p + 7 q = 8$

The equation 3p + 7q = 8 is a first degree equation and consists of two variables, p and q.

Thus, it is a linear equation in two variables.

Consider alternative 2).

$\frac{3 x^{2}}{4 y} = \frac{7 x^{2}}{5}$

$\Rightarrow \frac{3}{4 y} = \frac{7}{5}$

$\Rightarrow 28 y - 15 = 0$

Since the equation 28y − 15 = 0 consists of only one variable, y, it is not a linear equation in two variables.

Consider alternative 3).

$\frac{9 a}{2 b} = \frac{11 b}{13 a}$

$\Rightarrow 9 a \times 13 a = 11 b \times 2 b$

$\Rightarrow 117 a^{2} - 22 b^{2} = 0$

The equation $117 a^{2} - 22 b^{2} = 0$ consists of two variables, a and b, but its degree is 2.

Hence, it is not a linear equation in two variables.

Consider alternative 4).

The equation 15x + 17 = 0 consists of only one variable x, so it is not a linear equation in two variables.

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