If $6 x + 5 y - 7 = 0$ and $2 p x + 5 y + 1 = 0$ are parallel lines, find the value of $p$ .

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Solution

Step1: Calculation of slope of line $6 x + 5 y - 7 = 0$ .
The slope-intercept form of the equation of a line is given by $y = m x + c$ , where $m$ is the slope and $c$ is the $Y$ -intercept.
Simplify the equation $6 x + 5 y - 7 = 0$ to convert it into slope-intercept form.
$6 x + 5 y - 7 = 0 \Rightarrow 5 y = 7 - 6 x (a d d i n g 7 - 6 x t o b o t h s i d e s) ∴ y = - \frac{6}{5} x + \frac{7}{5} (d i v i d i n g b o t h s i d e s b y 5) - e q u a t i o n (1)$
Comparing equation (1) with equation $y = m x + c$ , we get the slope of the line $6 x + 5 y - 7 = 0$ as $m_{1} = - \frac{6}{5}$ .
Step2: Calculation of slope of line $2 p x + 5 y + 1 = 0$ .
Simplify the equation $2 p x + 5 y + 1 = 0$ to convert it into slope-intercept form.
$2 p x + 5 y + 1 = 0 \Rightarrow 5 y = - 1 - 2 p x (s u b t r a c t i n g 2 p x + 1 f r o m b o t h s i d e s) ∴ y = - \frac{2 p}{5} x - \frac{1}{5} (d i v i d i n g b o t h s i d e s b y 5) - e q u a t i o n (2)$
Comparing equation (2) with equation $y = m x + c$ , we get the slope of the line $2 p x + 5 y + 1 = 0$ as $m_{2} = - \frac{2 p}{5}$ .
Step3: Calculation of the value of $p$ .
Since, the lines $6 x + 5 y - 7 = 0$ and $2 p x + 5 y + 1 = 0$ are parallel, their slopes will be equal.
Equate the slopes $m_{1} = - \frac{6}{5}$ and $m_{2} = - \frac{2 p}{5}$ and solve for $p$ .
$- \frac{2 p}{5} = - \frac{6}{5} \Rightarrow 2 p = 6 (m u l t i p l y i n g b o t h s i d e s b y - 5) ∴ p = 3 (d i v i d i n g b o t h s i d e s b y 2)$
Hence, the value of $p$ is $3$ .

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