# How do you find the slope of the line parallel to and perpendicular to 2y-x=7?

Aug 5, 2017

The equation of the line parallel to the given line is -

$- x + 2 y = 14$
The equation of the perpendicular line is -
$2 x + y = 7$

#### Explanation:

Given -

$2 y - x = 7$

Rewrite it as to suit our convenience.

$- x + 2 y = 7$

The slope of the line is given by the formula

$m = \frac{- a}{b}$

Where -

$a -$ is the coefficient of x
$b -$ is the coefficient of y

The slope of the given line ${m}_{1} = \frac{- a}{b} = \frac{- \left(- 1\right)}{2} = \frac{1}{2}$

For a line to be parallel, it also must have the same slope.

So it is enough if you change the value of the Constant term.

Let us replace 7 with 14. There is no hard and fast rule in assigning any other value. A line can have any number of parallel lines.

The equation of the line parallel to the given line is -

$- x + 2 y = 14$

For a line to be perpendicular, the product of the slopes of the two lines must be equal to ${m}_{1} \times {m}_{2} = - 1$

There is short cut to find the equation of a perpendicular line.

Step 1
Interchange the coefficients of $x$ and $y$

$2 x - 1 y = 7$

Step 2

Change the sign of the $y$ coefficient

$2 x + 1 y = 7$

Now write the equation

The equation of the perpendicular line is -

$2 x + y = 7$

Its slope is ${m}_{2} = \frac{2}{1} = 2$

${m}_{1} \times {m}_{2} = \frac{- 1}{2} \times 2 = - 1$