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

Given -

#2y-x=7#

Rewrite it as to suit our convenience.

#-x+2y=7#

The slope of the line is given by the formula

#m=(-a)/b#

Where -

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

The slope of the given line #m_1=(-a)/b=(-(-1))/2=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+2y=14#

For a line to be perpendicular, the product of the slopes of the two lines must be equal to #m_1xxm_2=-1#

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

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

#2x-1y=7#

Step 2

Change the sign of the #y# coefficient

#2x+1y=7#

Now write the equation

The equation of the perpendicular line is -

#2x+y=7#

Its slope is #m_2=2/1=2#

Find the product to acertain the correctness of your answer

#m_1 xx m_2=(-1)/2xx2=-1#

Hence the equation of the perpendicular line is correct.