Math Problem--stumped, please help?

Question

The equation of line m is 8x-14y+3=0

a) for what value of k is the graph of kx-7y+10=0 parallel to line m?

b) what is k if the graphs of m and kx-7y+10=0 are perpendicular?

Any help is appreciated. If you don't feel like giving me the answer, a jump-start to it would also help. Best answer goes to the person who provides the most information on how to solve the problem. Thanks.

Answer 1

First thing you always have to to in linear equations in standard form is to convert them into the slope y-int. form. (this is just a complicated way of say isolate 'y' on one side of the equation)

8x-14y+3 = 0

8x+3=14y                       move y to the other side

8/14 x + 3/14 = y            divide both sides by 14 to get 'y' alone

4/7 x + 3/14 = y              simplify

y = 4/7 x + 3/14

okay so 4/7 is the slope and 3/14 is the y-intercept

Now do the same thing for the other equation

kx-7y+10 = 0                

kx + 10 = 7y                move y to the other side

k/7 x + 10/7 = y           divide both sides by 7 to get 'y' alone

y = k/7 x + 10/7

Now... in parellel lines, the the slope of both lines are equal (other wise they won't be parallel!)

Therefore:

4/7 = k/7         The slope of both lines are equal

k=4                 The '7' cancels since it is on both sides of the equation.

So part a) is solved: k=4

Part b) is the same idea:

The only difference here is that the line 'y = 4/7 x + 3/14' is perpendicular to 'y = k/7 x + 10/7'.

In perpendicular lines, the slopes of both lines are "negative reciprocals" of one another. That means if you take one slope, flip the fraction, and reverse the sign; you get the perpendicular slope.

So, the perpendicular slope of 4/7 is:

-7/4            Remember, flip the fraction and reverse the sign

Now it becomes similar to the last question

k/7 = -7/4         (basically this equation is saying that k/7 is perpendicular to 4/7, the original equation of the line)

k = -7/4 * 7       Isolate for k

k = -49/4          Simplify

Therefore k = -49/4 when the equations are parallel.

Hope that helps :)