Line l passes through (12,4)and line p is the graph of 4x-3y=4.

Question

Line L passes through (12,4)and line P is the graph of 4x−3y=4.
If L ⊥ P,what is the slope-intercept form equation of L?
Complete the equation of the line in slope-intercept form. If any number in the equation is not an integer, enter it as a fraction in simplest form.

y=

Answer 1

First find the slope of 4x - 3y  =  4; by inspection it is 4/3, but we can also find that by converting to slope-intercept form which might be the focus of the question.

 

4x - 3y             =           4

 

     +3y  -4        = +3y   -4

-------------           ---------

 

4x         -4         =  3y

 

 

i.e.

 

 

3y   =    4x   -    4               '(closer to our usual habit of recognizing slope-intercept form)

 

 

3y   =    4x   -    4

----        -----------

3                  3

 

 

 

y     =    4/3x -  4/3          ' slope intercept form of the given line

 

y     =     m x + b             'the formula so as to communicate to you that m = 4/3

 

Now for the perpendicular line we want m⊥.   A bit of drawing on dot paper reveals that m⊥ = -3/4.

 

 

Next we put the slope into point-slope form:

 

y  -   y1    =      m⊥(x   -   x1)

 

 

y  -    4     =     -3/4(x   -   12)         and that might be the answer.

 

However the question somewhat unclearly probably said the final answer

should be in slope intercept form.

 

Solve for y:

 

y  -    4    =     -3/4x    +   9

    +  4     =                 +  4

----------          ----------------

y             =      -3/4x    + 13

 

 

 

 

But if the focus is to work with slope intercept form, let's go directly to it.

 

y    =    m⊥ x    +    b     'general equation of slope.intercept

 

y    =   -3/4 x    +    b     'sub'd in the perpendicular slope

 

4    =   -3/4(12) +    b     'sub 4 for y, 12 for x, 'cause (x,y)=(12,4)on line

 

 

4    =    -9         +    b     'so now we can find b, the y.intercept

 

+9  =   +9

----      -----------------

 

13  =                      b      'we could have tracked it down from (12,4), but

                                      this is most always easier and should know this.

 

 

So the equation is y = -3/4 x  +  13

 

Answer 2

Line l is perpendicular to line p. This means that line l's slope is the negative reciprocal But before we find l's, we first need to find line p's:

4x - 3y = 4

First we need to isolate y:

-3y = 4 - 4x

Divide both sides by -3:

y = ⁴/₃x - ⁴/₃

 

To find the negative reciprocal, we flip the numerator and denominator, and switch neg ↔ pos. Since line p is positive, line l's slope will be negative. This will give us ^-3/₄

 

We now have a slope, and we have a point on the line--(12, 4). We can thus use the point-slope formula to get line l. We then can isolate y to create the slope-intercept form.

y - y₁ = m(x - x₁)

y - 4 = -3/4(x - 12)

y - 4 = -³/₄x + 9

y = -³/₄x + 13

 

We can check our answer:

4 = -3/4 * 12 + 13

4 = -3/4 * 12 + 13

4 = -9 + 13

4 = 4