# Thread: Find an equation for the line

1. ## Find an equation for the line

Hi! I am trying to study for my final and am really stuck on a problem. I'm not really sure how to start this one off--

A line goes through the point (p,q) where p satisfies (1/p - 2/3) over (1/4-1/3) = 7 and q is the imaginary part of 2-3i over 4+5i. The slope of the line is the extraneous solution to x + the square root of (x+1)=5. Find an equation for the line (No caculator approximations. Do exact work)

Thank you in advance for your help... this is kind of a tough one!

Ashley

firstly p:

$\displaystyle \frac{\frac{1}{p}-\frac{2}{3}}{\frac{1}{4}-\frac{1}{3}}=7$

$\displaystyle \frac{1}{p}-\frac{2}{3} = 7(\frac{1}{4}-\frac{1}{3})$

$\displaystyle \frac{1}{p}-\frac{2}{3} = 7(\frac{3}{12}-\frac{4}{12})$

$\displaystyle \frac{1}{p}-\frac{2}{3} = \frac{-7}{12}$

$\displaystyle \frac{1}{p} = \frac{-7}{12}+\frac{2}{3}$

$\displaystyle \frac{1}{p} = \frac{-7}{12}+\frac{8}{12}$

$\displaystyle \frac{1}{p} = \frac{1}{12}$

$\displaystyle p=12$

now q:

$\displaystyle q=Im(\frac{2-3i}{4+5i})$

we need to find the imaginary part of $\displaystyle \frac{2-3i}{4+5i}$ multiply through by the complex conjugate.

$\displaystyle \frac{2-3i}{4+5i}\times\frac{4-5i}{4-5i}$

$\displaystyle \frac{8-10i-12i+15i^2}{16-25i^2}$

$\displaystyle \frac{8-22i-15}{16+25}$

$\displaystyle \frac{-7-22i}{41}$

therefore $\displaystyle q= Im(\frac{-7-22i}{41}) = \frac{-22i}{41}$

3. Woah.... I never would have gotten that far! Thank you so much for getting me started. This may sound like a dumb question, but when it says find an equation for the line- does that mean in y=mx+b formula. I just don't understand where p and q fit in or how you can have a slope that is an extraneous solution... this final is going to get me

Thanks for the help!
Ashley

4. Without seeing the actual question or knowing the context it was asked in I would assume that the line does have the model $\displaystyle y=mx+b$

And we now have the following information (assuming its correct!)

$\displaystyle (p,q) = (x,y) = (12, \frac{-22i}{41})$ and $\displaystyle m= 7$

Substituting these into your model we can solve for b and then have the equation.

$\displaystyle y=mx+b$

with $\displaystyle (p,q)=(x,y) =(12, \frac{-22i}{41})$ and $\displaystyle m= 7$

gives $\displaystyle \frac{-22i}{41}=7\times 12+b$

gives $\displaystyle \frac{-22i}{41}=84+b$

gives $\displaystyle b = \frac{-22i}{41}-84$

so the linear model is

$\displaystyle y=7x+\left(\frac{-22i}{41}-84\right)$

or in context of the problem

$\displaystyle q=7p+\left(\frac{-22i}{41}-84\right)$

5. You have no idea how much that helps!!! I think I can make myself understand this now! You have done a great job of explaining this... maybe this final wont kill me after all haha

Thank you!!!!!
Ashley