Help! Write the equation of the line that is perpendicular to the line y=2x+2 and passes through the point (6, 3).

Question

austinl · Answer

Ok, we need to keep a couple things in mind with this one. The slope of a line that is perpendicular to another is it's negative inverse.

What do you think will be the slope of the perpendicular line in this case?

anonymous · Answer

I really don't understand slope, at all.

austinl · Answer

Slope is $\displaystyle \frac{rise}{run}~or~\frac{\Delta y}{\Delta x}$

Make sense?

A slope of $\displaystyle \frac{5}{2}$ would mean that you would go up five units and right 2 units from your y-intercept.

anonymous · Answer

Got it.

anonymous · Answer

So, how would I find the slope in this case?

austinl · Answer

In this case we are given this equation, which is in slope-intercept form,

$y=2x+2$

It's basic form is $y=mx+b$, where m is slope.

What is the slope of this line?

anonymous · Answer

Oh! 2!

austinl · Answer

Correct, now we need to find the slope of the line that is perpendicular. Which will be the negative reciprocal.

For example, lets say we have a slope of $\displaystyle \frac{a}{b}$.

What would be the negative reciprocal of that?

anonymous · Answer

negative a/b?

anonymous · Answer

Or -a/b?

austinl · Answer

No, a reciprocal of a fraction is when you swap the numerator and the denominator. What do you think now?

anonymous · Answer

b/a?

austinl · Answer

$\displaystyle -\frac{b}{a}$

Now, we have a slope of $\displaystyle 2~or~\frac{2}{1}$

What do you think that will be?

anonymous · Answer

What do I think what will be?

austinl · Answer

The negative reciprocal of the slope from the given problem.

anonymous · Answer

-1/2?

austinl · Answer

Correct!

So the basic structure of our new line will be,

$\displaystyle y=-\frac{1}{2}+b$

Now to find b, the y-intercept, we need to plug in the points given.

$(6,3)\rightarrow(x,y)$

Do you think you could do that? Solve for b with those givens?

anonymous · Answer

Absolutely, however. Where do I put x? After -1/2?

austinl · Answer

$(\color{red}{6},\color{blue}{3})\rightarrow(\color{red}{x},\color{blue}{y})$

$\displaystyle\color{blue}{y}=-\frac{1}{2}\color{red}{x}+b$

Make sense?

anonymous · Answer

Yes. Thank you so much!

austinl · Answer

Very cool! Good job!

austinl · Answer

Ok, to answer your concerns, you PLUG IN THE VALUES into the equation with the +b on the end. And you solve for b and then replace it with it.

anonymous · Answer

Oh, okay. Thanks!