slope with graph stuff

Question

Which equation describes a line that passes through  and is perpendicular to the line described by ?

a)

b)

c)

d)

Answer 1

Hi Rayna,

You must approach problems like this one methodically. At your level (I'll explain that in a moment), you want to use the tools you already know. So, in this problem, what major information are you given? 1) The desired line passes through (6, -8) and it is perpendicular to the line of 4x - 2y = 6 . What can you work on first? NOT that it passes through (6, -8): an infinite number of different lines pass through any point! So, on the "perpendicular to the line of 4x - 2y = 6 : Now, what do you need to do to get a slope out of that? You need to know the slope OF 4x - 2y = 6. So transform that into "slope-intercept" form:

-2y = 6 - 4x

y = 3 + 2x

y= 2x + 3

OK, so this line has a slope of +2. You should know by now, that if line A has slope m, that the perpendicular line B has slope -1/m . In short, your desired line has slope -1/(2) = -(1/2) .

So your desired line has form:

y = -(1/2)x + b -- but, you don't know what the intercept b should be!

But, you do know that the line has to contain the point (6, -8): that means that that point must satisfy the equality of the line equation. So plug in that (x,y) pair:

-8 = -(1/2)6 + b

-8 = -3 + b

b = -5

Can you assemble the proper equation now? You have figured out all the pieces (m and b) .

Now, you should look back on the steps necessary to analyze and utilize the initial information in the problem. You will see this type of problem many more times, so it's useful to get real familiar with the solution process.

By the way, I said "at your level" above in the setup. By which I meant, when you get very well acquainted with graphs and rotational transformations of graphs, you may be able to skip the calculation of the (given) line into standard form, and write directly the equivalent (open) form of the target line. For a rotation by 90 degrees, you can: 1) substitute x with -y (as a multiplicand) AND substitute y with x. Drop them into the format of the (given) line, that is, make:

(original =) 4(x) - 2(y) = 6

(rotated =) 4(-y) - 2(x) = c , where c is unknown; drop the given point into this equation to determine c:

4( - -8) - 2(6) = c ; 32 - 12 = c ; c=20 (But, you might reduce this equation to: -2y - x = 10 )

Let's check that out, to make sure it gives the same slope-intercept form answer:

-4y - 2x = 20

4y = -20 - (2/4)x

y = -(1/2)x - 5

Yep, got the same answer!

Hope this helps!

-- Cheers, Mr. d.

Answer 2

Equation a

Your choices are in Slope Intercept Form

Put your line in Slope Intercept Form by solving for y

Perpendicular lines have inverse slopes with opposite signs.

Use the coordinates of the point given to find your perpendicular line

4x - 2y = 6

Put this in Slope Intercept form

-2y = -4x + 6

Divide both sides by negative 2

y = 2x -3

The slope m is 2

The negative inverse of 2 is -1/2 and the slope of the perpendicular line.

m = -1/2 rules out choices c and d

You have only choice a and b to work with. Test a and b with the coordinates (6, -8)

a) y = (-1/2)x - 5

-8 = (-1/2)6 - 5

-8 = -3 - 5

-8 = -8

Choice a satisfies the requirements

b) y = (-1/2)x - 3

-8 = (-1/2)6 - 3

-8 = -3 - 3

-8 ≠ -6

Choice b does not satisfy the requirements

I hope you find this useful.