SAT Question

[UnsungZeros]

I was trying to help out my friend today with an SAT practice problem, but it ended up stumping me just as much as it stumped her. If anyone can solve it and provide a detailed explanation on how you arrived at that answer, I would greatly appreciate it. Here's the problem:

In an x-y plane, a line L passes through the origin and is perpendicular to another line whose equation is 4x + y = k, where k is a constant. If the lines intersect at (t, t+1), what is the value of t?

a) - 4/3
b) -5/4
c) 3/4
d) 5/4
e) 4/3

 

[Jeff Moote]

1) set the two equations for the lines in y = ax + b format

y = -4x + k
y = x/4

2) Substitute the intersection into the first one to get k

t + 1 = -4t + k
k = 5t + 1

3) Solve the system of equations for t

-4t + 5t + 1 = t/4
3t = -4
t = -4/3


so the answer is (a)

 

[Brad Barker]

this is a bit of a silly question, what with that "k" and all...

anyway, here are the steps:

1) use point-slope form to find the equation of the line in question

(that is, y-y_0 = m (x - x_o) ).

i) the author tells you that the line passes through the origin
ii) the author gives you an equation of a line that is perpindicular to this line.

that should give you (i) a point and (ii) the slope...

2) plug in "t" for x and "t+1" for y in the equation you just solved.

you can solve for "t" that way.

note that you don't even have to worry about the line that is perpindicular to L anymore!

would have been a better question to solve for k. (which i got to be -17/3.)


and, for comparison's sake, i got t = -4/3.


hopefully i got it right. i'm a math major, after all. (although i did better on the english section of the sat's myself. )

 

[Brad Barker]

1) set the two equations for the lines in y = ax + b format

y = -4x + k
y = x/4

2) Substitute the intersection into the first one to get k

t + 1 = -4t + k
k = 5t + 1

3) Solve the system of equations for t

-4t + 5t + 1 = t/4
3t = -4
t = -4/3


so the answer is (a)

the way i did it, you don't have to worry about two simultaneous equations.

 

[UnsungZeros]

Thanks a lot you guys.

 

[Brad Barker]

Thanks a lot you guys.

whose way did you like better?

 

[UnsungZeros]

MontyP explained it first and it was cleaner, so he wins.

 

[Jeff Moote]

the way i did it, you don't have to worry about two simultaneous equations.

Very true... I didn't really think about the most efficient way to do it. I almost always throw linear systems together and solve since I'm used to having like 20 unknowns and 20 equations.

whose way did you like better?

Yeah... I'm curious to know which way is easier to comprehend for a non math major / engineering student.

edit: you posted the answer to that while I was typing

 

[Brad Barker]

MontyP explained it first and it was cleaner, so he wins.

it's a popularity contest!