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# full question below: In the xy-plane, line L passes through the origin and is perpendicular to the line 4x+y=k

In the xy-plane, line L passes through the origin and is perpendicular to the line 4x+y=k, where k is a constant. If the two lines intersect at the point (t,t+1), that is the value of t?

I tried a long time to figure this out. It is from a textbook so I don't think there is a problem with the question.

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First off, we need the slope of the line L.

Since L is perpendicular to 4x+y=k, we find the latter's slope: m = -B/A (since the equation's in the form Ax + By = C).

m = -B/A = -4/1 = -4

L's slope should be the negative reciprocal of m: 1/4.

We know it's y-intercept is 0, since it goes through the origin (0,0).

So L's slope-intercept equation is y = (1/4)·x +0, or simply y = (1/4)·x

To solve for t, we put the point (t,t+1) into either equation. But since the second equation (4x + y = k) introduces another unneeded variable, use the one we found for L:

y = (1/4)·x

Substitute:

t+1 = (1/4)·t

t = -1 + t/4

(3/4)·t = -1

t = -4/3

t+1 = -1/3

Source(s): Me, a math teacher.
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• First rewrite the perpendicular line in slope intercept form:

y = -4x + k

So the line L's slope must be the negative reciprocal of -4 and since it passes thru the origin it's y-intercept (b) = 0

So the equation of line L is y = (1/4)x

To find t, plug (t, t+1) in to the equation y = (1/4)x

t+1 = (1/4)t

1 = (-3/4)t

-4/3 = t

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• First, it's perpendicular to 4x+y=k. That means its slope is the negative reciprocal of 4x+y=k. I'll figure that out first.

4x+y=k

y=-4x+k

Slope is -4

Slope of desired line is +1/4

My desired line looks like this:

y=1/4x +b

My line passes through (0,0)

Therefore 0=(1/4)0+b, b=0

My desired line is y=1/4x

The slope of my line between(t,t+1) and (0,0) is 1/4

Therefore [(t+1)-0]/[t-0] = 1/4

(t+1)/t=1/4

4t+4=t

3t = -4

t = -4/3

The value of t is -4/3

Verification: y=1/4x

(-1/3) = 1/4(-4/3)

-1/3 = -1/3

Thank you for an extremely interesting problem. If I knew how to give you stars or 10 points from this stupid computer, I would!

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• The original line is y = -4x+k, so has slope -4. That means L has slope 1/4. Two points on L are (0,0) and (t,t+1), so they must give the same slope: (t+1-0)/(t-0)=1/4. Now just solve for t.

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