In the xy-plane, line l passes through the origin and is perpendicular to the line 3x + 2y = k. k is constant.If the 2 lines intersect at (t, t + 1), what is t?

Im confused for the fact that I dont know the value of k.

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You don't actually need k:

3x+2y=k

You can rewrite this as:

2y=-3x+k , or:

y = -1.5 x + (k/2)

The slope of this line is -1.5 (or -3/2)

Line l is perpendicular to this line, which means it has slope +2/3 (the negative reciprocal of -3/2)

Line l passes through the origin, so it has y-intercept of 0

So, line l has equation:

y = (2/3)x + 0

or just

y = (2/3)x

If the two lines intersect at (t,t+1), then l has to go through the point (t,t+1). That means, we can plug in x=t and y=t+1, and the equation should be true. So:

(t+1)=(2/3)(t)

multiplying both sides by 3:

3t+3 = 2t

1t+3=0

t=-3

**Thus, t=-3.**

PS

You could, from here, if you wanted, use all of this information to figure out k, since you now know that the first line must pass through the point (-3,-2). But you don't actually need k to figure out t.

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