A student draws 35 lines in the XY plane, none of which are vertical

Yes , it will be 0 or negative...
You are correct. Doesn't effect the answer so didn't mention it.

There are 8 categories of lines

(i) positive slope, positive y-intercept - x

(ii) positive slope, passing through origin - y

(iii) positive slope, negative y-intercept - z

(iv) negative slope & positive y-intercept - a

(v) negative slope & passing through origin - b

(vi) negative slope & negative y-intercept - c

(vii) parallel to x - axis & with positive y-intercept - p

(viii) parallel to x - axis & with negative y-intercept - q

Lines x, y, z, a, p pass through Quadrant I, we are asked to find (b + c + q)

Given x + y + z + a + b + c + p + q = 35

x + y + z = 15

x = 15/3 = 5, hence y + z = 10

y + z + b + c + q = 23

hence b + c + q = 23 - 10 = 13


Answer C.


Thanks,
GyM

Total 35 lines
23 of those lines have y intercept \(\leq\) 0
5 of those lines have y intercept \(>\) 0
what about the remaining seven lines ?
if they don't have a zero y intercept, or negative y intercept or positive y intercept , what kind of intercept do they have ?

Is it possible to have a line with NO y intercept? Isn't it the same as having a zero Y intercept?
if those 7 lines pass through the origin , then y intercept should be 0 and we should have 23 +7 lines having y intercept \(\leq\)0

Please account for those 7 lines .
Thanks.

Here is my approach.
Line contains no point in Quadrant I means Line has Y-intercept ≤ 0 AND negative slope
Of the 15 lines with positive slope, 1/3 also have a positive Y-intercept --> remaining 2/3 * 15 = 10 lines have Y-intercept ≤ 0
Of the 23 lines have Y-intercept ≤ 0, 10 lines have Y-intercept ≤ 0 AND positive slope --> remaining 23-10 = 13 lines Y-intercept ≤ 0 and negative slope
Answer C.

That can be easily understood , can you help me understand , about the y intercept of the remaining 7 lines , positive 5 y intercept ,zero and less = 23 Y intercept , that makes 28, what about the y intercept of the remaining 7 lines ?

5 of the 15 lines which have a positive slope have y intercept \(>\) 0, not 5 of the total 35 lines have y intercept \(>\) 0

In general, it is possible to have lines with NO y-intercept. Those lines never intersect the y-axis, or in other words, those lines are vertical (parallel to the y-axis). It is not the same as having a zero y-intercept. If a line is vertical and has a zero y-intercept, it is y-axis.
In this question, the stem clearly states that "A student draws 35 lines in the XY plane, none of which are vertical..." so it's not possible to have lines with NO y-intercept in this case.

Hope it's clear.

I think you did not understand my question correctly.

May be I am missing something
total 35 lines , 15 have positive slope and 20 have 0 or negative slope ( 20 + 15 = 35)
out of 15 having positive slope, 5 have positive Y intercept
So 10 having positive slope have 0 or negative Y intercept
So total 15 lines have positive slope = 5 lines have positive y intercept and 10 lines have , zero or negative y intercept.
also we know total 23 lines have y intercept \(\leq\) 0

so 23 + 5 = 28 , what kind of y intercept do the remaining 7 lines have ?
hope now my question is clear.

Edit:
Thank you , understood it , the answer I was looking for was this :

Those 7 lines have positive y intercept but negative slope .

So here is the break up

Total 12 have positive y intercept out of which 5 have positive slope and 7 have zero or negative slope
total 23 have zero or negative Y intercept out of which 10 have positive slope and 13 have 0 or negative slope .

total 35 lines are now accounted for .

Drawing a 2x2 table may help you see it easier

Matrix method was easier here

---------------------(+ive Slope)---------(-ive slope)-----Total
+ive Y Intercept----------5----------------------7------------12
-ive Y Intercept----------10--------------------13-----------23
----------------------------15--------------------20-----------35

-ive slope and -ive Y Intercept will be responsible for answering how many lines contain no point in Quadrant I
13

C

out of 15 lines with posivitive slop, 5 lines have positive intercept on Y, 10 lines has negative or 0.

so out of the 23 lines with a Y-intercept less than or equal to zero,
there are must contains 10 positive slop which mentioned in positive slop.
and then 23-10=13 left with negative slop.
those 13 line with negative slop and Y intercept <=0 must not cross the quadrant I.

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