Line M has a y-intercept of –4, and its slope must be an integer-multi

Let m = slope of Line M
Using the y-intercept and the given points, we can find that \(-\frac{2}{5} < m < \frac{3}{4}\)
Since we need to think in multiples of \(\frac{1}{7}\), let's convert this to \(-\frac{14}{35} < m < \frac{21}{28}\)

Multiplying \(\frac{1}{7}\) by the consecutive integers between -2 and 5, inclusive, will give us values that fall in this range.
The correct answer is C.

Please elaborate the last sentence....how can values faill in the range after multiplying 1/7 between -2 and 5.

Hi Sterling,

Thanks for your answer.Could be please elaborate it ?

Regards

Trying to make sense of it :

-14/35 < m < 21/28

so m must be less than 21/28
20/28 = 5/7 so we can take values from 20/28 or 5/7 to 0
ie 5/7, 4/7, 3/7, 2/7, 1/7 and 0 (6 values)

now m must be > -14/35 ie -2/5
take m as -10/35 ie - 2/7
so two values -2/7 and -1/7

total values = 6 + 2 (8) (C)

But if the slope must be a integer multiple of 1/7 then how can there be 8 slopes? this would mean that slope can only be +/- 1,2,3 and so on.. Can anyone resolve this?

I am confused here too. I got to the point of -2/5 < m < 3/4. After that I couldn't figure how to get integer multiple of 1/7.

what is meant by 'integer multiple' of 1/7? does that mean multiples such as 1 2 3 4, or -1, -2, -3, -4 or something else? Thanks

Step 1.

Find range for possible slope values i.e., -0.4(-2/5) to 0.75(3/4)

Step 2.

We know from question stems that the limits of the above range are not included in our target values and that the values of slope have to be multiples of 0.14(1/7). Hence, only possible values are

-0.28, -0.14, 0, 0.14, 0.18, 0.42, 0.56 and 0.70

a total of 8 values.

Line M has y-intercept = -4, which means one of the points is (0,-4),

And it is given that the slope m=n \frac{1}{7}, where n is an integer.
Therefore the equation of the line M is (y+4)=(n/7)(x-0)
This can be simplified for n, n=7(y+4)/x

Given that the line is below (4,-1) and above (5,-6) implies that the values of n should lie between -14/5 and 21/4 or -56/20 and 105/20


Since n is an integer, there are only 8 possible values -> -40/20, -20/20, 0/20, 20/20, 40/20, 60/20, 80/20 and 100/20

Hence Answer C

SO Close. Completely Forgot the Horizontal Line with Slope = 0 at Y = -(4).

I see lots of others did as well.


GOAL: Find the UPPER and LOWER Boundaries of the Possible Slopes of Lines that could "FIT" within the Points Given.



(1st) The Line must Pass ABOVE the Point (5 , -6) and Pass through Y-Intercept (0 , -4)


If the Line Actually DID Pass through Point (5 , -6), the Slope of the that Line would be:

(-4 - -6)/ (0 - 5) = +2 / -5 = -(2/5)

The Upper Boundary of our Slope (Let's call it M) must be: M > -(2/5)



(2nd) The Line must Pass BELOW the Point of (4 , -1) and Pass through Y-Intercept (0 , -4)

The UPPER Boundary of the Possible Values Slope M can take would be the Slope that actually DOES Pass through Point (4 , -1)

(-4 - -1) / (0 - 4) = (-3) / (-4) = +(3/4)

Out Slope M must be: M < +(3/4)



(3rd) Find the "Integer-Multiples" of 1/7 that would Satisfy the Conditions



The Given Range that our Slope M can take is the following:

-(2/5) < M < +(3/4)


and we are told that M must be an "Integer-Multiple" of 1/7

I took this to mean that the Slope must have an INTEGER in the NUM and the DEN must be 7

Examples: +1/7 , -1/7 , +2/7 , -2/7, etc.

However, do NOT forget that 0 is an INTEGER Also. Thus, the Slope could be: M = 0/7 ---- or the Slope of a Horizontal Line Parallel to the X-Axis


(4th) Find the L.C.D. so we can compare the Boundaries with the Possible Values Slope - M can Take:

-(2/5) = -(56/140)

+(3/4) = +(105/140)

1/7 = 20/140
2/7 = 40/140
3/7 = 60/140
4/7 = 80/140
5/7 = 100/140



Given the Range that our M-Slope must fall within:

-(56/140) < M < +(105/140)


M can take the Following Values:

-(40/140)
-(20/140)

0 ----- (Horizontal Line: y = -(4) )

20/140
40/140
60/140
80/140
100/140


Solution:

There are 8 Possible Slopes of a Line that would meet the Conditions in the Question

-C-

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