Slope

[grandh01]

Lines m and n lie in the xy-plane and intersect at the point
(-2, 4). Is the slope of line m less than the slope of line n ?
(1) The x-intercept of line m is greater than the x-intercept
of line n.
(2) The y-intercept of line n is greater than the y-intercept
of line m.

OA IS D

[Anurag@Gurome]

Lines m and n lie in the xy-plane and intersect at the point
(-2, 4). Is the slope of line m less than the slope of line n ?
(1) The x-intercept of line m is greater than the x-intercept
of line n.
(2) The y-intercept of line n is greater than the y-intercept
of line m.

OA IS D

Equation of a line in point intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept of the line, (-b/m) is the x-intercept of the line.

We are given two lines: y(m) = mx + b and y(n) = nx + c. They intersect at the point (-2, 4).
So, 4 = -2m + b and 4 = -2n + c
b = 4 + 2m and c = 4 + 2n

Question is asking if the slope of line m less than the slope of line n, OR m < n?

(1) The x-intercept of line m is greater than the x-intercept of line n implies (-b/m) > (-c/n), which implies -(4 + 2m)/m > -(4 + 2n)/n OR (1/n) - (1/m) > 0.
Now if n = 2, m = -4, then m < n
If n = 2, m = 4, then m > n; NOT sufficient.

(2) The y-intercept of line n is greater than the y-intercept of line m implies c > b or 4 + 2n > 4 + 2m or n > m; SUFFICIENT.

The correct answer is B.

[svd.kumar]

Hi Anurag,

Answer seems D only, could you please evaluate my explanation.

Both lines passes through (-2,4)

1) x-intercept of line m is greater than x-intercept of line n
x-intercept is nothing but value of x when y=0. so lets assume it is (m1, 0) for line m and (n1, 0) for line n

we know m1 > n1

slope of line m = y2-y1/x2-x1 = 0-4/m1+2 = -4/m1+2
slope of line n = 0-4/n1+2 = -4/n1+2

so slope of line n > slope of line m
(also we need to ignore the sign here, because we are not checking which number is great, we are checking which line has more slope i.e diff in y by diff x)
Sufficient

2) Y-intercept of line n is greater than the y-intercept of line m
y-intercept is value of y when x=0. so lets assume it is (0, m2) for line m and (0, n2) for line n

we know n2>m2
slope of line m = y2-y1/x2-x1 = 0-4/m1+2 = m2-4/2
slope of line n = 0-4/n1+2 = n2-4/2

slope of line n > slope of line m
Sufficient

Thanks,
SVD

[Ian Stewart]

1) x-intercept of line m is greater than x-intercept of line n
x-intercept is nothing but value of x when y=0. so lets assume it is (m1, 0) for line m and (n1, 0) for line n

we know m1 > n1

slope of line m = y2-y1/x2-x1 = 0-4/m1+2 = -4/m1+2
slope of line n = 0-4/n1+2 = -4/n1+2

so slope of line n > slope of line m
(also we need to ignore the sign here, because we are not checking which number is great, we are checking which line has more slope i.e diff in y by diff x)
Sufficient

No, Statement 1 is not sufficient here. If, say, the x-intercept of line N is at -6, then the slope of line N is +1. Now, if the x-intercept of line M is at, say, -4, then the slope of line M is +2, which is greater than the slope of line N. But if the x-intercept of line M is at, say, 0, then the slope of line M is negative, so is certainly smaller than the slope of line N.

There are a lot of questions like this one, and while you can usually do some kind of algebra, it is pretty much always a lot faster to just draw out the various scenarios on the coordinate plane.

[svd.kumar]

Hi Ian,

Thanks for your reply, seems like I am little bit confused with slope concept here here. Could you please clarify below point

If two lines p and q have slopes 10, -10 respectively, which line has more slope.

My understanding is slope of both line are equal, because slope is measure of inclination of line. We cannot say line with slope 10 is more inclined than line with slope -10 here.

Thanks,
SVD

[Ian Stewart]

Thanks for your reply, seems like I am little bit confused with slope concept here here. Could you please clarify below point

If two lines p and q have slopes 10, -10 respectively, which line has more slope.

The slope of a line is just a number, so if we are asked to compare the slopes of two lines, we just need to know, numerically, which of the two slope values is greater. So in your example, the line with the slope of 10 would have the greater slope, since 10 is greater than -10. I think you're comparing the absolute values of the slopes, which is not correct.

The wording you've used might be leading you to your incorrect interpretation here. A question can never say "which line has more slope?" That would be like asking, when looking at two sets, "which set has more median?". Neither of those questions make mathematical sense. The slope, like the median, is a number, so if we are comparing two slopes or two medians, we need to ask which is greater, and not which is 'more'.

[svd.kumar]

Thanks Ian got it