Algebraic approach:Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p?We have two lines: \(y_n=m_1x+b_1\) and \(y_p=m_2x+b_2\). Q: is \(m_1<m_2\)?

(1) Lines n and p intersect at the point (5,1) --> \(1=5m_1+b_1=5m_2+b_2\) --> \(5(m_1-m_2)=b_2-b_1\). Not sufficient.

(2) The y-intercept of line \(n\) is greater than the y-intercept of line \(p\) --> y-intercept is value of \(y\) for \(x=0\), so it's the value of \(b\) --> \(b_1>b_2\) or \(b_2-b_1<0\). Not sufficient.

(1)+(2) \(5(m_1-m_2)=b_2-b_1\), as from (2) \(b_2-b_1<0\) (RHS), then LHS (left hand side) also is less than zero \(5(m_1-m_2)<0\) --> \(m_1-m_2<0\) --> \(m_1<m_2\). Sufficient.

Answer: C.

Graphic approach:Lines n and p lie in the xy plane. Is the slope of line n less than the slope of line p?(1) Lines n and p intersect at (5,1)

(2) The y-intercept of line n is greater than y-intercept of line p

The two statements individually are not sufficient.

(1)+(2) Note that

a higher absolute value of a slope indicates a steeper incline.

Now, if both lines have positive slopes then as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line p is

steeper hence its

slope is greater than the

slope of the line n:

Attachment:

1.PNG [ 14.29 KiB | Viewed 3048 times ]
If both lines have

negative slopes then again as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line n is

steeper hence the

absolute value of its

slope is greater than the

absolute value of the

slope of the line p, so the

slope of n is

more negative than the

slope of p, which means that the

slope of p is greater than the

slope of n:

Attachment:

2.PNG [ 13.66 KiB | Viewed 3047 times ]
So in both cases the

slope of p is greater than the

slope of n. Sufficient.

Answer: C.

For

more on this topic check Coordinate Geometry Chapter of Math Book:

math-coordinate-geometry-87652.htmlHope it helps.

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