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:
1.PNG [ 14.29 KiB | Viewed 3045 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:
2.PNG [ 13.66 KiB | Viewed 3044 times ]
So in both cases the slope of p is greater than the slope of n. Sufficient.
For more on this topic check Coordinate Geometry Chapter of Math Book: math-coordinate-geometry-87652.html
Hope it helps.
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