In the xy-coordinate system, points (2, 9) and (-1, 0) lie on line k. If the point (n, 21) lies on line k, what is the value of n?

A. 6
B. 7
C. 8
D. 9
E. 10
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Slope-intercept approach: Find the equation of line \(k\) in slope-intercept form
\(y=mx+b\)
\(m\) = slope
\(b\) = y-intercept

1) Find slope. Use the coordinates of the two given points, (2,9) and (-1,0)

Slope: \(\frac{rise}{run}=\frac{y_2-y_1}{x_2-x_1}=\frac{0-9}{-1-2}=\frac{-9}{-3}=3\)

Plug slope into equation: \(y=3x+b\)

2) Find \(b\). Plug (x,y) of either given point into equation. Example (-1,0):
\(y=3x+b\)
\(0=3(-1)+b\)
\(0=-3+b\)
\(b=3\)
Plug value of \(b\) into equation

3) Full equation of line \(k\) is \(y=3x+3\). Every point (x, y) [and (n, 21)] on a line must satisfy the equation of the line.

4) Point (n, 21) lies on line \(k\)
\(n =?\) (n = x-coordinate). Plug in (n, 21)
\(y=3x+3\)
\(21=3n+3\)
\(18=3n\)
\(n=6\)

Answer A
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Simply use the concept of slope (which many of us think of as Rise/Run) - It is the change in y co-ordinate for a unit change in x co-ordinate)
(2, 9), (-1 , 0) - When x co-ordinate reduces by 3 units, y co-ordinate reduces by 9 units (3 times).
(-1, 0), (n, 21) - So when y co-ordinate increase by 21 units, x co-ordinate will increase by 7 units (1/3).
So n = 6

Answer (A)

For more, check: https://www.veritasprep.com/blog/2016/0 ... line-gmat/
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