In the xy-coordinate system, points (2, 9) and (-1, 0) lie on line k.

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.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2016/0 ... line-gmat/
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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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Equation of the line passing through points (2, 9) and (-1, 0):

\(y-9=\frac{0-9}{-1-2}*(x-2)\)
Or, y-9=3(x-2), this is the equation of the line k

If the point (n, 21) lies on line k, then this point must satisfy the above equation.
So, 21-9=3n-6
or, 3n-6=12
or, 3n=18
or, n=6

Ans. (A)
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Solution

Given:

• Points (2, 9) and (-1, 0) lie on line k

To find:

• The value of n such that point (n, 21) also lies on line k.

Approach and Working:

• Points (2, 9) and (-1, 0) lie on line k.

o Hence, slope of line k=\(\frac{{9-0}}{{2- (-1)}}\)= \(\frac{9}{3}\)= 3

• Now, if point (n, 21) also lies on line k then the slope of (n, 21) and(-1, 0) will also be equal to 3.

o \(\frac{{21-0}}{{n+1}}\)= 3
o n=6

Hence, the correct answer is option A.

Answer: A
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Key concept: If you take any portion of a particular line, the slope between ANY two points on the line will always be the same.

So, for example, the slope between points (2, 9) and (-1, 0) must be equal to the slope between points (2, 9) and (n, 21)

Slope between (2, 9) and (-1, 0) = (9 - 0)/(2 - (-1)) = 9/3 = 3

This means the slope between points (2, 9) and (n, 21) must also equal 3

We can write: (21 - 9)/(n - 2) = 3
Simplify: 12/(n - 2) = 3
Multiply both sides by (n - 2) to get: 12 = 3(n - 2)
Divide both sides by 3 to get: 4 = n - 2
Solve: n = 6

Answer: A

Cheers,
Brent

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

For this question, let's keep in mind the equation y = mx + b where m is the slope of a given line, b is the y-intercept and then x and y would be any values for x or y on a given line.

With the information given, we can use the two points to find the slope first. (9) - (0) / (2) - (-1) -> 9/3 -> m = 3.

Then, we would like to solve for b. We can use either of the two points given.
9 = 3(2) + b
3 = b

Lastly, let's plug in (n,21) into the equation with each value that we found, and that will give us the value for n

21 = 3(n) + 3
18 = 3n
6 = n

ANSWER is A

Solution:

The slope of line k is:

(9 - 0) / (2 - (-1)) = 9/3 = 3

Therefore, if (n, 21) lies on line k, it must have a slope of 3 with any of the two given points. Let’s use the latter and create the equation:

(21 - 0) / (n - (-1)) = 3

21 / (n + 1) = 3

21 = 3n + 3

18 = 3n

6 = n

Answer: A
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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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7 units from -1 of x co-ordinate will lead to x = 6....great approach. How to intuitively think of slope as every 'a' value increase of x, y increases by 'b' value while solving questions in time crunch? The brain intuitively chooses the rise/run as a formula (as ingrained in the brain since school days)