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?

Let the equation of the line be y = mx + c, where m is the slope of the line and c the y intercept

Slope and line method

We know points (2,9) and(-1,0) lie on the line so slope of the line, which is defined by the ratio of difference in Ordinates of two points to difference in Abscissae of two points is
(9-0)/(2-(-1)) = 3

We now know that the equation = y = 3x + c
Substitute the value of (x,y)=(-1,0) in the equation y = 3x + c you get
0 = -3 + c
c = 3 (I get the question but just a very silly question i cant seem to figure out why point (2,9) is used to find the (y=mx+c) C?? why cant we use (-1,0)?) Hope this answers your question.

The equation of the line is y = 3x + 3 and If the point (n, 21) lies on the line then it should satisfy the equation y = 3x + 3. Substitute the value of (x,y)=(n,21) in the equation y = 3x + 3 you get
21 = 3n + 3
n = 18/3 = 6

Slope method

The points (2, 9),(-1, 0),(n, 21) lie on the same line.So, the slope of line segment joining (2, 9) and (-1, 0) is equal to the Slope of line segment joining (n, 21) and (-1, 0)

Slope of line segment joining (2, 9) and (-1, 0) = (9-0)/(2-(-1)) = 3
Slope of line segment joining (n, 21) and (-1, 0) = (21-0)/(n-(-1)) = 21/(n+1)
Equating the above - 21/(n+1) = 3 => n+1 = 7 => n = 6

Points substitution method

Let the equation of the line be y = mx + c, where m is the slope of the line and c the y intercept
We know that points (2, 9),(-1, 0) lie on the line. So substituting the values of x,y -(2, 9),(-1, 0)we get two equations.
9 = 2m + c
0 = -m + c
Solving the equations for m and c,we get the value of m = 3 and the value of c = 3
We now know that the equation of the line is y = 3x + 3 and If the point (n, 21) lies on the line then it should satisfy the equation y = 3x + 3. Substituting the value of (x,y)=(n,21) in the equation y = 3x + 3 you get 21 = 3n + 3 => n = 18/3 = 6

thank you so much!,