A line goes through point $A(9, 1)$, point $B(19, k)$ and point $C(7, 0)$. What is the value of $k$?
To find the value of $k$, I need to use the fact that the slope of the line is constant for any two points on it. I can pick any two points, but I will use $A$ and $C$ because they have simpler coordinates. The slope of the line through $A$ and $C$ is $\frac{0 - 1}{7 - 9} = \frac{-1}{-2} = \frac{1}{2}$. This means that the slope of the line through $A$ and $B$ must also be $\frac{1}{2}$. I can use the slope formula again, but this time with $A$ and $B$. The slope of the line through $A$ and $B$ is $\frac{k - 1}{19 - 9} = \frac{k - 1}{10}$. Since this slope must equal $\frac{1}{2}$, I can set up an equation and solve for $k$. I have $\frac{k - 1}{10} = \frac{1}{2}$. I can cross-multiply to get $2(k - 1) = 10$. Expanding the parentheses, I get $2k - 2 = 10$. Adding $2$ to both sides, I get $2k = 12$. Dividing both sides by $2$, I get $k = 6$. This is the value of $k$ that makes the line go through all three points. 