### Instruction: A line contains the points $(-1, 6)$, $(6, k)$ and $(20, 3)$. What is the value of $k$?### Response: To find the value of $k$, I need to use the fact that the slope of a line is constant for any two points on the line.
I can choose any two pairs of points and set their slopes equal to each other.
I'll start with $(-1, 6)$ and $(6, k)$.
The slope between these two points is $\frac{k - 6}{6 - (-1)}$.
I'll simplify this expression by combining the denominator: $\frac{k - 6}{7}$.
Next, I'll use $(6, k)$ and $(20, 3)$.
The slope between these two points is $\frac{3 - k}{20 - 6}$.
I'll simplify this expression by combining the denominator: $\frac{3 - k}{14}$.
Now, I can set the two slopes equal to each other and cross-multiply to get an equation: $\frac{k - 6}{7} = \frac{3 - k}{14}$.
Multiplying both sides by 14, I get $2(k - 6) = 3 - k$.
Expanding the left side, I get $2k - 12 = 3 - k$.
Adding $k$ to both sides, I get $3k - 12 = 3$.
Adding 12 to both sides, I get $3k = 15$.
Dividing both sides by 3, I get $k = 5$.