Question:

A line goes through points (-3, 9) and (2, 4). Find the equation of the linear line.

Answer:

There are two methods. You can use a line equation with two unknowns, \(a\) and \(b\):

\[  y = ax + b  \]

where \(a\) is the slope and \(b\) is the intercept. Plug in the coordinates to solve the following simultaneous equations.

\[  9 = -3a + b \]

\[  4 = 2a + b  \]

They give \(5 = -5a\). Thus, \(a=-1\). Plug back in either equation and we get \(b=6\). The line equation is

\[  y = -x + 6  \]

The other method is to utilize following formula:

\[  (y-y_1)=\frac{y_2-y_1}{x_2-x_1}(x-x_1)  \]

The factor, \(\frac{y_2-y_1}{x_2-x_1}\), corresponds to the slope. After plugging in the coordinates, we can obtain the same result.