Find the slope of the line through $(1, 2)$ and $(3, 8)$.

How we read your problem

Find the slope (rate of change) of the line through two points.

Step-by-step solution

1. 1

   Label the points

   $$(x_1, y_1) = (1, 2),\quad (x_2, y_2) = (3, 8)$$
2. 2

   Find the change in y (rise)

   $$y_2 - y_1 = 8 - 2 = 6$$
3. 3

   Find the change in x (run)

   $$x_2 - x_1 = 3 - 1 = 2$$
4. 4

   Divide rise by run

   $$m = \frac{6}{2} = 3$$
5. 5

   Find the y-intercept using the first point

   $$b = y_1 - m x_1 = 2 - 3 \cdot 1 = -1$$
6. 6

   Write the equation of the line

   $$y = 3x - 1$$

Formulas used

• Slope

  $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

• Slope-intercept form

  $$y = mx + b$$

The slope is 3: for every 1 unit increase in x, y changes by 3. The line rises from left to right.

Verification

• Second point: 3·(3) + -1 = 8 (expected 8)