Equation of a Line

Step-by-step solution

1. 1

   The given information

   The line passes through two points.

   $$(x_1, y_1) = \left(1,\ 2\right) \qquad (x_2, y_2) = \left(3,\ 8\right)$$
2. 2

   Slope

   The slope is rise over run: the change in y divided by the change in x.

   $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6}{2} = 3$$
3. 3

   Point-slope form

   Substitute the slope m = 3 and the point (1, 2) into the point-slope form y − y₁ = m(x − x₁).

   $$y - 2 = 3\left(x - 1\right)$$
4. 4

   Slope-intercept form

   Solve for y: distribute the slope and collect the constant term. The constant is the y-intercept b = -1, giving y = 3x - 1. Check with the second point: m·x₂ + b = 8, which matches y₂ ✓

   $$\begin{aligned} b &= y_1 - m \cdot x_1 \\ &= 2 - 3 \cdot 1 \\ &= -1 \\[2pt] y &= 3x - 1 \end{aligned}$$

   • b = y₁ − m·x₁ = 2 − 3·1 = -1
5. 5

   Standard form

   All coefficients are already integers; rearrange y = mx + b into Ax + By = C by moving the x-term to the left side.

   $$y = 3x - 1 \;\Longrightarrow\; 3x - y = 1$$
6. 6

   Intercepts

   Set x = 0 to find the y-intercept: y = -1. Set y = 0 and solve 0 = mx + b for x: x = −b/m = 1/3. The line crosses the axes at (0, -1) and (1/3, 0).

   $$\text{y-intercept: } \left(0,\ -1\right) \qquad \text{x-intercept: } \left(\frac{1}{3},\ 0\right)$$
7. 7

   Result

   The line has slope 3 and crosses the y-axis at (0, -1). Equation: y = 3x - 1.

   $$y = 3x - 1$$