Slope and Equation of a Line: Finding the Equation from Two Points

Slope and Equation of a Line

Finding the Equation of a Line Given Two Points

When given two points on a Cartesian plane, you can determine the equation of the line that passes through them by following a systematic process. This is a foundational skill in precalculus and analytic geometry.

• Step 1: Identify the Coordinates Let the two points be $(x_1, y_1)$ and $(x_2, y_2)$. For example, the points (1, 6) and (5, 14).

• Step 2: Calculate the Slope (m) The slope of the line is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Example: For points (1, 6) and (5, 14): $m = \frac{14 - 6}{5 - 1} = \frac{8}{4} = 2$

• Step 3: Use the Point-Slope Form The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Example: Using point (1, 6): $y - 6 = 2(x - 1)$

• Step 4: Simplify to Slope-Intercept Form The slope-intercept form is $y = mx + b$. Example: $y - 6 = 2(x - 1)$ $y - 6 = 2x - 2$ $y = 2x - 2 + 6$ $y = 2x + 4$

Key Terms

• Slope (m): Measures the steepness of the line; calculated as the ratio of the change in $y$ to the change in $x$ between two points.

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: $y = mx + b$, where $b$ is the $y$-intercept.

Example Application

• Given points (1, 6) and (5, 14):

  • Find the slope: $m = 2$

  • Equation: $y = 2x + 4$

Graphical Representation

• Plot both points on the coordinate plane.

• Draw the straight line passing through both points.

• The line will have a slope of 2 and cross the $y$-axis at (0, 4).