Study Notes on Writing Linear Equations in Slope-Intercept Form

Equation of a Line in Slope-Intercept Form

Key Concepts

• Slope-Intercept Form Definition: The slope-intercept form of a linear equation is expressed as: $y = mx + b$ where:
  • $m$ represents the slope of the line.
  • $b$ represents the y-intercept (the point where the line crosses the y-axis).

Steps to Write the Equation

1. Identify Points: If a graph is provided, identify two points ($(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$) on the line.
2. Calculate the Slope (m): The slope is calculated using the formula:
   $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
3. Determine b (Y-Intercept): After calculating the slope, use one of the points and the slope to solve for $b$ in the equation:
   $b = y - mx$
4. Construct the Equation: Substitute the values of $m$ and $b$ back into the slope-intercept form.
   • Resulting equation: $y = mx + b$

Example Breakdown

Given a hypothetical line with points identified at coordinate values:

• Point 1: $(1, 2)$
• Point 2: $(3, 4)$

1. Calculate Slope:
   $m = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1$
2. Use one point for Y-Intercept: By substituting point $(1, 2)$ into the y-intercept formula:
   • $2 = 1(1) + b$
   • Hence, $b = 2 - 1 = 1$
3. Final Equation:
   • Substitute $m$ and $b$ back into the equation:
     $y = 1x + 1$
     or simply $y = x + 1$

Graphical Representation

• Additional graphical points can be derived or plotted based on the specified equation to visualize the line and confirm accuracy in slope and intercept calculations.