Notes on Linear Equations

1. Slope-Intercept Form

• Formula: y = mx + b
  • M: Slope of the line
  • B: Y-intercept
• Intercepts:
  • X-intercept: Point where the graph touches the X-axis (e.g., (-3, 0))
  • Y-intercept: Point where the graph touches the Y-axis (e.g., (0, 4))
• Finding X-Intercept:
  • Formula: x_i = b / m
  • Set y = 0 and solve for x

2. Calculating Slope from Two Points

• Formula: slope = (y2 - y1) / (x2 - x1)
  • Rise: Change in Y-values
  • Run: Change in X-values
• Example Calculation:
  • Points: (2, 3) and (5, 7)
  • Rise = 7 - 3 = 4
  • Run = 5 - 2 = 3
  • Slope = 4/3

3. Point-Slope Form

• Formula: y - y1 = m(x - x1)
  • M: Slope
  • (x1, y1): A point on the line

4. Standard Form

• Formula: Ax + By = C
• Finding Slope from Standard Form:
  • Slope = A / B
• Finding Y-Intercept:
  • Set x = 0, solve for y: y = C / B
• Finding X-Intercept:
  • Set y = 0, solve for x: x = C / A
• Derivation to Slope-Intercept Form:
  • Rearrange to y = mx + b to identify slope and intercepts

5. Intercept Form

• Formula: x/a + y/b = 1
  • a: X-intercept
  • b: Y-intercept
  • Slope: -b/a

6. Parallel and Perpendicular Lines

• Parallel Lines:
  • Same slope: m1 = m2
• Perpendicular Lines:
  • Slopes are negative reciprocals: m1 = -1/m2
  • Example: If m1 = 3/5, then m2 = -5/3

7. Horizontal and Vertical Lines

• Horizontal Line:
  • Slope = 0
  • Equation: y = k (where k is a constant)
• Vertical Line:
  • Slope = Undefined
  • Equation: x = h (where h is a constant)

Example Equations

• Horizontal Line Example: y = 2 (y-intercept of 2)
• Vertical Line Example: x = 3 (x-intercept of 3)

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