Key Concepts of Linear Relationships

1. Recognizing Linear Relationships

• Equation: Example - $y = 2x + 1$.
  • Linear relationships are variations where Y = constant * X + constant.
  • Slope coefficient is the constant multiplied by X.
• Table of Values:
  • Choose evenly spaced values for X, e.g., -2 to 2.
  • Calculate Y using the equation.
  • Constant difference in consecutive Y values indicates linearity.
• Graph:
  • Points from the table form a straight line when connected.*

2. Understanding Slope

• Definition: Slope (m) = Rate of change.
• Calculation from Graph:
  • Use two points, calculate change in Y over change in X (rise/run).
  • Positive rise for upward, negative rise for downward.
• Calculation from Table:
  • Algebraically calculate difference in Y values over X values.
  • Use formula $(Y_2 - Y_1) / (X_2 - X_1)$.

3. Slope-Intercept Form

• Formula: $y = mx + b$.
  • $m$ is slope, $b$ is Y-intercept.
• Graph to Equation:
  • Identify slope and Y-intercept from graph to form equation.
• Equation to Graph:
  • Plot Y-intercept, use slope to plot other points.

4. Standard Form

• Formula: $Ax + By + C = 0$.
• Example: $3x + 6y - 12 = 0$.
• Converting Forms:
  • To Slope-Intercept: Isolate Y.
  • From Slope-Intercept: Eliminate fractions, rearrange terms.

5. Point-Slope Form

• Formula: $y - b = m(x - a)$.
  • Used when a specific point (a, b) on the line is known.
• Graph to Equation:
  • Identify slope, use a known point.
• Equation to Graph:
  • Use the given point and slope to plot.

6. Finding Equation from Two Points

• Use points to calculate slope.
• Substitute in slope-intercept form or point-slope form.
• Convert between different forms if needed.

7. Parallel and Perpendicular Lines

• Parallel lines: Same slope, never meet.
• Perpendicular lines: Slopes are negative reciprocals.

8. Horizontal and Vertical Lines

• Horizontal Line: $y = b$.
  • Slope is 0.
• Vertical Line: $x = a$.
  • Slope is undefined.

9. Finding Point of Intersection by Graphing

• Graph both lines, identify intersection point.

10. Finding Point of Intersection Algebraically

• Methods: Substitution and Elimination.
• Substitution: Solve for one variable, substitute into the other.
• Elimination: Align equations, subtract to eliminate one variable.