Linear Equations: Point-Slope Form, Graphing, Parallel & Perpendicular Lines

Introduction to Linear Equation Forms

• Upcoming lessons include point-slope form, standard form, graphing from all forms, and finding points of intersection.
• A review of similar triangles will follow before the unit test.
• The current focus is on mastering graphing skills, with similar triangles to be revisited later.

Derivation of Point-Slope Form

• Understanding the Slope Formula: The slope ($m$) between two points $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$ is defined as the change in $y$ divided by the change in $x$:
  $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
• Adapting for Point-Slope Form: To derive the point-slope form, one of the two points is kept as a generic variable point $(x, y)$, while the other is a known specific point $(x<em>1, y</em>1)$. Substituting these into the slope formula gives:
  $m = \frac{y - y<em>1}{x - x</em>1}$
• Rearranging to Point-Slope Form: To eliminate the denominator and isolate the $(y - y<em>1)$ term, both sides of the equation are multiplied by $(x - x</em>1)$. It's important to remember that the fraction bar implies invisible brackets around the numerator and denominator.
  $m(x - x<em>1) = \frac{y - y</em>1}{x - x<em>1} \cdot (x - x</em>1)$
  The $(x - x<em>1)$ terms on the right side cancel out, resulting in the point-slope form: $y - y</em>1 = m(x - x_1)$

Understanding the Components of Point-Slope Form

• $m$: Represents the slope or rate of change of the line. It is a specific numerical value.
• $(x<em>1, y</em>1)$: Represents any single known point that lies on the line. $x<em>1$ is its x-coordinate, and $y</em>1$ is its y-coordinate.
• $(x, y)$: These are generic variables and represent all points on the line. They are left as $x$ and $y$ in the equation.
• The Minus Signs: The minus signs in $(y - y<em>1)$ and $(x - x</em>1)$ are inherent to the formula, originating from the calculation of