Lines, Slopes, and Their Equations

Determining a Line

• A straight line in a plane is uniquely determined by exactly two distinct points.
• Once two points $P<em>1(x</em>1,y<em>1)$ and $P</em>2(x<em>2,y</em>2)$ are fixed, every other point on the line lies on the infinite extension through these two.

Slope: Definition & Interpretation

• Fundamental definition:
  • $m = \dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x} = \dfrac{y<em>2 - y</em>1}{x<em>2 - x</em>1}.$
• “Rise” = vertical change $\Delta y$; “Run” = horizontal change $\Delta x$.
• Geometric meaning: measures steepness/inclination.
• Units: A pure number; need not be an integer—fractions and negatives are common.

Classes of Slopes

• m>0 ⇒ line rises left-to-right.
• m<0 ⇒ line falls left-to-right.
• $m=0$ ⇒ horizontal (all $y$‐values identical).
• Vertical line: $\Delta x=0$, slope is undefined (division by zero) — “no slope.”

Example 1 – Slope from Two Points

• Given points $(-1,4)$ and $(3,16)$
  • $m = \dfrac{16-4}{3-(-1)} = \dfrac{12}{4}=3.$
• Concludes slope is $3$.

Equation of a Line

To create an algebraic relation between any $x$, $y$ on the line.

Point–Slope Form

• Start with known point $(x<em>1,y</em>1)$ and slope $m$:
  • $y-y<em>1 = m(x-x</em>1).$
• Derivation used in lecture: multiply both sides of $m=\dfrac{y-y<em>1}{x-x</em>1}$ by $x-x_1$.
• Nickname: “point–slope form.”

Example 2 – Equation Through a Point with Given Slope

• Given slope $m=3$ and point $(-1,4)$
  • $y-4 = 3\bigl(x-(-1)\bigr) \Rightarrow y-4 = 3(x+1).$

Example 3 – Equation Through Two Points

• Points $(2,5)$ and $(4,15)$

1. Find slope:
   • $m = \dfrac{15-5}{4-2}=\dfrac{10}{2}=5.$
2. Choose either point (lecture chose smaller numbers $(2,5)$):
   • $y-5 = 5(x-2).$

• Either point produces the same line.

Slope–Intercept Form

• Re-arrange point–slope when the intercept is known: take $x<em>1=0,\;y</em>1=b$.
• Algebra:
  • $y-b = m(x-0) \implies y = mx + b.$
• Directly reveals:
  • Slope $m$ = coefficient of $x$.
  • y-intercept $b$ = constant term (point $(0,b)$).

Example 4 – Given Slope & y-Intercept

• Slope $3$, y-intercept $4$
  • Equation: $y = 3x + 4.$ (No additional work.)
• Graphing using slope:

1. Plot intercept $(0,4).$
2. From that point, move run $+1$, rise $+3$ (since $m=3=\tfrac{3}{1}$) to point $(1,7).$
3. Draw the unique line through the two points.

• Had the slope been $-3$, the “rise” would be $-3$ → move down three.

General (Standard) Form

• Written as $Ax + By + C = 0$ with $A,B,C \in \mathbb R$.
• Any linear equation can be rearranged into this form.

Example 5 – Analysis of General Form

Given $3x + 4y - 12 = 0$

1. Convert to slope–intercept:
   • $4y - 12 = -3x \Rightarrow 4y = -3x + 12 \Rightarrow y = -\tfrac34 x + 3.$
2. Extract data:
   • Slope $m = -\tfrac34.$
   • y-intercept point $(0,3).$
3. x-intercept (set $y=0$):
   • $3x - 12 = 0 \Rightarrow x = 4.$
   • x-intercept point $(4,0).$
4. Sketch: Mark $(0,3)$ and $(4,0)$; draw the line between/through them.

Special Lines

• Horizontal line: form $y = k$
  • Slope $m = 0$, all points share constant $y$.
• Vertical line: form $x = h$
  • Undefined slope; cannot compute $m$.

Parallel & Perpendicular (Preview)

• Mentioned as upcoming topic.
  • Parallel lines share identical slope $m$.
  • Perpendicular slopes satisfy $m<em>1 m</em>2 = -1$ (negative reciprocal).

Graphing Strategies & Practical Remarks

• For any line, two points suffice; additional points are redundant but can check accuracy.
• Slope offers a quick “rise/run” method to locate a second point when one is known.
• Instructor emphasized importance across disciplines (biology, physics, psychology) → many empirical relations linearize.

Classroom / Assessment Notes

• Lowest quiz grade mentioned was $7$ (contextual but indicates grading scale/expectations).
• Typical exam tasks:
  • Find slope from two points.
  • Write equation in requested form (point–slope, slope–intercept, or general).
  • Determine and plot x- and y-intercepts.
  • Graph using minimal points.
• Final