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 P1(x1,y1) and P2(x2,y2) 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{y2 - y1}{x2 - x1}.
• “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 (x1,y1) and slope m:
  • y-y1 = m(x-x1).
• Derivation used in lecture: multiply both sides of m=\dfrac{y-y1}{x-x1} 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 x1=0,\;y1=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 m1 m2 = -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