Straight-Line Equations & Graphs – Comprehensive Study Notes

Basics of the Cartesian Plane

• The Cartesian ("xy" or simply "the") plane is an infinite, perfectly flat 2-D surface on which a rectangular coordinate system (René Descartes) is super-imposed.
  • Two perpendicular number lines intersect at the mutual $0$ location called the origin $(0,0)$.
  • Horizontal axis → x-axis (positive to the right). Vertical axis → y-axis (positive upward).
  • Together they are the axes (plural of axis).
• Quadrants (Roman numerals, counter-clockwise):
  1. Quadrant I (upper-right) : x>0,\ y>0.
  2. Quadrant II (upper-left) : $x<0,\ y>0$.
  3. Quadrant III (lower-left) : x<0,\ y<0.
  4. Quadrant IV (lower-right): x>0,\ y<0.
  • Points on one or both axes are quadrantal points; at least one coordinate equals $0$ (e.g.
    origin has both $0$).
• Ordered pair $(a,b)$:
  • $a$ = x-coordinate (= abscissa) → horizontal positioning.
  • $b$ = y-coordinate (= ordinate) → vertical positioning.
  • Order matters: $(a,b)\neq (b,a)$ in general.
• Plotting = placing a dot at the location given by an ordered pair.
• A finite list of ordered pairs is a relation; if no two points share the same $x$, the relation is a function.
  • Domain = set of all x-coordinates. Range = set of all y-coordinates.

Distance & Midpoint Formulas

• Given $P<em>1(x</em>1,y<em>1)$ and $P</em>2(x<em>2,y</em>2)$:
  • Horizontal leg length $|x<em>2-x</em>1|$; vertical leg length $|y<em>2-y</em>1|$.
  • Distance formula (from Pythagorean theorem):
    $D = \sqrt{(x<em>2-x</em>1)^2 + (y<em>2-y</em>1)^2}$.
    • Example: distance between $(5,-4)$ and $(-3,7)$ equals $\sqrt{(-3-5)^2+(7+4)^2}=\sqrt{64+121}=\sqrt{185}\approx13.601$.
  • Midpoint formula:
    $M\Big(\dfrac{x<em>1+x</em>2}{2}\,,\; \dfrac{y<em>1+y</em>2}{2}\Big)$.
    • Example: midpoint of $(7,-2)$ and $(-1,5)$ is $\big(3,\;\tfrac{3}{2}\big)$.
• Fractional-shift (translation) of shapes: Add the same values to x- and y-coordinates of every vertex. Example polygon shifted up 4, left 3 ⇒ $x\to x-3,\;y\to y+4$.

Straight Lines in the Plane

Intercepts

• x-intercept: intersection with x-axis → point $(a,0)$.
  • Not present for horizontal lines coincident with x-axis (infinite intercepts) or horizontal lines not crossing x-axis (none).
• y-intercept: intersection with y-axis → point $(0,b)$.
  • Absent for vertical lines on y-axis or vertical lines not crossing y-axis.
• The rare case where both intercepts coincide is when the (non-vertical, non-horizontal) line passes through the origin $(0,0)$.

Slope (Tilt Quantifier)

• Symbol $m$.
• Rise-over-Run (visual counting): $m = \dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x}$ with run always taken positive, rise signed (+ up, − down).
  • Horizontal line ⇒ $m=0$.
  • Vertical line ⇒ slope undefined (never say "no slope").
  • Positive slope ⇒ line rises left→right. Negative slope ⇒ line falls left→right.
• Slope formula (non-visual): $m = \dfrac{y<em>2-y</em>1}{x<em>2-x</em>1} = \dfrac{y<em>1-y</em>2}{x<em>1-x</em>2}$ (order must match in numerator & denominator).
  • Works even with distant points or fractional coordinates.
  • Example: between $(\tfrac34,7)$ and $(\tfrac13,\tfrac56)$ ⇒ $m = \dfrac{\tfrac56-7}{\tfrac13-\tfrac34}=\dfrac{-\tfrac{29}{6}}{-\tfrac{5}{12}}=\tfrac{58}{5}$.

Parallel & Perpendicular Lines via Slope

• Parallel ⇔ slopes equal $m<em>1=m</em>2$.
• Perpendicular ⇔ either one line is vertical & the other horizontal or product of slopes $m<em>1m</em>2=-1$ (negative reciprocal relationship).

Algebraic Forms of Linear Equations

1. General Form

• $Ax+By+C=0$ with integers $A,B,C$ having no common factor, $A\ge0$ and at least one of $A,B$ non-zero.
• Serves as a universal reference; any line’s equation can be rearranged into this form.

2. Slope-Intercept Form (preferred for graphing)

• $y = mx + b$.
  • Slope $m$ evident; y-intercept is $(0,b)$.
  • If x-term missing ⇒ $m=0$ (horizontal line). Example: $y=5$.
  • Graphing steps: plot $(0,b)$; use rise/run from slope to get second point; draw line.

3. Point-Slope Form

• $y-y<em>1 = m(x - x</em>1)$ using any known point $(x<em>1,y</em>1)$ on the line and known slope $m$.
  • Derived directly from slope definition to avoid recomputing $m$ later.
  • Easily rearranged to any other form.

4. Two-Point Form

• Replace $m$ in point-slope with slope formula: $y-y<em>1 = \dfrac{y</em>2-y<em>1}{x</em>2-x<em>1}\,(x - x</em>1)$.
  • Useful when two points are known and slope has not yet been computed.

5. Two-Intercept Form

• $\dfrac{x}{a} + \dfrac{y}{b} = 1$ where intercepts are $(a,0)$ and $(0,b)$ (both non-zero).
  • Rearranged from general form by dividing through by $-C$.
  • Handy for quick graphing when both intercepts are known.

Equivalence & Infinite Representations

• Multiplying an existing equation by any non-zero constant yields a different-looking but equivalent equation representing the same geometric line.

Applications of Linear Models

• The slope’s ratio interpretation translates to "per" in real-world units.
  • Example 1: Satellite dishes in U.S. grew from $5.9$ million (2001) to $22.3$ million (2007).
    $m = \dfrac{22.3-5.9}{2007-2001} = \dfrac{16.4}{6}\approx2.73$ ⇒ growth rate $\approx2.73$ million dishes per year.
  • Example 2: IT employment rose from $3.54$ million (2001) to $3.84$ million (2006).
    $m = \dfrac{0.30}{5} = 0.06$ ⇒ about $60{,}000$ additional workers per year.
• Ethical/practical implication: Linear trend projection assumes constant rate; real-world factors (policy, technology shifts) may invalidate long-term extrapolation.

Solving Systems of Two Linear Equations (Graphical Method)

• A system = two linear equations examined together. Possible graphical relationships:
  1. Intersect at one point → consistent (unique solution).
  2. Parallel (no intersection) → inconsistent (no solution).
  3. Coincident (same line) → consistent & dependent (infinitely many solutions).
• Procedure (graphing version):
  1. Convert each equation to a convenient graphing form (slope-intercept or two-intercept).
  2. Sketch both lines on same plane.
  3. Read intersection(s) visually; verify by substitution.
• Limitations: labor-intensive, estimation error, ambiguous when lines nearly coincide. Algebraic methods (substitution, elimination, matrices) avoid these drawbacks (explored in Ch. 5).

Summary of Key Formulas (for rapid recall)

• Distance between $P<em>1$ and $P</em>2$: $D = \sqrt{(x<em>2-x</em>1)^2 + (y<em>2-y</em>1)^2}$.
• Midpoint: $\big(\tfrac{x<em>1+x</em>2}{2},\tfrac{y<em>1+y</em>2}{2}\big)$.
• Slope (general): $m = \dfrac{y<em>2-y</em>1}{x<em>2-x</em>1}$.
• General line form: $Ax + By + C = 0$ with conditions on $A,B,C$.
• Slope-Intercept: $y = mx + b$.
• Point-Slope: $y-y<em>1 = m(x-x</em>1)$.
• Two-Point: $y-y<em>1 = \dfrac{y</em>2-y<em>1}{x</em>2-x<em>1}\,(x-x</em>1)$.
• Two-Intercept: $\dfrac{x}{a} + \dfrac{y}{b} = 1$.

Glossary (Condensed)

• Abscissa – x-coordinate.
• Ordinate – y-coordinate.
• Domain/Range – sets of x’s & y’s in a relation/function.
• Scatter plot – graph of discrete points.
• Rise/Run – vertical/horizontal displacement between two line points.
• Slope (m) – $\tfrac{\text{rise}}{\text{run}}$, tilt measure.
• Line segment / Ray / Line – finite / semi-infinite / bi-infinite straight path.
• Consistent / Inconsistent / Dependent systems – solution classifications as above.

Exercise & Practice Themes (for exam prep)

• Identify quadrants, axes, intercepts, plot given points.
• Determine function vs. mere relation & give domain/range.
• Compute distances, midpoints, fractional-length points along a segment.
• Calculate slope visually & algebraically; recognize slope sign from sketch.
• Convert between all equation forms; extract slope/intercepts; graph lines.
• Write equations from given data (point + slope, two points, intercepts).
• Analyze parallel/perpendicular relationships through slopes.
• Solve 2×2 linear systems by graphing; classify as unique, none, or infinitely many solutions.
• Real-world modeling: derive linear equations from temporal data; interpret slope as rate; extrapolate judiciously.

Tip for exams:
• Always present intercept answers as ordered pairs $(a,0)$ or $(0,b)$.
• Show algebraic checks when claiming lines are parallel/perpendicular.
• Label axes, quadrants, and scales on sketches for full credit.