Chapter 6 – Linear Equations & Graphs (Complete Study Notes)

Core format: $y = mx + b$
- $m$ = slope (“rise over run”, rate of change)
- $b$ = $y$ -intercept (point $\,(0,b)\,$ )
Information the form gives instantly
- Direction & steepness (sign & magnitude of $m$ )
- Initial / starting value when the independent variable is $0$
- Whether the line rises, falls, is horizontal ( $m=0$ ), or vertical (cannot be written in this form)
Finding $m$ and $b$ from a graph
- Pick two clear lattice points
- $m = \dfrac{\Delta y}{\Delta x} = \dfrac{y2-y1}{x2-x1}$
- Read $b$ where the graph crosses the $y$ -axis
Writing an equation when you know $m$ and $b$
- Substitute directly into $y=mx+b$
Re-writing another linear form into slope–intercept
- Isolate $y$ with algebra (add/subtract terms, then divide by the $y$ coefficient)
Graphing quickly
- Plot $(0,b)$
- From that point move “rise $m{num}$ , run $m{den}$ ”
- Draw the straight line through the points
Typical problem types
- Predicting unknown values (substitute $x$ or $y$ )
- Comparing rates (slope)
- Interpreting intercepts in context (initial height, starting cost, etc.)
Classroom lab (elastic-cup & marbles) highlights
- Independent variable: number of marbles $x$
- Dependent variable: apparatus length $y$
- Slope = extension per marble (cm / marble)
- $y$ -intercept = original length with no marbles
- Equation example: $y = mx + b \;\Rightarrow\; y = 2x + 1$ (if data matched that)

Core format: $Ax + By + C = 0$
- $A,B,C$ are integers; by convention A>0
- Valid for
- All non-vertical & vertical lines
- Horizontal lines as well
Converting from slope–intercept
- Rearrange so RHS = 0
- Clear any fractions by multiplying through by the LCM of denominators
- Ensure A>0
Finding intercepts rapidly
- $x$ -intercept: set $y=0$ → solve for $x$
- $y$ -intercept: set $x=0$ → solve for $y$
- Plot $(xi,0)$ and $(0,yi)$ and draw the straight line
Special cases
- Vertical line: $x = k$ ⇒ $x + 0y - k = 0$ (infinite $y$ -intercepts)
- Horizontal line: $y = c$ ⇒ $0x + y - c = 0$ (infinite $x$ -intercepts)
Why general form?
- Symmetry in algebraic manipulation
- Useful for elimination in systems
- Both intercepts appear by inspection
Example summary
- Rewrite $y = -\dfrac13x + 6$
- Multiply by 3: $3y = -x + 18$
- Bring all terms left: $x + 3y - 18 = 0$
- Graph $2x - 3y - 6 = 0$
- $y$ -int: $2(0)-3y-6=0→y=-2$
- $x$ -int: $2x-3(0)-6=0→x=3$
Intercepts in applications
- Disk-space question: $T + 4M = 60$
- $T$ -int 60 shows max TV shows with zero movies
- $M$ -int 15 shows max movies with zero TV shows

Core format (non-vertical lines): $y - y1 = m\,(x - x1)$
- $m$ = slope
- $(x1,y1)$ = any known point on the line
Derivation recap
- Start with slope definition $m = \dfrac{y-y1}{x-x1}$
- Cross-multiply to obtain the form
When is it the best tool?
- When slope and one point are given
- When two points are given (find $m$ , then plug one point)
- When writing families of parallel/perpendicular lines
Converting to other forms
- Distribute $m$ and isolate $y$ to get slope–intercept
- Clear fractions & gather terms to get general form
Worked example
- Given slope $-3$ through $(-2,5)$
- $y-5 = -3(x+2)$
- Expand ⇒ $y = -3x - 1$
- General ⇒ $3x + y + 1 = 0$
Two-point procedure
1. Compute $m$
2. Substitute either point
3. Simplify as desired

Parallel criteria
- Same slope ( $m1 = m2$ ) but different intercepts
- All vertical lines are mutually parallel; all horizontal lines are mutually parallel
Perpendicular criteria
- Slopes are negative reciprocals: $m1 m2 = -1$
- Vertical ( $m$ undefined) ⟂ Horizontal ( $m=0$ )
Building a parallel line
1. Copy slope $m$ of reference line
2. Substitute new point in $y = mx + b$ or point–slope
3. Rearrange to desired form
Building a perpendicular line
1. Take $m’ = -\dfrac1m$ (or undefined/0 swap)
2. Proceed as above
Example recap
- Parallel to $y = 2x + 4$ through $(1,-6)$
- $y+6 = 2(x-1)\;\Rightarrow\;y = 2x - 8$
- Perpendicular to $3x + 2y - 6 = 0$ with $x$ -int 9
- Original slope $-\tfrac32$ ⇒ perpendicular slope $\tfrac23$
- Use $(9,0)$ ⇒ $y = \tfrac23(x-9)$ ⇒ $2x - 3y - 18 = 0$

Slope calculations
- From table → pick any two rows
- From formula → use $\Delta$ notation
- From general form → rearrange to isolate $y$ first
Intercept extraction
- From slope–intercept: $b$ already visible; $x$ -intercept at $y=0$
- From general: quick substitution technique
Graphing strategies
- Slope–intercept: intercept + slope steps
- General: both intercepts
- Point–slope: start at given point, count off slope run/rise
Parallel/perpendicular tests in coordinate proofs
- Equal slopes ⇒ opposite sides of a quadrilateral parallel ⇒ parallelogram
- Negative-reciprocal slopes ⇒ right angle ⇒ verify right triangle or altitude
Common real-world rates portrayed as slopes
- Price per unit, speed, pressure change with depth, temperature lapse rate, heart-rate zones, cricket chirps vs. temperature, hair growth, etc.

Convert $\,\;4x - y + 3 = 0$ to $y = mx + b$
Write the general-form equation of a horizontal line through $(0,-5)$
A saving plan starts with $\$120$ and grows $\$25$ each month
- Equation? How long to reach $\$745$ ?
Determine if the segments joining $A(2,5)$ – $B(-1,1)$ and $C(4,-2)$ – $D(1,-6)$ are parallel, perpendicular, or neither
Find the altitude from vertex $B$ of $\triangle ABC$ with $A(-2,1),B(1,4),C(4,-2)$

Arithmetic-sequence formula $an = a1 + (n-1)d$ mirrors $y = mx + b$ :
- $a_1$ like $b$ (start value), $d$ like $m$ (common difference / rate)
Choice of form is a “tool selection” task
- Need rate & initial value quickly? → Slope-intercept
- Need both intercepts or to use elimination? → General
- Know a point and slope or constructing parallel/perpendicular? → Point-slope
Ethical & practical lens
- Interpreting slope/intercept tells us feasibility (e.g.
  max guests within budget, safe temperature zones, pressure limits for submarines)
- Inaccurate tool choice wastes time or mis-models real data (analogy: ruler vs. tape measure)