o Standard Form
-> Standard form is Ax + By = C with A, B not both zero.
-> Convention: A >= 0; A, B, C integers; gcd 1.
-> gcd is greatest common factor; clear fractions when rational.
-> Example: y = (2/5)x + 14 becomes 2x - 5y = -70.
-> Subtract (2/5)x: -(2/5)x + y = 14.
-> Multiply by -5: 2x - 5y = -70.
-> Here A = 2, B = -5, and C = -70.
-> Check: -2x + 2y = 2 is not standard; divide by 2.
-> It reduces to -x + y = 1; multiply by -1: x - y = -1.
o Slope Intercept
-> Slope-intercept form is y = mx + b.
-> m is the slope; b is the y-intercept, not standard B.
-> Solve the equation for y before reading either value.
-> Example: 12x - 4y = 24.
-> Subtract 12x: -4y = -12x + 24.
-> Divide by -4: y = 3x - 6, so m = 3 and b = -6.
-> Check 4x = -2y + 22.
-> Add 2y: 4x + 2y = 22, then 2y = -4x + 22.
-> Thus y = -2x + 11; a positive 2 is wrong.
o Point Slope
-> Point-slope form is y - y1 = m(x - x1).
-> m is the slope and (x1, y1) is one point on the line.
-> Example: slope 7 through (2, -3).
-> Substitute: y - (-3) = 7(x - 2).
-> Simplify the left side: y + 3 = 7(x - 2).
-> Expand and solve for y to reach slope-intercept form.
-> y + 3 = 7x - 14 gives y = 7x - 17.
-> Check (2, -3): 7(2) - 17 = -3, as required.
o Two Points
-> Given two points, first compute the slope.
-> m = (y2 - y1)/(x2 - x1), provided x2 != x1.
-> Example: (1, 4) and (5, 16).
-> m = (16 - 4)/(5 - 1) = 12/4 = 3.
-> With (1, 4): y - 4 = 3(x - 1).
-> With (5, 16): y - 16 = 3(x - 5).
-> Both simplify to y = 3x + 1.
-> Check (1, 4): 3 + 1 = 4.
-> Check (5, 16): 15 + 1 = 16.
o Context Models
-> Define the input and output with units before writing.
-> Example: a gym charges $40 to join and $25 per month.
-> Let x be months and y be total cost in dollars.
-> The fixed fee is the y-intercept: b = 40.
-> The monthly rate is the slope: m = 25.
-> Model: y = 25x + 40.
-> After 6 months, y = 25(6) + 40 = 190 dollars.
-> The point (6, 190) satisfies the model.
-> Use the model only while the rate and fee stay fixed.
o Model Limits
-> A linear model assumes a constant rate of change.
-> Real prices, populations, and demand often change rates.
-> Extrapolation far beyond the data is a claim, not a fact.
-> Rounded slopes make predictions approximate, not exact.
-> Example: slope 4/3 rounded to 1.33 loses precision.
-> From (0, 2), x = 30 gives (4/3)(30) + 2 = 42 exactly.
-> The rounded model gives 1.33(30) + 2 = 41.9.
-> State the domain and the rounding when reporting a result.