Math: Linear Equations

WHAT IS A LINEAR EQUATION?

A linear equation is an equation where the variable (like x, n, or t)

has no exponent higher than 1. When graphed, it makes a straight line.

  Examples: C = 10n + 40    y = 2x + 10    t/3 - 4 = 14

SLOPE-INTERCEPT FORM: y = mx + b

This is the most important form to recognize.

  m = the SLOPE (rate of change — how much y changes for each unit of x)
  b = the Y-INTERCEPT (the starting value — the value when x = 0)

  Example: C = 10n + 40
    Slope (rate) = 10 → cost goes up $10 per gigabyte
    Y-intercept = 40 → base cost is $40 (even with 0 gigabytes)

WRITING EQUATIONS FROM WORD PROBLEMS

Identify two things:

  1. What is the FIXED cost / starting amount? → that's b
  2. What is the PER-UNIT cost / rate? → that's m

  Example: Cellphone plan charges $40/month flat + $10 per GB.
    Fixed = $40 → b = 40
    Per unit = $10/GB → m = 10
    Equation: C = 10n + 40

  Example: $35 setup fee + $32.25 per jersey.
    Fixed = $35 → b = 35
    Per unit = $32.25 → m = 32.25
    Equation: C = 32.25n + 35

  COMMON MISTAKE: Mixing up which number is the rate and which is the
  fixed cost. The rate is always "per something." The fixed cost is
  always there regardless of quantity.

WRITING EQUATIONS FROM TABLES

  Step 1: Find the RATE OF CHANGE (slope)
    Pick any two rows and calculate: (change in y) / (change in x)

  Step 2: Find the STARTING VALUE (y-intercept)
    Use: b = y - mx (plug in any row)

  Example: 1 movie = $12, 2 = $14, 3 = $16, 4 = $18
    Rate = (14 - 12) / (2 - 1) = $2 per movie
    Starting value = 12 - 2(1) = 10
    Equation: C = 2n + 10

  VERIFY: Plug in another row. 2(3) + 10 = 16 ✓

SOLVING ONE-STEP EQUATIONS

Do the opposite operation to both sides.

  t/3 = 18        → multiply both sides by 3 → t = 54
  2m = 24         → divide both sides by 2 → m = 12
  x + 5 = 12      → subtract 5 from both sides → x = 7
  x - 3 = 10      → add 3 to both sides → x = 13

SOLVING TWO-STEP EQUATIONS

Undo operations in reverse order: deal with addition/subtraction first,

then multiplication/division.

  t/3 - 4 = 14
    Step 1: Add 4 to both sides → t/3 = 18
    Step 2: Multiply both sides by 3 → t = 54

SOLVING EQUATIONS WITH DISTRIBUTION

Distribute (multiply) first, then solve.

  48 = 2(12 + m)
    Step 1: Distribute the 2 → 48 = 24 + 2m
    Step 2: Subtract 24 → 24 = 2m
    Step 3: Divide by 2 → m = 12

  3(7 - x) = 27
    Step 1: Distribute the 3 → 21 - 3x = 27
    Step 2: Subtract 21 → -3x = 6
    Step 3: Divide by -3 → x = -2

SOLVING EQUATIONS WITH FRACTIONS

Multiply both sides by the denominator to eliminate the fraction.

  b/4 = b - 12
    Multiply everything by 4: b = 4b - 48
    Subtract 4b: -3b = -48
    Divide by -3: b = 16

  (7/3)x - 6 = 22
    Add 6: (7/3)x = 28
    Multiply by 3/7: x = 28 × 3/7 = 84/7 = 12

VERIFYING SOLUTIONS BY SUBSTITUTION

Plug your answer back into the original equation to check.

  Is x = 12 a solution to (7/3)x - 6 = 22?
    (7/3)(12) - 6 = 84/3 - 6 = 28 - 6 = 22 ✓ Yes!

  Is x = 12 a solution to 3(x - 17) = 19?
    3(12 - 17) = 3(-5) = -15 ≠ 19 ✗ No.

SOLVING WORD PROBLEMS WITH EQUATIONS

  Step 1: Define the variable (what are you solving for?)
  Step 2: Write the equation
  Step 3: Solve

  Example: 2,500 points per purchase + 7,500 bonus. Need 70,000 total.
    Let n = number of purchases
    7,500 + 2,500n = 70,000
    2,500n = 62,500
    n = 25 purchases

KEY THINGS TO REMEMBER

  y = mx + b → m is the rate, b is the starting value
  Word problems: identify the fixed amount and the per-unit rate
  Tables: rate of change = (change in y) / (change in x)
  Solving: do the opposite operation to isolate the variable
  Distribution: multiply the outside number by everything in brackets
  Fractions: multiply both sides by the denominator
  Always verify: plug your answer back in to check