{
 "source": "https://peabody.kids/flashcards/math-algebra-rules",
 "id": "math-algebra-rules",
 "title": "Algebra Rules and Properties",
 "category": "math",
 "subject": "Math",
 "levels": [
  "6-8",
  "9-12"
 ],
 "description": "Properties of numbers, rules for solving equations and inequalities, and common factoring patterns.",
 "tags": [
  "algebra",
  "properties",
  "rules"
 ],
 "frontLabel": "Rule",
 "backLabel": "Statement",
 "cards": [
  [
   "Commutative property of addition",
   "a + b = b + a"
  ],
  [
   "Commutative property of multiplication",
   "ab = ba"
  ],
  [
   "Associative property of addition",
   "(a + b) + c = a + (b + c)"
  ],
  [
   "Associative property of multiplication",
   "(ab)c = a(bc)"
  ],
  [
   "Distributive property",
   "a(b + c) = ab + ac"
  ],
  [
   "Additive identity",
   "a + 0 = a"
  ],
  [
   "Multiplicative identity",
   "a × 1 = a"
  ],
  [
   "Additive inverse",
   "a + (−a) = 0"
  ],
  [
   "Multiplicative inverse",
   "a × (1/a) = 1, for a not zero"
  ],
  [
   "Zero property of multiplication",
   "a × 0 = 0"
  ],
  [
   "Reflexive property",
   "a = a"
  ],
  [
   "Symmetric property",
   "If a = b, then b = a"
  ],
  [
   "Transitive property",
   "If a = b and b = c, then a = c"
  ],
  [
   "Addition property of equality",
   "If a = b, then a + c = b + c"
  ],
  [
   "Multiplication property of equality",
   "If a = b, then ac = bc"
  ],
  [
   "Substitution property",
   "If a = b, then b can replace a in any expression"
  ],
  [
   "Rule for multiplying two negatives",
   "The product is positive"
  ],
  [
   "Rule for multiplying a negative and a positive",
   "The product is negative"
  ],
  [
   "Rule for dividing two negatives",
   "The quotient is positive"
  ],
  [
   "Subtracting a negative",
   "a − (−b) = a + b"
  ],
  [
   "Solve x + 7 = 12",
   "x = 5"
  ],
  [
   "Solve 3x = 21",
   "x = 7"
  ],
  [
   "Solve 2x − 5 = 9",
   "x = 7"
  ],
  [
   "Solve x/4 = 6",
   "x = 24"
  ],
  [
   "Solve 5x + 3 = 2x + 15",
   "x = 4"
  ],
  [
   "Rule for inequalities",
   "Multiplying or dividing by a negative number flips the inequality sign"
  ],
  [
   "Solve −2x > 6",
   "x < −3"
  ],
  [
   "Solve |x| = 5",
   "x = 5 or x = −5"
  ],
  [
   "Solve |x| < 3",
   "−3 < x < 3"
  ],
  [
   "Solve |x| > 3",
   "x < −3 or x > 3"
  ],
  [
   "FOIL",
   "First, Outer, Inner, Last: multiplying two binomials"
  ],
  [
   "Expand (x + 3)(x + 2)",
   "x² + 5x + 6"
  ],
  [
   "Expand (x + 4)²",
   "x² + 8x + 16"
  ],
  [
   "Expand (x − 5)²",
   "x² − 10x + 25"
  ],
  [
   "Expand (x + 3)(x − 3)",
   "x² − 9"
  ],
  [
   "Factor x² + 7x + 12",
   "(x + 3)(x + 4)"
  ],
  [
   "Factor x² − 9",
   "(x − 3)(x + 3)"
  ],
  [
   "Factor x² − 5x + 6",
   "(x − 2)(x − 3)"
  ],
  [
   "Factor x² + 2x + 1",
   "(x + 1)²"
  ],
  [
   "Factor 2x² + 6x",
   "2x(x + 3)"
  ],
  [
   "Sum of cubes",
   "a³ + b³ = (a + b)(a² − ab + b²)"
  ],
  [
   "Difference of cubes",
   "a³ − b³ = (a − b)(a² + ab + b²)"
  ],
  [
   "Binomial cube",
   "(a + b)³ = a³ + 3a²b + 3ab² + b³"
  ],
  [
   "Solve x² = 49",
   "x = 7 or x = −7"
  ],
  [
   "Solve x² − 5x + 6 = 0",
   "x = 2 or x = 3"
  ],
  [
   "Midpoint of two numbers",
   "Their average: (a + b)/2"
  ],
  [
   "Cross-multiplication",
   "If a/b = c/d, then ad = bc"
  ],
  [
   "Adding fractions with unlike denominators",
   "Use a common denominator first"
  ],
  [
   "Slope between two points",
   "(y₂ − y₁)/(x₂ − x₁)"
  ],
  [
   "Perpendicular to slope 2",
   "Slope −1/2"
  ],
  [
   "Distance, rate, time",
   "d = r × t"
  ],
  [
   "Percent change",
   "(new − old) / old × 100%"
  ],
  [
   "Simple interest",
   "I = P r t"
  ],
  [
   "Average (mean) formula",
   "Sum of values divided by number of values"
  ],
  [
   "Order of solving with variables on both sides",
   "Collect variable terms on one side, constants on the other, then divide"
  ],
  [
   "Check a solution",
   "Substitute it back into the original equation"
  ]
 ]
}