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Algebra Mastery: The 3 Pillars of Equations

Pillar 1: The 'Undo' Method (Multi-Step) #

To isolate xx, you have to work backward through the order of operations. Think of it as unwrapping a gift: undo the addition/subtraction first, then the multiplication/division.

Quick Example: Solve: 4x+11=354x + 11 = 35

  • 1.
    Subtract 1111: 4x=244x = 24
  • 2.
    Divide by 44: x=6x = 6

    • Practice Drill: Solve: 5x−8=225x - 8 = 22

    Single choice

    Pillar 2: The Handshake (Distributive Property) #

    When you see parentheses, the number outside must "shake hands" (multiply) with every term inside before you start moving things across the equals sign.

    Quick Example: Solve: 3(x+4)=213(x + 4) = 21

  • 1.
    Distribute the 33: 3x+12=213x + 12 = 21
  • 2.
    Subtract 1212: 3x=93x = 9
  • 3.
    Divide by 33: x=3x = 3

    • Practice Drill: Solve: 2(3x−5)=202(3x - 5) = 20

    Fill in blanks

    Pillar 3: The Great Migration (Variables on Both Sides) #

    If xx is on both sides, it's messy. Pick a "variable side" (usually the left) and a "constant side" (the right). Move terms using inverse operations until xx is alone.

    Quick Example: Solve: 9x−5=4x+209x - 5 = 4x + 20

  • 1.
    Subtract 4x4x from both sides: 5x−5=205x - 5 = 20
  • 2.
    Add 55 to both sides: 5x=255x = 25
  • 3.
    Divide by 55: x=5x = 5

    • Practice Drill: Solve: 7x+2=3x+227x + 2 = 3x + 22

    Single choice
    Info

    Pro Tip: Always plug your answer back into the original equation. If the left side doesn't equal the right side, go back and check your signs—especially during distribution!

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