Math

Solving Equations
Pre-Algebra · Chapter 2 study tool
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The golden rule

⚖️ An equation is a balance

The goal is to get the variable alone. Whatever you do to one side, you MUST do to the other side, so it stays balanced. Draw a line down the middle to keep the two sides straight.

To undo something, use the opposite (inverse) operation:

+ ⟷ −
× ⟷ ÷
One-step

1️⃣ One-step equations

Ask: what is happening to the variable? Then do the opposite to both sides.

a + 3 = −143 is added → subtract 3 from both sides → a = −17 −9r = −18r is multiplied by −9 → divide by −9 → r = 2 r3 = −15r is divided by 3 → multiply by 3 → r = −45 23m = −16multiply by 3 → 2m = −48 → divide by 2 → m = −24
Watch out: −a = −24 means "the opposite of a is −24", so a = 24. Divide both sides by −1.
Two-step

2️⃣ Two-step equations: GEMA in reverse

Order of operations is GEMA. To solve, you go backwards: undo A first, then M.

GGrouping
EExponents
MMult / Div
2nd
AAdd / Sub
1st
Hint: cover up what's with the variable. In 2x − 3 = −5, cover the 2x. What's left on that side is −3, so undo that first.
2x − 3 = −5start 2x = −2add 3 to both sides x = −1divide both sides by 2
Keep the sign with its term. In −3x + 5 = 2, the term is −3x, not 3x. After subtracting 5: −3x = −3, so x = 1.
Can't combine unlike terms. In −16 = 9 − 5c, the 9 and −5c can't be mashed together. Subtract 9 from both sides: −25 = −5c, so c = 5.
Whole group on top of a fraction? In x + 43 = −5, the whole top is divided by 3, so multiply first: x + 4 = −15, then x = −19.
Distributive property

📦 The box method

The number outside multiplies EVERY term inside the parentheses. Put the outside number on the side and the inside terms on top, then multiply into each square.

Try it. Change the numbers:

outside
x term
number
Three things to check:
① Multiply, don't add (−5 × 3v = −15v).
② Hit both terms: 8(e − 1) = 8e − 8.
③ Signs: negative × negative = positive, so −3(4x − 11) = −12x + 33.
Multi-step

🪜 Multi-step recipe

  1. Distribute (box method) to get rid of parentheses
  2. Combine like terms on the same side (like 3x + 36x = 39x)
  3. Undo + or − on both sides
  4. Undo × or ÷ on both sides
  5. Check by plugging your answer back in
−6(−5 + r) = 66start 30 − 6r = 66−6 × −5 = 30, −6 × r = −6r −6r = 36subtract 30 from both sides r = −6divide both sides by −6
Integer rules

➕➖ Sign rules (most mistakes happen here!)

OperationRuleExample
× or ÷Same signs → positive
Different signs → negative		−6 × −7 = 42
18 ÷ −4 = −4.5
AddSame signs: add, keep the sign
Different signs: subtract, keep the sign of the bigger number		−8 + −5 = −13
−9 + 4 = −5
SubtractAdd the opposite3 − (−7) = 3 + 7 = 10
Last step

✅ Always check your answer

Put your answer back into the original equation. If both sides match, you're right.

2 − 7x = 142you got x = −20 2 − 7(−20)= 2 + 140 = 142 ✓

Every problem from the worksheets and notebook pages. Try each one, and tap Next step if you get stuck. The steps show one at a time so you can check your work as you go.

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