Algebra · Linear equations
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Solving linear equations
Why an equation is a balance, why every move must be the INVERSE operation applied to both whole sides, and how brackets and unknowns on both sides fall to the same two moves.
Algebra · Linear equations
Solving linear equations
Why an equation is a balance, why every move must be the INVERSE operation applied to both whole sides, and how brackets and unknowns on both sides fall to the same two moves.
Why it works
An equation is a balance. says: the two sides are the same number. Anything done to one whole side must be done to the other, or the balance — and the information — is destroyed. Solving is peeling the free one operation at a time, always by the inverse: the is undone by ADDING 7 (not by moving it across unchanged), and the by dividing:"Move it to the other side and flip the sign" is shorthand for exactly this — the flip IS the inverse. Check by substituting back: . ✓
Divide the WHOLE side, not just one term. From , dividing by 3 gives — every term gets divided. (Cleaner: subtract 6 first.) Halving only the breaks the balance.
Unknowns on both sides: collect them on one side first. — subtract from both sides so all the 's stand together: , then , . The -terms *combine by subtraction* across the equals sign; adding them () double-counts.
Brackets: expand first (or divide both sides by the multiplier). is either — the 4 hits both terms — or, since the whole left side is 4 times a package, . Both give . What's illegal is : a half-expanded bracket.
A negative -term isn't special — the same moves work. : subtract 12 to get , divide by to get . (Or add to both sides first so the -count is positive: .) Sign discipline, not a new method.
Solutions aren't always whole numbers. or are perfectly good answers — resist the urge to force an integer, and let the substitution check be the judge.