Worked examples — Linear equations in one variable — solving, transposition method

Before anything, one reminder of the whole game. An equation is a balance scale. The left pan and the right pan weigh the same. To keep them equal, any move you make on one pan you must make on the other. "Transposing" is just the fast way of writing that move.

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The scenario matrix

Here is every type of linear equation. Each row is a "cell" — a distinct thing that can go wrong or need care. The worked examples below are each tagged with the cell they cover.

Cell	What makes it special	Danger it hides
A	Positive coefficient, positive constant	none — the warm-up
B	Variable on both sides	which side do the $x$'s go?
C	Negative coefficient of $x$	dividing by a negative flips signs
D	Fraction (variable in numerator over a number)	must multiply the whole other side
E	Brackets to expand first	forgetting to distribute
F	Word problem (real world)	turning English into $ax+b=c$
G	Degenerate: no solution ($a=c$, $b\ne d$)	you get a false statement
H	Degenerate: infinite solutions ($a=c$, $b=d$)	you get a always-true statement
I	Zero on a side / answer is zero	$x=0$ is a real answer, not "no answer"

We now hit every cell.

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Cell A — the warm-up (positive everything)

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Cell B — variable on both sides

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Cell C — negative coefficient (the sign trap)

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Cell D — a fraction with the variable on top

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Cell E — brackets first (distribute)

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Cell F — a word problem (English → equation)

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Cell G — no solution (a lie appears)

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Cell H — infinitely many solutions (always true)

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Cell I — the answer is zero (don't panic)

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The whole map in one diagram

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Connections

• Linear equations in one variable — solving, transposition method — the parent method these examples drill
• 2.1.01-Variables-and-constants — what the letters $x,a,b,c$ mean
• 2.1.06-Forming-equations-from-word-problems — building Cell F from English
• 2.1.08-Linear-equations-in-two-variables — when one letter isn't enough
• 2.3.01-Linear-inequalities — same moves with $<$, where dividing by a negative flips the sign
• 2.5.02-Balancing-chemical-equations — the same balancing logic in chemistry
• 3.2.04-Graphical-solution-of-linear-equations — Cells G/H/I as pictures of lines

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Recall Quick self-test

An equation reduces to $0=0$. How many solutions? ::: Infinitely many (identity, Cell H). An equation reduces to $7=-2$. How many solutions? ::: None (contradiction, Cell G). An equation reduces to $3x=0$. How many solutions? ::: Exactly one, $x=0$ (Cell I). When collecting variables on both sides, which side should you pick? ::: The side that keeps the $x$-coefficient positive, to avoid sign errors.