Foundations — Linear equations in one variable — solving, transposition method

Before you can solve $ax+b=cx+d$, you must be sure of what each mark on the page means. This page builds every one of those marks from nothing — no symbol is used before it is drawn. Read top to bottom; each idea leans on the one above it.

1. A number and the number line

The very first picture. A number is a position on a straight ruler that stretches forever both ways.

• Picture: the amber dot sits at $+3$, three equal steps right of zero.
• Why the topic needs it: the answer to an equation is one such position. When we finally write $x=3$, we are pointing at exactly this dot. Positive/negative also decides every sign flip later, so we anchor it now.

2. Constants: numbers that stay put

• Picture: a nailed-down marker on the number line. It cannot slide.
• Why the topic needs it: in $ax+b=c$, the letters $a,b,c$ are stand-ins for constants — real numbers you already know but that we're too lazy to write out while explaining the general rule. See 2.1.01-Variables-and-constants for the fuller story.

3. The variable $x$: a number we don't know yet

• Picture: a covered box on the number line with a question mark. Somewhere under that box is a real position — solving the equation lifts the lid.
• Why "$x$"? Just tradition. It could be $y$, a box, or a smiley. What matters: one letter = one unknown number.
• Why the topic needs it: "in one variable" literally means one such box. If a second box $y$ appeared (as in 2.1.08-Linear-equations-in-two-variables), it would be a different game — you'd need two clues, not one.

4. Coefficient: how many copies of the box

Writing $5x$ means "five boxes side by side," i.e. $x+x+x+x+x$.

• Picture: five identical boxes stacked in a row — whatever weight is in one, the row weighs five times that.
• Why the topic needs it: the last step of every solution divides by this coefficient to shrink "five boxes" down to "one box." If the coefficient were $0$, there'd be no box at all — that's exactly why the parent insists $a\ne 0$.

5. The operation signs $+-\times ÷$ and their inverses

Every symbol here comes in a pair: an action and the action that undoes it.

• Picture: two arrows on the number line pointing opposite ways and cancelling to land you back where you started.
• Why the topic needs it: "transposition" is nothing but applying the inverse to both sides at once, written as a shortcut. Every rule in the parent's [!formula] box is one row of this table.

6. The equals sign $=$ and the balance scale

This is the heart. The most misread symbol in all of algebra.

• Picture: a balance scale, perfectly level. Left pan = right pan, in weight.
• The golden rule made visual: if you add weight to one pan you must add the same to the other, or the scale tips and the statement stops being true.
• Why the topic needs it: this is why transposition is allowed. "Move $+5$ across as $-5$" is short for "subtract $5$ from both pans." The scale never tips, so equality survives every step.

7. Fractions and the fraction bar

• Picture: one whole cut into $q$ equal strips; you keep $p$ of them.
• Why the topic needs it: Example 3 in the parent, $\frac{2x-1}{3}=5$, has the whole left pan divided by $3$. To undo it you multiply both pans by $3$ — otherwise the scale tips. Forgetting to multiply the whole other side is a classic slip (parent's Mistake 2).

8. Highest power = 1 (what "linear" means)

• Picture: $x^1$ is a single flat step — it draws a straight line on a graph (see 3.2.04-Graphical-solution-of-linear-equations). $x^2$ would bend into a curve.
• Why the topic needs it: "linear in one variable" = the highest power of $x$ is exactly $1$. No $x^2$, no $\sqrt{x}$. That flatness is what guarantees a single clean answer and lets transposition work in a fixed number of steps.

Prerequisite map

Read it as a bottom-up build: the number line feeds everything; constants and the variable combine into the coefficient; operations give us inverses; inverses plus the balance-scale meaning of $=$ give us transposition; the "power $=1$" idea labels the whole thing linear.

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, re-read that section before touching the parent note.