Linear equations in one variable — solving, transposition method

What Is a Linear Equation in One Variable?

Why "linear"? The graph of $y = ax + b$ is a straight line. A solution is where that line crosses a horizontal line at height $c$.

Why "one variable"? Only one letter to find. If we had $2x + 3y = 7$, that's two variables — different game.

The Transposition Method: Step-by-Step

Goal: Isolate $x$ on one side so we get $x = \text{something}$.

Why does this work? We're actually performing the inverse operation on both sides:

• $x + 5 = 12$ → subtract $5$ from both sides → $x = 12 - 5$
• Writing "$+5$ moves to the other side as $-5$" is the shortcut

Let's derive the transposition rules from scratch.

Derivation from First Principles

Starting point: An equation is a statement of equality. The golden rule: do the same thing to both sides.

Case 1: Addition/Subtraction

Start: $x + a = b$

Apply the inverse operation (subtract $a$ from both sides): $x + a - a = b - a$ $x = b - a$

The shortcut: "$+a$ on the left becomes $-a$ on the right."

Case 2: Multiplication/Division

Start: $ax = b$ (where $a \neq 0$)

Apply the inverse operation (divide both sides by $a$): $\frac{ax}{a} = \frac{b}{a}$ $x = \frac{b}{a}$

The shortcut: "$\times a$ on the left becomes $\div a$ on the right."

What if we have both? Undo operations in reverse order of PEMDAS. If the equation built up as "multiply then add," we undo as "subtract then divide."

Worked Examples

Standard Algorithm

The 4-Step Process:

1. Simplify each side independently (combine like terms, expand brackets)
2. Collect all variable terms on one side, constants on the other (transpose as needed)
3. Simplify to get $ax = b$
4. Divide both sides by coefficient: $x = \frac{b}{a}$

Order matters: Alwaysundo addition/subtraction before multiplication/division (reverse of PEMDAS).

Common Mistakes

Key Formulas & Facts

Connections

• 2.1.01-Variables-and-constants — what $x$ and $a$ mean
• 2.1.06-Forming-equations-from-word-problems — where these equations come from
• 2.1.08-Linear-equations-in-two-variables — next level: two unknowns
• 2.3.01-Linear-inequalities — what if we have $<$ instead of $=$?
• 2.5.02-Balancing-chemical-equations — same transposition logic in chemistry
• 3.2.04-Graphical-solution-of-linear-equations — geometric view

Practice Recall

#flashcards/maths

What is a linear equation in one variable? :: An equation where the variable appears only to the first power, in the form $ax + b = c$ or equivalent, with $a \neq 0$.

State the transposition rule for multiplication. :: When a coefficient multiplies the variable, it crosses as division (and division crosses as multiplication).

Concept Map

Hinglish (regional understanding)