Lecture · 13 min · basic

Linear equations

Balance, inverses, and a unique solution

Euler · simple equations

A linear equation is a balanced scale. Whatever you do to one pan you must do to the other — not because a teacher said so, but because equality is an equivalence relation preserved by invertible operations.

On the board today

—Solve $a x + b = c$ by undoing operations in reverse order.
—Know when a linear equation has one, none, or infinitely many solutions.
—Translate a word sentence into $a x + b = c$ .

The general line

a x + b = c (a \neq = 0) ⟹ x = \frac{c - b}{a}

If $a = 0$ , the equation is no longer about $x$ . It becomes the statement $b = c$ : either always true (infinitely many solutions) or never (none). The coefficient of $x$ is what makes the unknown visible. Geometrically, $y = a x + b$ is a non-horizontal line only when $a \neq = 0$ ; it hits $y = c$ exactly once.

Worked example · Clearing a fraction

Solve $\frac{2 x - 1}{3} = 5$ .

1.Multiply both sides by $3$ : $2 x - 1 = 15$ .
2.Add $1$ : $2 x = 16$ .
3.Divide by $2$ : $x = 8$ .

$x = 8$ . Check: $(16 - 1) /3 = 5$ .

Pause the lecture

How many solutions does $0 \cdot x + 4 = 4$ have? How many does $0 \cdot x + 4 = 5$ have?

Common mistake. Dividing by an expression that might be zero silently loses solutions or invents them. If you divide by $x - 1$ , you must check $x = 1$ separately.

Practice this lecture

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