Linear equations

An equation states that two expressions are equal and contains an unknown. To solve an equation means to find every number that makes the statement true when substituted for the unknown.

An equation is linear when the unknown appears in the first power — not squared, not in a denominator, not under a root:

A balance scale, or equivalent equations

The equals sign works like a balance scale: both sides weigh the same. As long as we do exactly the same thing to both sides, the balance holds and the set of solutions does not change. Equations related that way are called equivalent.

Two operations preserve equivalence:

The condition $c \neq 0$ is not a formality: multiplying both sides by zero turns any equation into the true $0 = 0$, and dividing by zero cannot be done at all.

This is where the school phrase "move it across and change the sign" comes from. It is not a separate rule but shorthand: from $x + 5 = 12$ we subtract $5$ from both sides and are left with $x = 7$.

Solving step by step

The plan is always the same: tidy up both sides, gather the unknowns on one side and the numbers on the other, then divide by the coefficient of the unknown.

Checking is part of the solution, not decoration: it costs two multiplications and catches every lost sign.

The solution on a graph

The equation $2x + 3 = 11$ can be read as a question: for which $x$ does the line $y = 2x + 3$ reach the height $11$?

The lines cross at a single point, so the equation has exactly one solution. The same picture explains both special cases below: parallel lines never meet, while lines that coincide share every point.

When there is no solution — and when there are infinitely many

Sometimes the unknowns vanish along the way. What is left is a statement about numbers alone, and that statement decides:

An equation as a model

The power of equations comes from describing real situations. A taxi charges 8 for the ride plus 3 per kilometre; how far does 29 take us?

Seven kilometres. The hard part of such a task is rarely the arithmetic — it is writing the equation down.

In the exercises below, enter the solution in the form $x = 5$ (a bare $5$ is accepted too).

Common mistakes

• Moving a term across without changing its sign — $x + 5 = 12$ gives $x = 7$, not $x = 17$.
• Operating on one side only — when you divide, divide both sides.
• Losing a sign at a bracket — $-2(x - 5)$ is $-2x + 10$.
• Dividing by an expression containing the unknown — it may be zero, so solutions get lost.
• Skipping the check — the one step that catches an arithmetic slip.