One-Step Equations

An equation is a statement that two expressions are equal, like $x + 7 = 12$ . Solving an equation means finding the value (or values) of the variable that make the statement true. A one-step equation needs only a single operation to isolate the variable.

Inverse Operations

The key idea is that every operation has an inverse that undoes it, and whatever you do to one side of an equation, you must do to the other side too, to keep it balanced:

Addition and subtraction undo each other.
Multiplication and division undo each other.

To solve $x + a = b$ , subtract $a$ from both sides: $x = b - a$ . To solve $x - a = b$ , add $a$ to both sides: $x = b + a$ . To solve $ax = b$ (with $a \neq 0$ ), divide both sides by $a$ : $x = \frac{b}{a}$ . To solve $\frac{x}{a} = b$ , multiply both sides by $a$ : $x = ab$ .

Worked Examples of Each Type

$x + 9 = 4$ : subtract $9$ from both sides to get $x = 4 - 9 = -5$ .
$x - 6 = 15$ : add $6$ to both sides to get $x = 21$ .
$7x = -35$ : divide both sides by $7$ to get $x = -5$ .
$\frac{x}{4} = 9$ : multiply both sides by $4$ to get $x = 36$ .

Always Check Your Solution

After solving, it's good practice to substitute your answer back into the original equation to confirm both sides match. This habit catches arithmetic slips before they cost you points on a contest. For $7x = -35$ with solution $x = -5$ : substituting gives $7(-5) = -35$ , which checks out.

Example

Solve for $x$ : $-3x = 21$ .

Step 1: The variable is being multiplied by $-3$ , so divide both sides by $-3$ to undo that: $\frac{-3x}{-3} = \frac{21}{-3}$

Step 2: Simplify. Dividing a positive by a negative gives a negative: $x = -7$ .

Step 3 (check): substitute back in: $-3(-7) = 21$ . Since $21 = 21$ , the solution is correct.

Key Takeaways

A one-step equation is solved by applying a single inverse operation to both sides.
Addition undoes subtraction (and vice versa); multiplication undoes division (and vice versa).
Whatever operation you apply, apply it to both sides to keep the equation balanced.
Always check your solution by substituting it back into the original equation.

Practice problems

2000 Fermat Problem 6 Fermat

1 pts

2000 Pascal Problem 1 Pascal

1 pts

2000 Pascal Problem 2 Pascal

1 pts

2000 Pascal Problem 3 Pascal

1 pts

2000 Pascal Problem 4 Pascal

1 pts