Solving One-Step Equations

An equation is like a balance scale: both sides have to be equal for it to be true. Solving an equation means finding the value of the variable that keeps both sides balanced.

One-step equations are the simplest type — they only require a single operation to isolate the variable. If you can do basic arithmetic and grasp the idea of "doing the opposite," you can solve them.

The Golden Rule: What You Do to One Side, You Must Do to the Other

This is the most important concept in all of algebra. If you have an equation and you add 5 to the left side, you must add 5 to the right side too. If you divide the left side by 3, you must divide the right side by 3. This keeps the equation balanced.

Think about it with a real balance scale. If you have equal weights on both sides and you add a weight to just one side, it tips over. To keep it balanced, you have to add the same weight to both sides.

Addition and Subtraction Equations

Start with equations that need addition or subtraction to undo.

Example: Solve $x + 7 = 12$.

The goal is to get $x$ by itself. Right now, there's a $+7$ attached to it. The opposite of adding 7 is subtracting 7, so subtract 7 from both sides:

$$x + 7 - 7 = 12 - 7$$

Simplify: $$x = 5$$

Check: $5 + 7 = 12$ ✓. So $x = 5$ is correct.

Example: Solve $y - 9 = 15$.

The variable has $-9$ attached to it. The opposite of subtracting 9 is adding 9.

Add 9 to both sides: $$y - 9 + 9 = 15 + 9$$

Simplify: $$y = 24$$

Check: Does $24 - 9 = 15$? Yes!

Example: Solve $23 = n + 8$.

Notice the variable is on the right side this time. That's fine — the same rules apply. Subtract 8 from both sides.

$$23 - 8 = n + 8 - 8$$ $$15 = n$$

Or you can write it as $n = 15$ if you prefer. Both mean the same thing.

Multiplication and Division Equations

When the variable is being multiplied or divided, the opposite operation undoes it.

Example: Solve $5x = 30$.

The variable is being multiplied by 5. What operation undoes multiplication? Show answerDivision. Divide both sides by 5: $\frac{5x}{5} = \frac{30}{5}$, so $x = 6$. Check: $5(6) = 30$ ✓

Example: Solve $\frac{m}{4} = 7$.

The variable is being divided by 4. The opposite of dividing by 4 is multiplying by 4.

Multiply both sides by 4: $$4 \cdot \frac{m}{4} = 4 \cdot 7$$

Simplify: $$m = 28$$

Check: Does $\frac{28}{4} = 7$? Yes!

Example: Solve $-3y = 21$.

The variable is being multiplied by -3. Divide both sides by -3.

$$\frac{-3y}{-3} = \frac{21}{-3}$$

When you divide a positive by a negative, you get a negative: $$y = -7$$

Check: Does $-3(-7) = 21$? Yes, because negative times negative is positive.

Equations with Fractions

Sometimes the coefficient (the number in front of the variable) is a fraction. The move is the same: multiply both sides by the reciprocal of that fraction.

Example: Solve $\frac{2}{3}x = 10$.

The variable is being multiplied by $\frac{2}{3}$. To undo this, multiply both sides by the reciprocal, which is $\frac{3}{2}$.

$$\frac{3}{2} \cdot \frac{2}{3}x = \frac{3}{2} \cdot 10$$

The left side simplifies to $x$ because $\frac{3}{2} \cdot \frac{2}{3} = 1$.

$$x = \frac{3}{2} \cdot 10 = \frac{30}{2} = 15$$

Check: Does $\frac{2}{3}(15) = 10$? Let's see: $\frac{2 \cdot 15}{3} = \frac{30}{3} = 10$. Yes!

Practice Problems

Try solving these on your own:

$x + 12 = 20$ Show answer$x = 8$ (subtract 12 from both sides)
$y - 5 = 14$ Show answer$y = 19$ (add 5 to both sides)
$6m = 42$ Show answer$m = 7$ (divide both sides by 6)
$\frac{n}{8} = 3$ Show answer$n = 24$ (multiply both sides by 8)
$-4p = 32$ Show answer$p = -8$ (divide both sides by -4)
$17 = a + 9$ Show answer$a = 8$ (subtract 9 from both sides)
$\frac{3}{5}t = 15$ Show answer$t = 25$ (multiply both sides by $\frac{5}{3}$)
$x - 7 = -2$ Show answer$x = 5$ (add 7 to both sides)

What Comes Next?

Three things to keep an eye on as you practice. First, whatever you do to one side, you must do to the other; that's the rule that keeps the equation balanced. Second, the operation you apply is the opposite of what's already being done to the variable: addition undoes subtraction, multiplication undoes division. Third, watch the negative signs: $-3x = 15$ gives $x = -5$, not $x = 5$. When in doubt, plug your answer back into the original equation.

Once one-step equations feel routine, you're ready for multi-step equations, where two or more operations have to be undone in sequence. The core idea of keeping the equation balanced doesn't change — there's just more bookkeeping.