Solving One-Step Equations

An equation is like a balance scale. Both sides need to be equal for the equation to be true. When we solve an equation, we're finding the value of the variable that makes both sides balance perfectly.

One-step equations are the simplest type because they only require one operation to isolate the variable. If you can do basic arithmetic and understand the idea of "doing the opposite," you can solve these.

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

Let's start with equations where we need to add or subtract.

Example: Solve $x + 7 = 12$.

We want to get $x$ by itself. Right now, there's a $+7$ attached to it. To get rid of the $+7$, we do the opposite operation—we subtract 7.

Subtract 7 from both sides: $$x + 7 - 7 = 12 - 7$$

Simplify: $$x = 5$$

Let's check: Does $5 + 7 = 12$? Yes! 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. We need to 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, we use the opposite operation to solve.

Example: Solve $5x = 30$.

The variable is being multiplied by 5. The opposite of multiplying by 5 is dividing by 5.

Divide both sides by 5: $$\frac{5x}{5} = \frac{30}{5}$$

Simplify: $$x = 6$$

Check: Does $5(6) = 30$? Yes!

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. Don't panic—just remember to multiply by the reciprocal.

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:

$x + 12 = 20$
$y - 5 = 14$
$6m = 42$
$\frac{n}{8} = 3$
$-4p = 32$
$17 = a + 9$
$\frac{3}{5}t = 15$
$x - 7 = -2$

Answers:

$x = 8$ (subtract 12 from both sides)
$y = 19$ (add 5 to both sides)
$m = 7$ (divide both sides by 6)
$n = 24$ (multiply both sides by 8)
$p = -8$ (divide both sides by -4)
$a = 8$ (subtract 9 from both sides)
$t = 25$ (multiply both sides by $\frac{5}{3}$)
$x = 5$ (add 7 to both sides)

What Trips People Up

The most common mistake is forgetting to do the same operation on both sides. If you subtract 5 from the left, you absolutely must subtract 5 from the right.

Getting confused about which operation to use happens a lot. Just remember the key principle: do the OPPOSITE of what's being done to the variable. If it's being added, you subtract. If it's being multiplied, you divide.

Sign errors with negatives are another frequent problem. When you have $-3x = 15$ and divide both sides by -3, the answer is $x = -5$, not $x = 5$.

Always check your answer by plugging it back into the original equation. This catches careless errors and helps you build confidence in your work.

What Comes Next?

One-step equations are the foundation. Once you're comfortable with these, you'll move on to multi-step equations, where you might need to do several operations to isolate the variable. But the core principle—keeping the equation balanced—never changes.