Why do you have to do the same thing to both sides of an equation?

An equation is like a balanced scale. If you add or do something to only one side, the scale tips and they're no longer equal. Doing the same operation to both sides keeps them balanced and maintains equality. That's how you preserve the truth of the equation while solving it.

How do I know which inverse operation to use?

Look at what is being done to the variable. If the variable is being added to, subtract. If it's being subtracted from, add. If it's being multiplied, divide. If it's being divided, multiply. Ask yourself: "What operation is stuck to my variable?" Then do the opposite.

What if my answer is a decimal or fraction? Is that okay?

Yes, absolutely. Solutions can be whole numbers, decimals, or fractions. As long as your answer is nonnegative (zero or positive) and makes the original equation true when you substitute it back, it's correct. Always check by plugging your answer back into the original equation.

Can a one-step equation have more than one solution?

No. A one-step equation of the form x + p = q or px = q has exactly one solution. The equation is asking for one specific value that makes it true. (Later, you'll learn about equations with no solutions or infinitely many solutions, but those look different.)

M6MATH-8.2

Solving One-Step Equations

Learn to solve one-step equations by using inverse operations to isolate the variable and find its value.

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What this lesson covers

An equation is a mathematical sentence that shows two quantities are equal. Solving an equation means finding the value of the variable that makes the equation true. In this lesson, you'll learn to solve one-step equations—equations that require just one operation to find the variable's value. This skill is essential for algebra and helps you answer real-world questions like "How much did each item cost?" or "How far did I travel?".

Understanding One-Step Equations

A one-step equation has a variable on one side and requires exactly one operation to solve it. The most common forms are addition/subtraction equations (like

x + 5 = 12

x - 3 = 8

) and multiplication/division equations (like

3x = 15

\frac{x}{4} = 6

). In each case, something is being done to the variable, and your job is to undo it. The equation

x + 5 = 12

means "some number plus 5 equals 12." To find

x

, you ask yourself: what operation is being done to

x

? Addition. So you undo it with subtraction. The solution is

x = 7

because

7 + 5 = 12

. Similarly, in

3x = 15

, the variable is being multiplied by 3, so you undo it by dividing both sides by 3 to get

x = 5

Using Inverse Operations

Inverse operations are operations that undo each other. Addition and subtraction are inverses; multiplication and division are inverses. To solve an equation, you perform the inverse operation on both sides to keep the equation balanced. Think of an equation like a balanced scale: if you add weight to one side, you must add the same weight to the other side to keep it balanced. For the equation

x + 7 = 15

, subtract 7 from both sides:

x + 7 - 7 = 15 - 7

, which simplifies to

x = 8

. For

6x = 24

, divide both sides by 6:

\frac{6x}{6} = \frac{24}{6}

, which gives

x = 4

. The key is always performing the same operation on both sides of the equals sign. This preserves the equality and ensures your answer is correct.

Working with Nonnegative Rational Numbers

At this level, the numbers in your equations can be whole numbers, fractions, or decimals—all nonnegative (zero or positive). For example, you might solve

x + 0.5 = 2.3

\frac{1}{2}x = 3

. When working with fractions, use the same inverse operations. In

x + \frac{1}{4} = \frac{3}{4}

, subtract

\frac{1}{4}

from both sides to get

x = \frac{1}{2}

. With decimals, the process is identical: for

x + 1.5 = 4.2

, subtract 1.5 from both sides to find

x = 2.7

. The inverse operation method works the same way whether you're using whole numbers, decimals, or fractions. The main difference is practicing careful arithmetic with those number types.

Checking Your Solution

After solving an equation, always substitute your answer back into the original equation to verify it works. This is called checking your solution. For

x + 9 = 16

, if you found

x = 7

, plug 7 back in:

7 + 9 = 16

✓. For

5x = 35

, if you found

x = 7

, check:

5(7) = 35

✓. If your answer doesn't work when you substitute it back, you know to look for an arithmetic error. Checking teaches you to be confident in your work and catches mistakes before you move forward.

Key terms

One-step equation.: An equation that requires exactly one inverse operation to solve for the variable.
Inverse operations.: Operations that undo each other: addition and subtraction are inverses; multiplication and division are inverses.
Isolate the variable.: To rewrite an equation so the variable stands alone on one side of the equals sign.
Solution.: The value of the variable that makes an equation true when substituted back into the equation.
Nonnegative rational numbers.: Numbers that are either zero or positive and can be expressed as fractions, decimals, or whole numbers.

Worked example

Solve the equation

2.5x = 10

. Show your work and check your answer.

Step 1: Identify what operation is being done to the variable. The variable

x

is being multiplied by 2.5.

Step 2: Apply the inverse operation to both sides. The inverse of multiplication is division. Divide both sides by 2.5:

\frac{2.5x}{2.5} = \frac{10}{2.5}

Step 3: Simplify.

x = 4

Step 4: Check your solution. Substitute

x = 4

back into the original equation:

2.5(4) = 10

10 = 10

✓

The solution is

x = 4

. When you substitute it back, the equation is true, so your answer is correct.

Practice questions

Solve:

x + 6.5 = 11.2

Answer: $x = 4.7$

The variable

x

has 6.5 added to it, so subtract 6.5 from both sides:

x + 6.5 - 6.5 = 11.2 - 6.5

. This simplifies to

x = 4.7

. To check:

4.7 + 6.5 = 11.2

✓.

Solve the equation

\frac{x}{3} = 8

. Show all your work and verify your solution.

Answer: $x = 24$

The variable

x

is being divided by 3. To undo division, multiply both sides by 3:

3 \cdot \frac{x}{3} = 3 \cdot 8

, which gives

x = 24

. Check by substituting back:

\frac{24}{3} = 8

✓. The solution is correct.

Which equation is solved correctly?

$x - 5 = 12$ , so $x = 7$
$4x = 20$ , so $x = 5$
$x + 3 = 10$ , so $x = 13$
$\frac{x}{2} = 6$ , so $x = 3$

Answer: $4x = 20$ , so $x = 5$

Check each: (1)

7 - 5 = 2

, not 12, so incorrect. (2)

4(5) = 20

✓, correct. (3)

13 + 3 = 16

, not 10, incorrect. (4)

\frac{3}{2} = 1.5

, not 6, incorrect. The second equation is solved correctly.

FAQ

Why do you have to do the same thing to both sides of an equation?: An equation is like a balanced scale. If you add or do something to only one side, the scale tips and they're no longer equal. Doing the same operation to both sides keeps them balanced and maintains equality. That's how you preserve the truth of the equation while solving it.
How do I know which inverse operation to use?: Look at what is being done to the variable. If the variable is being added to, subtract. If it's being subtracted from, add. If it's being multiplied, divide. If it's being divided, multiply. Ask yourself: "What operation is stuck to my variable?" Then do the opposite.
What if my answer is a decimal or fraction? Is that okay?: Yes, absolutely. Solutions can be whole numbers, decimals, or fractions. As long as your answer is nonnegative (zero or positive) and makes the original equation true when you substitute it back, it's correct. Always check by plugging your answer back into the original equation.
Can a one-step equation have more than one solution?: No. A one-step equation of the form $x + p = q$ or $px = q$ has exactly one solution. The equation is asking for one specific value that makes it true. (Later, you'll learn about equations with no solutions or infinitely many solutions, but those look different.)

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