Why do I have to do the same thing to both sides?

Because the equal sign means both sides currently have the same value, like a balanced scale. If you change one side only, the two sides stop being equal and your new equation no longer has the same solution as the original. Applying the identical operation to both sides keeps the balance and keeps the solution unchanged.

Is checking my answer really necessary if I am confident?

Yes, and it takes about ten seconds. Substitution is the only way to know for sure, and it catches the two most common slips: using the same operation instead of the inverse, and mishandling a negative sign. It also means you never have to wonder whether an answer is right, because you have proven it.

Should I divide by a fraction or multiply by the reciprocal?

Both give the same answer, but multiplying by the reciprocal is usually faster and less error-prone. For 4/5x = 12, multiply both sides by 5/4 to get x = 15. Dividing by 4/5 requires flipping the fraction anyway, so you may as well flip it from the start.

What do I do when the equation says -x = 12?

Recognize that -x means -1 · x, so the coefficient is -1. Divide both sides by -1 to get x = -12. Check it: -(-12) = 12, which is true. The answer is not 12, because -x with x = 12 would give -12.

M7MATH-6.1

One-Step Equations

Learn to solve one-step equations like x + p = q and px = q using inverse operations, handle negative and fraction coefficients, and check every solution by substituting.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on One-Step Equations, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start One-Step Equations free

What this lesson covers

An equation is a math sentence claiming two expressions are equal. Solving one means finding the number that makes the sentence true. In a one-step equation, exactly one operation stands between the variable and its value — so exactly one move undoes it.

This lesson gives you a reliable routine: look at what is happening to the variable, undo it with the inverse operation on both sides, then substitute your answer back into the original equation to prove it works. You will handle addition and subtraction equations, multiplication and division equations, negative numbers, and coefficients that are fractions like

\frac{2}{3}x = 10

. These same moves power the two-step equations and word problems coming next in this unit, so building the habit of undoing carefully now saves a lot of confusion later.

What It Means to Solve an Equation

An equation like

x + 7 = 12

says "some number plus 7 equals 12." The solution is the value of the variable that makes both sides equal. Here the solution is 5, because

5 + 7 = 12

is a true statement.

Think of the equal sign as the center of a balance scale. Both sides currently weigh the same. If you add 4 pounds to the left pan, you must add 4 pounds to the right pan, or the scale tips and the equation is no longer true. This is the whole idea behind solving: whatever operation you do to one side, you must do to the other side.

Your goal is to get the variable alone on one side — this is called isolating the variable. You isolate it by undoing whatever is attached to it using an inverse operation.

Operation in the equation	Inverse used to undo it
Adding $p$	Subtract $p$ from both sides
Subtracting $p$	Add $p$ to both sides
Multiplying by $p$	Divide both sides by $p$
Dividing by $p$	Multiply both sides by $p$

A very common early mistake is doing the same operation instead of the inverse. For

x + 7 = 12

, adding 7 to both sides gives

x + 14 = 19

, which is still true but no closer to an answer. Ask yourself: "What is being done to

x

?" Then do the opposite.

Addition and Subtraction Equations: x + p = q

For the form

x + p = q

, subtract

p

from both sides. Write the subtraction under both sides so your work lines up:

x + 7 = 12

x + 7 - 7 = 12 - 7

x = 5

When the equation subtracts from the variable, add instead. For

m - 9 = 4

, add 9 to both sides to get

m = 13

.

Negative numbers show up constantly here, and this is where careful work matters. Consider

x + 8 = 3

. Subtracting 8 from both sides gives

x = 3 - 8 = -5

. Many students stop at 5 because they compute

8 - 3

instead. Check it:

-5 + 8 = 3

. True, so

-5

is right.

Now consider

y + (-4) = 10

, which can be rewritten as

y - 4 = 10

. Adding 4 to both sides gives

y = 14

. Rewriting "plus a negative" as subtraction before you start prevents sign errors.

One more form to recognize:

-6 = k - 2

. The variable is on the right, which is perfectly fine. Add 2 to both sides:

-4 = k

, so

k = -4

. You do not need to flip the equation around first, though you may if it feels clearer.

Decimals and fractions work identically. For

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

, subtract

\frac{1}{4}

from both sides to get

x = \frac{2}{4} = \frac{1}{2}

Multiplication and Division Equations: px = q

px = q

, the number

p

multiplied by the variable is the coefficient. To undo multiplication, divide both sides by that coefficient.

4x = 28

\frac{4x}{4} = \frac{28}{4}

x = 7

With a negative coefficient, keep the sign attached to the number you divide by. For

-5x = 35

, divide both sides by

-5

x = -7

. Dividing by positive 5 and then guessing the sign is how errors creep in.

If the variable is divided by a number, multiply. For

\frac{n}{3} = 12

, multiply both sides by 3 to get

n = 36

.

Fractional coefficients have a shortcut. Instead of dividing by

\frac{2}{3}

, multiply both sides by its reciprocal,

\frac{3}{2}

\frac{2}{3}x = 10

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

x = 15

The reciprocal times the original fraction equals 1, which leaves

x

alone. Notice that dividing by a fraction less than 1 makes the answer larger — that surprises students, but

\frac{2}{3}

of 15 really is 10.

Watch out for

-x = 9

. The hidden coefficient is

-1

, so divide both sides by

-1

to get

x = -9

. The answer is not 9.

Finally, do not mix the two families up. In

6x = 18

you divide, giving

x = 3

; you do not subtract 6, which would wrongly give 12.

Checking by Substitution

A solution is only confirmed once you substitute it back into the original equation and see both sides come out equal. Substituting into a line you rewrote will not catch a mistake you made while rewriting.

The format your teacher will look for has three parts. Replace the variable with your value, simplify each side separately, and state whether the two sides match.

Check

x = -7

-5x = 35

-5(-7) = 35

35 = 35 \;\checkmark

Use parentheses around a negative value you substitute. Writing

-5 \cdot -7

without them invites sign slips, and writing

-5-7

by accident changes the problem entirely.

If the two sides do not match, you have caught an error — that is the checking step doing its job. Go back and ask two questions: did I use the inverse operation, and did I apply it to both sides? Those two questions cover nearly every mistake in a one-step equation.

Equation	Step	Solution	Check
$x - 6 = -2$	Add 6	$x = 4$	$4 - 6 = -2$
$\frac{x}{-3} = 5$	Multiply by $-3$	$x = -15$	$\frac{-15}{-3} = 5$
$\frac{3}{4}x = 9$	Multiply by $\frac{4}{3}$	$x = 12$	$\frac{3}{4}(12) = 9$

Checking also builds number sense you will lean on later. When you meet two-step equations, a quick substitution tells you instantly whether both of your undoing steps worked.

Choosing the Right Move: A Decision Routine

Students rarely get stuck on the arithmetic. They get stuck deciding which operation to undo. Use this routine every time.

First, find the variable and cover it with your finger. Whatever is left touching it is what you must undo. In

x - 11 = 4

, what touches

x

-11

, so add 11. In

\frac{2}{5}x = 8

, what touches

x

is multiplication by

\frac{2}{5}

, so multiply by

\frac{5}{2}

.

Second, name the operation out loud: "

x

is being multiplied by negative 3." Naming it makes the inverse obvious.

Third, write the inverse operation on both sides, aligned under the equal sign. Skipping this written line is where careless errors live, because you end up doing the step to only one side in your head.

Fourth, substitute to check.

One subtlety:

x + p = q

and

px = q

look similar on the page but behave very differently. Compare

x + 5 = 20

(solution 15) with

5x = 20

(solution 4). The presence or absence of a plus sign changes everything, so read the equation before you start computing.

Also remember that a solution can be negative, a fraction, or zero. There is no rule saying answers must be whole numbers. For

8x = 6

, dividing gives

x = \frac{6}{8} = \frac{3}{4}

, and that is a perfectly good final answer — leave it as a reduced fraction rather than rounding a decimal.

Key terms

Equation.: A mathematical sentence stating that two expressions are equal, such as $x + 4 = 9$ .
Solution.: The value of the variable that makes an equation true. Substituting it produces equal values on both sides.
Inverse operation.: The operation that undoes another. Addition and subtraction are inverses; multiplication and division are inverses.
Isolate the variable.: To get the variable alone on one side of the equal sign, which reveals its value.
Coefficient.: The number multiplied by a variable. In $-7x$ , the coefficient is $-7$ ; in $-x$ , it is $-1$ .
Reciprocal.: The fraction flipped upside down. The reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$ , and a number times its reciprocal equals 1.
Substitution.: Replacing a variable with a number, used to check whether a proposed solution makes the original equation true.
Properties of equality.: The rules allowing you to add, subtract, multiply, or divide both sides of an equation by the same amount and keep it true.

Worked example

Solve each equation and check your solution: (a)

x + 13 = 6

(b)

-\frac{3}{4}m = 9

Part (a). Look at what touches

x

: it is

+13

. The inverse of adding 13 is subtracting 13, so subtract 13 from both sides.

x + 13 = 6

x + 13 - 13 = 6 - 13

x = -7

The right side is

6 - 13

, which is negative because we are subtracting a larger number from a smaller one. Check by substituting into the original equation:

-7 + 13 = 6

, and

6 = 6

, so

x = -7

is correct. Notice that answering 7 here would fail the check, since

7 + 13 = 20

, not 6.

Part (b). Here

m

is multiplied by

-\frac{3}{4}

. Rather than dividing by a fraction, multiply both sides by the reciprocal of

-\frac{3}{4}

, which is

-\frac{4}{3}

-\frac{3}{4}m = 9

-\frac{4}{3}\cdot\left(-\frac{3}{4}m\right) = -\frac{4}{3}\cdot 9

On the left,

-\frac{4}{3}\cdot-\frac{3}{4} = 1

, leaving just

m

. On the right,

-\frac{4}{3}\cdot 9 = -\frac{36}{3} = -12

m = -12

Check in the original:

-\frac{3}{4}(-12) = \frac{36}{4} = 9

. A negative times a negative is positive, and

9 = 9

, so

m = -12

is correct. The answer is negative because a negative coefficient times a negative value gives the positive 9 on the right.

Practice questions

What is the solution to

-6x = 42

$x = 7$
$x = -7$
$x = 36$
$x = -252$

Answer: $x = -7$

The variable is multiplied by

-6

, so divide both sides by

-6

\frac{-6x}{-6} = \frac{42}{-6}

, giving

x = -7

. A positive divided by a negative is negative. Checking confirms it:

-6(-7) = 42

. The answer 7 comes from ignoring the negative sign, 36 comes from subtracting 6 instead of dividing, and

-252

comes from multiplying instead of dividing.

Solve

\frac{5}{6}y = 20

and show the check.

Answer: $y = 24$

The coefficient is

\frac{5}{6}

, so multiply both sides by its reciprocal

\frac{6}{5}

. On the left,

\frac{6}{5}\cdot\frac{5}{6}y = y

. On the right,

\frac{6}{5}\cdot 20 = \frac{120}{5} = 24

. So

y = 24

. Check in the original equation:

\frac{5}{6}(24) = \frac{120}{6} = 20

, and

20 = 20

, so the solution is confirmed. The answer is larger than 20 because taking only

\frac{5}{6}

of the number must shrink it down to 20.

Maya solved

x - 9 = -4

and wrote

x = -13

. Explain her error and give the correct solution with a check.

Answer: She subtracted 9 instead of adding it; the correct solution is $x = 5$ .

In the equation, 9 is being subtracted from

x

, so the inverse operation is to add 9 to both sides:

x - 9 + 9 = -4 + 9

, which gives

x = 5

. Maya used the same operation as the equation rather than the inverse, computing

-4 - 9 = -13

. Substituting her answer exposes the problem:

-13 - 9 = -22

, not

-4

. Substituting the correct value works:

5 - 9 = -4

, a true statement.

FAQ

Why do I have to do the same thing to both sides?: Because the equal sign means both sides currently have the same value, like a balanced scale. If you change one side only, the two sides stop being equal and your new equation no longer has the same solution as the original. Applying the identical operation to both sides keeps the balance and keeps the solution unchanged.
Is checking my answer really necessary if I am confident?: Yes, and it takes about ten seconds. Substitution is the only way to know for sure, and it catches the two most common slips: using the same operation instead of the inverse, and mishandling a negative sign. It also means you never have to wonder whether an answer is right, because you have proven it.
Should I divide by a fraction or multiply by the reciprocal?: Both give the same answer, but multiplying by the reciprocal is usually faster and less error-prone. For $\frac{4}{5}x = 12$ , multiply both sides by $\frac{5}{4}$ to get $x = 15$ . Dividing by $\frac{4}{5}$ requires flipping the fraction anyway, so you may as well flip it from the start.
What do I do when the equation says $-x = 12$ ?: Recognize that $-x$ means $-1 \cdot x$ , so the coefficient is $-1$ . Divide both sides by $-1$ to get $x = -12$ . Check it: $-(-12) = 12$ , which is true. The answer is not 12, because $-x$ with $x = 12$ would give $-12$ .

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