Pre-Algebra · Algebra 1 · Grades 6, 7

How to Solve One-Step Equations

Quick answer

To solve a one-step equation, undo the operation attached to the variable by doing the opposite operation to both sides. If something is added, subtract it; if the variable is multiplied, divide. For x + 7 = 12, subtract 7 from both sides to get x = 5. Always check by substituting your answer back in.

What you'll learn

  • Solve one-step equations using inverse operations
  • Explain why the same operation must be applied to both sides
  • Check a solution by substituting it into the original equation

What a one-step equation is

An equation says that two expressions are equal. A one-step equation is one you can solve by doing a single operation to both sides.

x+7=12x + 7 = 12

Solving means finding the value of the variable that makes the statement true. Here that value is 55, because 5+7=125 + 7 = 12.

Definition

To isolate the variable means to get it alone on one side of the equals sign. That is the goal of every equation you will ever solve, from Algebra 1 through calculus.

Why you do the same thing to both sides

Picture a balance scale with the equals sign as the pivot. The equation x+7=12x + 7 = 12 says the two pans weigh the same.

If you remove 77 from the left pan only, the scale tips — the two sides are no longer equal, and the equation you are looking at is no longer the one you started with. Remove 77 from both pans and the scale stays level. The weights changed, but the balance did not, so the value of xx that made it true is still the same value.

That is the entire rule, and it never changes:

Whatever you do to one side of an equation, you must do to the other.

How to solve a one-step equation

  1. Look at what is being done to the variable.
  2. Do the inverse operation to both sides.
  3. Simplify.
  4. Check by substituting your answer into the original equation.

The four inverse pairs:

If the variable has…Undo it with…
something addedsubtraction
something subtractedaddition
something multiplieddivision
something dividedmultiplication

Why the answer is a point on the number line

An equation has one job: to pin the variable to a value. The answer to x+7=12x + 7 = 12 is a single place.

The solution of x + 7 = 12 A number line from 0 to 10 with a single filled dot on 5. 0 1 2 3 4 5 6 7 8 9 10 5
The solution of x + 7 = 12

That is worth noticing now, because it is what changes later. An inequality such as x+7<12x + 7 < 12 answers the same kind of question but shades a whole stretch of the line instead of marking one dot. Same method, different shape of answer.

Checking is not optional

Substituting your answer back into the original equation is the one step that tells you, on your own, whether you are right.

x=5⇒5+7=12  ✓x = 5 \quad\Rightarrow\quad 5 + 7 = 12 \;\checkmark

Check against the equation you were given, not against a line you wrote afterwards. If you copied a number down wrongly at step one, checking against your own copy will agree with the mistake.

Worked examples

Common mistakes

Where this goes next

Every equation you meet later is this one with more steps in front of it.

EquationWhat changedLesson
x+7=12x + 7 = 12one operationthis one
2x+7=122x + 7 = 12two operationstwo-step equations
2(x+3)+7=122(x + 3) + 7 = 12brackets and like terms firstmulti-step equations
2x+7=x+122x + 7 = x + 12variables on both sidesvariables on both sides

The goal never changes — isolate the variable — and neither does the rule that whatever you do to one side you do to the other. Only the amount of tidying before you get there.

Practice problems

  1. Solve x+9=15x + 9 = 15.

    Hint

    Something is added to xx. Do the opposite to both sides.

    Answer

    x=6x = 6

    Full solution

    Subtract 99 from both sides: x+9−9=15−9x + 9 - 9 = 15 - 9, so x=6x = 6.

    Check: 6+9=156 + 9 = 15 ✓

  2. Solve y−12=3y - 12 = 3.

    Answer

    y=15y = 15

    Full solution

    Add 1212 to both sides: y−12+12=3+12y - 12 + 12 = 3 + 12, so y=15y = 15.

    Check: 15−12=315 - 12 = 3 ✓

  3. Solve 8k=568k = 56.

    Answer

    k=7k = 7

    Full solution

    Divide both sides by 88: k=568=7k = \tfrac{56}{8} = 7.

    Check: 8(7)=568(7) = 56 ✓

  4. Solve p4=9\tfrac{p}{4} = 9.

    Answer

    p=36p = 36

    Full solution

    Multiply both sides by 44: p=4⋅9=36p = 4 \cdot 9 = 36.

    Check: 36÷4=936 \div 4 = 9 ✓

  5. Solve −5w=35-5w = 35.

    Hint

    The coefficient is −5-5. Divide by the whole thing, sign included.

    Answer

    w=−7w = -7

    Full solution

    Divide both sides by −5-5: w=35−5=−7w = \tfrac{35}{-5} = -7.

    Check: −5(−7)=35-5(-7) = 35 ✓

  6. Solve t+4.5=10t + 4.5 = 10.

    Answer

    t=5.5t = 5.5

    Full solution

    Subtract 4.54.5 from both sides: t=10−4.5=5.5t = 10 - 4.5 = 5.5.

    Check: 5.5+4.5=105.5 + 4.5 = 10 ✓

  7. Solve x−6=−2x - 6 = -2.

    Answer

    x=4x = 4

    Full solution

    Add 66 to both sides: x=−2+6=4x = -2 + 6 = 4.

    Check: 4−6=−24 - 6 = -2 ✓

  8. A bag holds cc marbles. After adding 1818 marbles there are 4545. Write and solve an equation for cc.

    Hint

    “Adding 18 gives 45” translates directly to an equation.

    Answer

    c+18=45c + 18 = 45, so c=27c = 27

    Full solution

    The equation is c+18=45c + 18 = 45.

    Subtract 1818 from both sides: c=45−18=27c = 45 - 18 = 27.

    Check: 27+18=4527 + 18 = 45 ✓ The bag started with 2727 marbles.

Frequently asked questions

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

An equation says two things are equal. If you change one side without changing the other, they stop being equal and the equation is no longer true. Doing the same operation to both sides keeps the balance, so the solution does not change.

How do I know which operation to undo with?

Use the inverse. Addition is undone by subtraction, subtraction by addition, multiplication by division, and division by multiplication. Look at what is being done to the variable, then do the opposite.

What is the difference between an equation and an expression?

An expression is a phrase, like 3x + 2, and there is nothing to solve. An equation has an equals sign, like 3x + 2 = 11, and it makes a claim that can be true or false. You can only solve something that makes a claim.

Is x = 5 the same as 5 = x?

Yes. The equals sign means both sides name the same number, and that does not depend on which side you write first. Answers are usually written with the variable on the left out of habit, not because the other form is wrong.

How do I check my answer?

Put your answer back into the original equation and work out both sides. If they come to the same number, the answer is right. This is the one check that never needs a teacher.

What to learn next

Key terms in this lesson

Coefficient
A coefficient is the number multiplying a variable in a term. In 3x the coefficient is 3, and in -5y it is -5, because the sign belongs to the number.
Equation
An equation is a statement that two expressions are equal, joined by an equals sign. Solving one means finding every value of the variable that makes the statement true.
Inverse operations
Inverse operations undo each other. Addition and subtraction are inverses, and so are multiplication and division. Solving an equation means applying the inverse of whatever was done to the variable.
Variable
A variable is a letter standing in for a number you do not know yet, or one that can change. In x + 3 = 10, the variable x represents the single number that makes the sentence true.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

  • CCSS.MATH.CONTENT.6.EE.B.7Expressions and EquationsSolve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
For teachersTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.