How can I solve a one-step equation using inverse operations?
Quick answer
Identify the operation acting on the variable, then undo it by doing the inverse operation to both sides. Undo addition with subtraction, subtraction with addition, multiplication with division, and division with multiplication. Simplify to isolate the variable, then substitute your answer into the original equation to check that both sides match. When dividing both sides to undo multiplication, use a nonzero number. [S1][S2]
The question, the details, the next step.
What does “inverse operation” mean when I’m solving a one-step equation?
An inverse operation undoes another operation: addition and subtraction undo each other, as do multiplication and division. The goal is to get the variable by itself while keeping the equation balanced. To preserve equality, do the same operation to both sides. [S1][S2]
A quick decision flow:
- Find the variable and the number or operation directly attached to it.
- Ask what is being done to the variable: added, subtracted, multiplied, or divided?
- Undo that operation on both sides, then simplify.
- Substitute the result into the original equation as a check. [S1]
How do I choose the right operation? I sometimes see a minus sign and add when I should subtract.
Read the equation as an operation being applied to the variable, rather than choosing based on the sign alone. For example, in x − 6 = 10, six is being subtracted from x, so add six to both sides. In x + 6 = 10, six is being added, so subtract six from both sides. These steps undo the operation and preserve equality. [S1]
For multiplication and division, use the same idea. In 4x = 20, x is multiplied by four, so divide both sides by four. In x/4 = 5, x is divided by four, so multiply both sides by four. When dividing both sides to undo multiplication, divide by the same nonzero number. [S2]
Could you show the steps and check the answers with examples?
Here are four examples, one for each operation:
- Addition: x + 7 = 12. Subtract 7 from both sides: x = 5. Check: 5 + 7 = 12.
- Subtraction: x − 3 = 8. Add 3 to both sides: x = 11. Check: 11 − 3 = 8.
- Multiplication: 6x = 30. Divide both sides by 6: x = 5. Check: 6 × 5 = 30.
- Division: x/4 = 3. Multiply both sides by 4: x = 12. Check: 12/4 = 3.
For each check, put the proposed value back into the original equation and evaluate both sides. The cited material supports balancing with the same operation on both sides and checking by substitution. [S1][S2]
What if I’m still unsure whether my answer works? Can I use a reusable layout?
Yes. Write the original equation, name the operation acting on the variable, and show the same inverse operation on each side. Then simplify and check in the original equation. For example, with y + 9 = 14:
Original: y + 9 = 14 Undo addition: y + 9 − 9 = 14 − 9 Simplify: y = 5 Check: 5 + 9 = 14
If your check gives different values on the two sides, revisit the step where you applied the inverse operation or did the arithmetic. For multiplication equations, dividing both sides to undo multiplication requires a nonzero number. [S2]
A practical checklist
Copy the equation accurately, including the sign and number attached to the variable.
Identify the operation applied directly to the variable.
Choose its inverse: addition/subtraction or multiplication/division.
Apply that same inverse operation to both sides and simplify.
Substitute the answer into the original equation and compare both sides.
A few more useful answers
Why do I have to do the same thing to both sides?
How can I check a solution without repeating every solving step?
Replace the variable in the original equation with your proposed answer, then evaluate each side. If the two sides match, the substitution checks out. [S1]
What if the variable appears in a longer expression?
This guide covers one-step equations where a single inverse operation isolates the variable. Longer expressions may require additional steps, so do not assume this one-step flow is enough.
Sources
References used for this guide.
- [S1] OpenStax Elementary Algebra 2e: Addition and Subtraction Properties of Equality openstax.org · Checked
- [S2] OpenStax Elementary Algebra 2e: Multiplication and Division Properties of Equality openstax.org · Checked