Solving One-Step Equations
Use inverse operations to isolate a variable in one move — the foundation of all equation solving.
Every equation you'll ever solve — a loan payment, a temperature conversion, a distance problem — is built on one core idea: get the variable alone. One-step equations teach you the fundamental mechanism: whatever operation is applied to the variable, you undo it using the opposite (inverse) operation. Once this click happens, every more complex equation is just multiple rounds of the same move.
- Identify the operation being applied to the variable in a one-step equation
- Apply the correct inverse operation to isolate the variable
- Solve equations involving addition, subtraction, multiplication, and division
An equation is a balance scale. Both sides are equal. To keep the scale balanced, every operation you apply to one side must also be applied to the other. The goal: get the variable alone on one side.
left side
right side
Subtract 5 from both sides: \(x + 5 - 5 = 12 - 5\), so \(x = 7\). Check: \(7 + 5 = 12\) ✓
Inverse Operation Reference
Undo the Operation to Isolate the Variable
| If the equation has | Apply this inverse | Example |
|---|---|---|
| x + a = b | Subtract a from both sides | x + 5 = 12 → x = 7 |
| x − a = b | Add a to both sides | x − 3 = 8 → x = 11 |
| ax = b | Divide both sides by a | 4x = 20 → x = 5 |
| x/a = b | Multiply both sides by a | x/3 = 6 → x = 18 |
Solve: \(x + 9 = 14\)
- Operation on x: adding 9 → inverse: subtract 9
- Subtract 9 from both sides: \(x + 9 - 9 = 14 - 9\)
- Simplify: \(x = 5\)
- Check: \(5 + 9 = 14\) ✓
Solve: \(\dfrac{y}{4} = 7\)
- Operation on y: dividing by 4 → inverse: multiply by 4
- Multiply both sides by 4: \(\dfrac{y}{4} \cdot 4 = 7 \cdot 4\)
- Simplify: \(y = 28\)
- Check: \(\dfrac{28}{4} = 7\) ✓
The formula for Celsius to Fahrenheit is \(F = 1.8C + 32\). When the temperature is 77°F, solve for Celsius using a one-step approach after substituting: \(77 = 1.8C + 32\).
- Subtract 32 from both sides: \(77 - 32 = 1.8C\) → \(45 = 1.8C\)
- Divide both sides by 1.8: \(C = \dfrac{45}{1.8} = 25\)
77°F = 25°C.
Solve each equation and verify your answer:
- \(n - 6 = 11\)
- \(3t = 27\)
- \(\dfrac{m}{5} = 8\)
▶ Show Answers
- \(n = 17\) — add 6 to both sides.
- \(t = 9\) — divide both sides by 3.
- \(m = 40\) — multiply both sides by 5.
- Applying the operation instead of the inverse: For \(x + 5 = 12\), don't add 5 again. The + is on x, so subtract 5.
- Only operating on one side: Whatever you do to the left, do to the right. Both sides must stay balanced.
- Skipping the check: Always substitute your answer back in. It takes 10 seconds and catches errors immediately.
- One-step equations are solved with a single inverse operation.
- Inverse pairs: + ↔ −, and × ↔ ÷.
- Balance rule: every operation applies to BOTH sides equally.
- Always check by substituting your solution back into the original equation.
Nurses solve one-step equations every shift: if a medication dose is 0.5 mg per kg and a patient weighs 70 kg, the equation \(0.5 \times 70 = d\) gives the dose directly. In the other direction: if a total dose of 150 mg is divided into equal 50 mg tablets, solving \(50t = 150\) tells you the patient needs 3 tablets. Precision here is life-critical.
Calculator Connection
Use the One-Step Equation Solver to check your work — enter any one-step equation and see the inverse operation applied with the solution shown.
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Solving One-Step Equations - Quiz
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