A one-variable equation is useful when one unknown quantity determines every other unknown expression. The key step is not solving the equation; it is defining the unknown and translating each relationship without losing units or meaning.
From a situation to one equation
Equation-building workflow
- Name the requested quantity
Underline what the final answer must report.
- Choose one unknown
Define a variable with units, preferably for the quantity that makes other expressions simplest.
- Express related quantities
Write every unknown amount in terms of the chosen variable.
- Find the balancing relationship
Use a total, difference, distance, value, or capacity statement to create equality.
- Solve
Apply algebra while tracking units.
- Verify
Substitute into the original relationships and check that the answer is realistic.
Cost, revenue, profit, and total value
| Quantity | Relationship | Unit check |
|---|---|---|
| Total value | \(\text{number of items}\times\text{value per item}\) | items × dollars/item = dollars |
| Revenue | \(\text{quantity sold}\times\text{selling price}\) | items × dollars/item = dollars |
| Profit | \(\text{revenue}-\text{cost}\) | dollars − dollars = dollars |
| Remaining quantity | \(\text{starting quantity}-\text{quantity used}\) | same quantity unit throughout |
Profit is money received from sales minus the total buying or production cost. Selling price alone is not profit.
Inventory and profit
A vendor buys \(x\) notebooks for \(\$4\) each. Six are damaged, and the remaining notebooks sell for \(\$9\) each. The profit is \(\$176\). Find \(x\).
- Define the unknown
Let \(x\) be the number originally purchased.
- Build revenue
The vendor sells \(x-6\) notebooks, so \(R=9(x-6)\).
- Build cost
The buying cost is \(C=4x\).
- Use profit
\(9(x-6)-4x=176\).
- Solve
\(9x-54-4x=176\), so \(5x=230\) and \(x=46\).
- Interpret
A whole-number result fits the item-count context.
Distance, rate, and time
Distance equals rate times time. Rearranging gives \(r=d/t\) and \(t=d/r\). Units must be compatible before substitution.
| Unknown | Formula | Example unit result |
|---|---|---|
| Distance \(d\) | \(d=rt\) | miles/hour × hours = miles |
| Rate \(r\) | \(r=d/t\) | kilometers ÷ hours = kilometers/hour |
| Time \(t\) | \(t=d/r\) | meters ÷ meters/second = seconds |
Equal-distance round trips
On an outbound-and-return trip along the same route, the two distances are equal even when the speeds and times differ. If outbound time is \(t\), outbound distance is \(r_1t\). The return distance must equal that same expression, not necessarily use the same time.
| Segment | Rate | Time | Distance |
|---|---|---|---|
| Outbound | \(48\) km/h | \(t\) hours | \(48t\) km |
| Return | \(36\) km/h | \(t+1\) hours | \(36(t+1)\) km |
Same route, different speeds
A cyclist rides outward at \(48\) km/h and returns along the same route at \(36\) km/h. The return takes one hour longer. Find the one-way distance.
- Define time
Let \(t\) be the outbound time in hours; return time is \(t+1\).
- Use equal distances
\(48t=36(t+1)\).
- Solve
\(48t=36t+36\), so \(12t=36\) and \(t=3\).
- Find requested distance
\(d=48(3)=144\) kilometers.
Other one-variable structures
- Total-budget equations combine all spending categories and the remaining amount.
- Capacity equations equate the amount currently present plus added amount to the final or full amount.
- Vote or inventory equations often express one count as a fixed amount more or less than another count.
- Sequential changes must be applied in order; a percentage of the remaining amount is not a percentage of the original amount.
A remaining-budget equation
A student spends one fifth of a \(\$750\) project budget on supplies, then pays \(\$42\) for each of \(n\) lab sessions, leaving \(\$96\). Find \(n\).
- Fixed spending
One fifth of \(750\) is \(150\) dollars.
- Write the balance
\(750-150-42n=96\).
- Solve
\(600-42n=96\), so \(42n=504\) and \(n=12\).
Check your understanding
A boat travels the same distance downstream at \(15\) km/h and upstream at \(10\) km/h. Which statement must be true?
- The two travel times are equal.
- The upstream time is longer.
- The average speed is exactly \(12.5\) km/h.
- The downstream distance is longer.
Show answer and explanation
Answer: The upstream time is longer.
For the same distance, \(t=d/r\). The smaller upstream rate produces the larger time.
What to remember
- Define one unknown and express every related unknown in terms of it.
- Profit equals revenue minus total cost, not selling price minus purchase price for one item unless quantities match appropriately.
- Use \(d=rt\) with compatible units and equal-distance equations for same-route trips.
- Always translate the solved variable into the quantity and units requested.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.