A mathematical model turns a real situation into a relationship that can answer questions. A linear model is appropriate when equal changes in the independent variable produce equal changes in the dependent variable.
Modeling vocabulary
| Term | Meaning in context | Question to ask |
|---|---|---|
| Independent variable | input chosen or observed | What changes or passes? |
| Dependent variable | output affected by the input | What quantity responds? |
| Rate of change | output change per one input unit | How much per unit? |
| Initial value | output when the input is \(0\) | What is present at the start? |
The coefficient \(m\) is the constant rate of change and \(b\) is the output at \(x=0\). Both meanings must be stated with units.
A reliable word-problem process
From context to answer
- Name the target
State exactly what the question asks and the required units.
- Organize the facts
Separate starting information, rates, later values, and restrictions.
- Define variables
Give each symbol one meaning and one unit.
- Check linearity
Look for a constant additive change over equal input intervals.
- Find the rate
Use \(m=(y_2-y_1)/(x_2-x_1)\) or read a stated per-unit change.
- Find the start
Use a known point in \(y=mx+b\) to determine \(b\) if it is not stated.
- Use the model
Evaluate or solve the equation for the requested quantity.
- Check context
Verify units, domain, sign, and reasonableness.
A decreasing linear model
A tank contains \(600\) liters and drains \(50\) liters each hour. Measuring volume in hundreds of liters gives \(V=6-0.5t\). The graph's intercept represents the starting \(600\) liters, and its slope represents a loss of \(50\) liters per hour.
| Time \(t\) (hours) | Volume \(V\) (liters) | Change from previous row |
|---|---|---|
| \(0\) | \(600\) | starting value |
| \(2\) | \(500\) | \(-100\) liters in \(2\) hours |
| \(5\) | \(350\) | \(-150\) liters in \(3\) hours |
| \(8\) | \(200\) | \(-150\) liters in \(3\) hours |
Building and using models
Membership cost
A study studio charges a \(\$35\) enrollment fee and \(\$18\) per month. Write the total-cost model and find the cost after \(7\) months.
- Define variables
Let \(m\) be months and \(C\) be total dollars paid.
- Identify model parts
The monthly rate is \(18\), and the starting fee is \(35\).
- Write the model
\(C=18m+35\).
- Evaluate
\(C(7)=18(7)+35=161\).
Recover an unknown starting value
A trail crew's elevation changes at \(75\) feet per hour. After \(4\) hours the crew is at \(1{,}520\) feet. Find the starting elevation.
- Model
Write \(E=75t+b\).
- Use the known point
\(1{,}520=75(4)+b\).
- Solve
\(b=1{,}520-300=1{,}220\).
Recognizing when a model is linear
- Linear: a savings balance increases by the same dollar amount each week.
- Linear: temperature falls by the same number of degrees each hour over the modeled interval.
- Not linear: a population grows by the same percentage each year, because the added amount changes.
- Not necessarily linear: a trip with changing speeds unless a constant-rate interval is specified.
Check your understanding
A device is worth \(D=840-55a\) dollars after \(a\) years. What does \(-55\) represent?
- Its starting value is \(\$55\).
- Its value falls \(\$55\) per year.
- It lasts \(55\) years.
- Its value is negative.
Show answer and explanation
Answer: Its value falls \(\$55\) per year.
The coefficient of the independent variable is the rate. Its negative sign indicates depreciation.
What to remember
- Use a linear model only when the rate of change is constant over the relevant interval.
- In \(y=mx+b\), interpret \(m\) as output units per input unit and \(b\) as the output at input zero.
- A model is not complete until its variables and units are defined.
- Check whether the computed answer addresses the requested contextual quantity.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.