5.4 Independent and Dependent Variables, Tables, & Two-Quantity Relationships
Key Takeaways
- The independent variable is the one you choose or control; the dependent variable is the one that responds, and it is always the output in a table, graph, or function.
- Writing an equation for a two-quantity relationship means expressing the dependent variable in terms of the independent one, as in d = 55t.
- A value substituted into an inequality or equation either makes it true or false; testing values is the assessed method for deciding which numbers belong to a solution set.
- The same relationship must read consistently across four representations — verbal description, table, graph, and equation — and TABE asks candidates to move between them.
Independent and Dependent Variables, Tables, & Two-Quantity Relationships
Standard 6.EE.9 asks candidates to "use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable" and to "analyze the relationship between the dependent and independent variables using graphs and tables." That is a mouthful, but the tested skill is simple: decide which quantity drives, then write the rule.
Which Variable Is Which?
| Independent variable | Dependent variable | |
|---|---|---|
| Role | the cause, the input, the thing you choose | the effect, the output, the thing that responds |
| Usual letter | $x$ (or $t$ for time, $n$ for count) | $y$ (or $d$, $C$, $P$…) |
| Table position | left column / top row | right column / bottom row |
| Graph position | horizontal axis | vertical axis |
| In an equation | appears on the right | stands alone on the left |
The test question to ask yourself: Which quantity would I set, and which one would then be determined?
| Situation | Independent | Dependent | Equation |
|---|---|---|---|
| Driving at 55 mph | hours driven, $t$ | distance, $d$ | $d = 55t$ |
| Gym: $30 join fee + $12/month | months, $m$ | total cost, $C$ | $C = 12m + 30$ |
| Paid $18.50 per hour | hours worked, $h$ | gross pay, $P$ | $P = 18.5h$ |
| Tank drains 3 gal/min from 90 gal | minutes, $t$ | gallons left, $V$ | $V = 90 - 3t$ |
You cannot choose your distance and have the clock respond to it, which is why distance is dependent and time is independent. When a situation genuinely runs both ways, TABE tells you which is which in the wording: "…as a function of…" names the independent variable second.
Building the Table
Once the equation exists, the table is mechanical. For $C = 12m + 30$:
| Months $m$ | 0 | 1 | 3 | 6 | 12 |
|---|---|---|---|---|---|
| Cost $C$ | $30 | $42 | $66 | $102 | $174 |
Two features to notice, because TABE asks about both:
- The value at $m = 0$ is the fixed cost — the join fee, the starting tank level, the flat charge. It is the number that stands alone in the equation.
- Each one-unit step in $m$ changes $C$ by the same amount, $12. Constant change per step is what makes the relationship linear.
Recognizing the Rule From a Table
Work in the other direction when the table comes first.
| Boxes $b$ | 2 | 5 | 8 | 11 |
|---|---|---|---|---|
| Weight $w$ (lb) | 19 | 40 | 61 | 82 |
- Find the step. From $b = 2$ to $b = 5$ is 3 boxes; weight rises $40 - 19 = 21$ lb. So $21 \div 3 = 7$ lb per box.
- Back up to zero. At $b = 2$ weight is 19; removing two boxes removes 14 lb, so at $b = 0$ weight is 5 lb. That is the empty pallet.
- Write it. $w = 7b + 5$.
- Check another row. $b = 11$: $7(11) + 5 = 82$. ✓
Proportional or not? If the value at zero is 0, the relationship is proportional and takes the form $y = kx$. Here it is 5, not 0, so this relationship is linear but not proportional — a distinction TABE tests explicitly at Level D.
Substitution Testing: Does This Value Work?
Standards 6.EE.5 and 6.EE.6 ask candidates to decide whether a specific number makes an equation or inequality true. The method is substitution, and it is worth points on multiple-choice items where solving algebraically would take longer than testing the four options.
Which value of $x$ satisfies $4x - 7 = 2x + 9$?
| Try $x$ | Left: $4x - 7$ | Right: $2x + 9$ | True? |
|---|---|---|---|
| 4 | 9 | 17 | no |
| 6 | 17 | 21 | no |
| 8 | 25 | 25 | yes |
| 10 | 33 | 29 | no |
For an inequality, testing tells you membership in a solution set:
Is $x = 5$ in the solution set of $3x + 4 \le 18$? Substitute: $3(5) + 4 = 19$, and $19 \le 18$ is false. So no. Is $x = 4$? $3(4) + 4 = 16 \le 18$ is true, so yes.
Reading the Graph of a Two-Quantity Relationship
Plot the independent variable horizontally and the dependent variable vertically, always. For the tank equation $V = 90 - 3t$:
- The line starts at 90 on the vertical axis, because that is the value when $t = 0$.
- It falls 3 units for every 1 unit right, because the rate of change is negative.
- It hits the horizontal axis at $t = 30$, because $90 - 3(30) = 0$ — the tank is empty after 30 minutes.
Every one of those three readings is a separate TABE item type. Practice extracting all three from any graph you meet.
A phone plan charges a $25 monthly fee plus $0.09 per minute of international calling. Which quantity is the dependent variable, and which equation models the monthly bill?
A table shows that a delivery service charges $14 for 2 packages, $26 for 5 packages, and $38 for 8 packages. Which equation models the cost C for p packages, and is the relationship proportional?
Which value of x makes the inequality 5x - 8 < 17 false?