Dependent vs Independent Variables Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Dependent vs Independent Variables.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The independent variable is chosen freely as input; the dependent variable's value is then determined by the function rule.

You choose the input (independent), and the function gives the output (dependent).

Read the full concept explanation β†’

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The independent variable is chosen freely; the dependent variable is whatever the function then returns.

Common stuck point: The procedure for dependent vs independent variables is the easy part; the trap is swapping the axes. Asking "Which quantity do I choose freely, and which one is then determined by it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Which quantity do I choose freely, and which one is then determined by it?

Worked Examples

Example 1

easy
In y=3x+2y = 3x + 2, which variable is independent and which is dependent?

Answer

xx is independent; yy is dependent.

First step

1
The value of xx is chosen freelyβ€”it is the independent variable.

Full solution

  1. 2
    The value of yy is determined by xxβ€”it is the dependent variable.
  2. 3
    We say 'yy depends on xx'.
The independent variable is the input (you choose it), and the dependent variable is the output (it is calculated from the input). In y=f(x)y = f(x), xx is always independent.

Example 2

medium
A factory's cost is C=500+8nC = 500 + 8n where nn is the number of units produced. Identify the variables.

Example 3

medium
A phone plan costs C=30+0.10mC = 30 + 0.10 m dollars, where mm is minutes used. Identify both variables and find CC when m=200m = 200.

Example 4

medium
Ohm's law states V=IRV = IR with RR fixed. With current II as independent, find the dependent value VV when I=3I = 3 A and R=4 ΩR = 4 \, \Omega.

Example 5

hard
In y=12xy = \frac{12}{x}, describe how the dependent variable responds as the independent variable triples.

Example 6

challenge
A car's stopping distance is d=0.05v2d = 0.05 v^2 in meters. If a driver doubles their speed, by what factor does the dependent value dd grow?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
In the equation d=60td = 60t (distance = speed Γ— time), which is dependent?

Example 2

medium
On a graph, which axis typically shows the independent variable?

Example 3

easy
In y=2xy=2x, which variable is independent?

Example 4

easy
In y=2xy=2x, which variable is dependent?

Example 5

easy
Distance =60t=60t (speed 60 mph). Which variable is independent?

Example 6

easy
In 'cost depends on number of items', which is dependent?

Example 7

easy
On a graph of yy vs xx, which axis holds the independent variable?

Example 8

easy
Temperature changes over time: T=f(t)T=f(t). Independent variable?

Example 9

easy
In A=Ο€r2A=\pi r^2, which variable is dependent?

Example 10

easy
If yy is a function of xx, can the same xx give two different yy values?

Example 11

medium
A taxi charges 3plus3 plus 2 per mile: C=3+2mC=3+2m. Identify both variables and their roles.

Example 12

medium
Given y=3xβˆ’1y=3x-1, find the dependent value when the independent value is 4.

Example 13

medium
In x=5βˆ’2yx=5-2y, if yy is independent, express the dependent variable explicitly.

Example 14

medium
A spring's length L=10+0.5wL=10+0.5w depends on weight ww. Find ww giving L=15L=15.

Example 15

medium
In the data table input 1,2,3 β†’ output 2,4,6, name the variables.

Example 16

medium
Plant height grows with days: h=2dh=2d. Which variable would you put on the xx-axis?

Example 17

medium
Is the relation x=3x=3 (a vertical line) a function of xx? Identify the issue.

Example 18

challenge
In P=kVP=\frac{k}{V} (fixed kk), describe how the dependent variable responds as the independent one doubles.

Example 19

challenge
A circle's area depends on radius. If the radius is the independent variable, find the rate-style ratio Ar2\frac{A}{r^2} and explain.

Example 20

challenge
Two quantities satisfy y=mxy=mx for unknown fixed mm. Using the point (2,6)(2,6), find mm, then state which symbol is the parameter.

Example 21

medium
Given C=5+2nC=5+2n, find the dependent value when n=6n=6.

Example 22

medium
In 'pressure depends on depth' for a diver, name independent and dependent variables.

Example 23

easy
In y=4xβˆ’7y = 4x - 7, name the independent variable.

Example 24

easy
In y=4xβˆ’7y = 4x - 7, name the dependent variable.

Example 25

easy
A water tank fills at a constant rate, V=5tV = 5t (liters at time tt minutes). Which is dependent?

Example 26

easy
In h=2+3dh = 2 + 3d where dd is days, find the dependent value when d=4d = 4.

Example 27

medium
In the equation 5x+2y=205x + 2y = 20, if we treat xx as independent, solve for the dependent variable.

Example 28

medium
In a science experiment, drug dose causes a change in blood pressure. Which variable should be on the xx-axis of the graph?

Example 29

medium
A coffee shop's daily revenue RR depends on customers nn: R=4.5nR = 4.5 n. Find nn when R=270R = 270.

Example 30

medium
In y=x2y = x^2, find the dependent value when the independent value is βˆ’3-3.

Example 31

medium
For a table with inputs 1,2,3,41, 2, 3, 4 giving outputs 5,8,11,145, 8, 11, 14, write the linear rule with xx independent.

Example 32

medium
A scientist plots height versus age. Which axis holds age?

Example 33

hard
In the formula P=nRTVP = \frac{nRT}{V} from the ideal gas law, with n,Rn, R fixed and you may control VV and TT, how many independent variables are there?

Example 34

hard
In an experiment to test how plant growth depends on sunlight and water, identify the dependent variable.

Example 35

hard
For the rule V=43Ο€r3V = \frac{4}{3}\pi r^3, if rr doubles, what happens to VV?

Example 36

hard
In y=mx+by = mx + b, which symbols are parameters and which is the independent variable?

Example 37

hard
In y=kxy = kx, the dependent value when x=4x = 4 is y=12y = 12. Find kk.

Example 38

challenge
A box's volume V=lwhV = lwh has ww and hh fixed. Which variable is independent and how does VV scale if ll doubles?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function definitionvariables