Key takeaway: The independent variable is the input quantity in a relationship, the one you choose or that changes on its own, and it is placed on the horizontal axis of a graph.
Grade 66.EE.C.9Expressions and Equations
Definition of Independent Variable
In a relationship between two changing quantities, the independent variable is the one that stands on its own. You pick its values, or time moves it along, and the other quantity responds.
At a bake sale, the number of cookies sold decides how much money comes in, so cookies sold, \(c\), is the independent variable. The money raised is the dependent one. A useful test is to ask “which quantity can I change freely?” Another clue is the word “depends”: the quantity that depends on the other is the second one. In tables, it fills the input column.
Formula and notation
With an equation such as
\[m = 3.5c\]
the input \(c\) sits on the right, where the rule acts on it, and the output \(m\) sits alone on the left. On a graph, the input goes on the horizontal axis.
Key facts
- Time is almost always the input, since it advances regardless of anything else.
- Its values form the input column of a table.
- Changing the independent variable causes the other quantity to change.
- One independent variable can control several different dependent quantities.
Independent Variable example
A bake sale charges \$3.50 per cookie, so the money raised is \(m = 3.5c\). Name the independent variable and fill in the money for 4, 10 and 15 cookies.
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The number of cookies \(c\) is chosen first, so it is the input.
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\(c = 4\): \(3.5 \times 4 = 14\). \(c = 10\): \(3.5 \times 10 = 35\).
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\(c = 15\): \(3.5 \times 15 = 52.5\).
The independent variable is \(c\); the amounts are \$14, \$35 and \$52.50.
Common mistake
Watch out
Brooke says the money is independent “because that is the goal of the sale.”
Why it is wrong
Importance does not decide the role. The amount of money is determined by the cookies sold, so it depends on them. The quantity you choose or that drives the change is the independent one.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Expressions and Equations