Many quantities change together: the more hours you work, the more you earn. You have seen pairs like these in ratio graphs, and you can now write equations for them.
Idea clave
The independent variable is the one that changes on its own or that we choose. The dependent variable changes because of it. An equation like $d = 50t$ gives the dependent variable ($d$) from the independent one ($t$). On a graph, the independent variable goes on the horizontal axis.
Por qué funciona
A table, a graph and an equation are three views of the same pairs. The table lists some pairs, the graph shows all of them as a line, and the equation is the rule that makes every pair. Putting the independent variable across the bottom lets you read the graph like a story: as time goes on (to the right), distance goes up.
| $t$ (hours) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| $d$ (miles) | 0 | 50 | 100 | 150 | 200 |
Ejemplo resuelto: Which is which?
A plant grows 3 cm each week. Let $w$ be the number of weeks and $g$ the growth in cm. Which variable is dependent? Write an equation.
- The growth depends on how many weeks have passed, so $g$ is dependent and $w$ is independent.
- Each week adds 3 cm: $g = 3w$.
Respuesta: $g$ is dependent; $g = 3w$.
Ejemplo resuelto: A starting amount
A pool already has 50 gallons and fills at 20 gallons per minute. Write an equation for the gallons $G$ after $m$ minutes, and find $G$ when $m = 6$.
- Start with 50 and add 20 for each minute: $G = 20m + 50$.
- When $m = 6$: $G = 20(6) + 50 = 170$.
Respuesta: $G = 20m + 50$; 170 gallons.
Error común
Putting the dependent variable on the horizontal axis. Quick check: ask "which one changes because of the other?" That one is dependent and goes up the side. Time is almost always independent.