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Dependent & Independent
Variables

Learn to identify which quantity you control and which quantity responds, then connect tables, graphs, and equations.

6th Grade Math

What Are Variables?

A variable is a letter or symbol that stands for a number that can change. In real life, many quantities change together: the number of hours you work and the money you earn, or the number of miles you drive and the gas you use.

Independent Variable
The quantity you choose or control. It stands on its own.
Dependent Variable
The quantity that depends on (responds to) the independent variable.

A Real-World Example

A car travels at a constant speed of 65 miles per hour. The longer you drive, the farther you go. Which quantity do you control?

Time (hours)
You decide how long to drive. This is the independent variable: \( t \).
Distance (miles)
Distance depends on how long you drive. This is the dependent variable: \( d \).

The Input-Output Machine

Input (Independent) Rule (Equation) Output (Dependent)

You feed the independent variable into the equation (the rule), and the dependent variable comes out. The equation is the machine that connects them.

The Core Relationship

1
Tables list ordered pairs \(( x, y )\) showing how \( y \) changes as \( x \) changes.
2
Graphs plot those ordered pairs on a coordinate plane: \( x \)-axis (horizontal) and \( y \)-axis (vertical).
3
Equations express the rule: \( y = \text{(some expression in } x) \).

Connecting Tables, Graphs, and Equations

Let's use \( d = 65t \). Build a table, then plot the points.

\( t \) (hours) \( d \) (miles)
00
165
2130
3195
4260

Graphing \( d = 65t \)

t (hours) d (mi) 0 1 2 3 4 0 65 130 195 260 (0, 0) (1, 65) (2, 130) (3, 195) (4, 260)

Independent vs. Dependent

Independent Dependent
Role The input you choose The output that results
In an equation Appears on the right side Appears alone on the left side
On a graph Horizontal (\( x \))-axis Vertical (\( y \))-axis
In a table First column / row Second column / row
Example (\( d = 65t \)) \( t \) (time) \( d \) (distance)

Step-by-Step Process

1
Read the problem. Ask: "What quantity do I choose or control?" That is the independent variable.
2
Find the dependent variable. Ask: "What quantity changes because of my choice?" That is the dependent variable.
3
Write the equation. Express the dependent variable in terms of the independent variable: \( y = \text{(rule using } x) \).
4
Make a table. Choose several values for the independent variable and compute the dependent variable.
5
Graph it. Plot the ordered pairs with the independent variable on the \( x \)-axis and the dependent variable on the \( y \)-axis.

Example 1: Lemonade Stand

You sell cups of lemonade for $2 each. Write an equation, make a table, and describe the variables.

\( c \) (cups) \( m \) (dollars)
00
510
1020
1530
2040

Example 2: Plant Growth

A plant is 3 cm tall and grows 2 cm per week. Write an equation for its height \( h \) after \( w \) weeks.

w (weeks) h (cm) 0 2 4 6 8 0 5 10 15 20 (0, 3) (2, 7) (4, 11) (6, 15) (8, 19)

Example 3: Movie Tickets

Movie tickets cost $9 each, plus a $5 online fee. Write an equation for total cost \( C \) for \( n \) tickets.

\( n \) (tickets) \( C \) (dollars) Calculation
114\( 9(1) + 5 = 14 \)
223\( 9(2) + 5 = 23 \)
332\( 9(3) + 5 = 32 \)
550\( 9(5) + 5 = 50 \)

Reading Context Clues

In word problems, certain phrases hint at which variable is which. Here are common patterns:

Independent Clues
"for every...", "each...", "per...", "the number of..."
Dependent Clues
"total...", "how much...", "the cost of...", "depends on..."

Analyzing Patterns in Tables

When you look at a table, find the rate of change: how much does the dependent variable change each time the independent variable increases by 1?

x (independent) y (dependent) 1 5 2 10 3 15 4 20 +5 +5 +5

Common Mistakes

✕ Swapping the Variables
Putting the dependent variable on the \( x \)-axis and independent on the \( y \)-axis. Remember: independent = horizontal.
✕ Confusing "Depends On"
Thinking "time depends on distance." Ask: "Which do I choose first?" You choose time; distance follows.
✕ Forgetting the Starting Value
In \( h = 2w + 3 \), forgetting the +3. The plant was already 3 cm tall before any weeks passed.

Summary Table

Representation Independent Dependent What to Look For
Equation Right side Left side (alone) \( y = \text{rule}(x) \)
Table 1st column 2nd column Constant change pattern
Graph Horizontal axis Vertical axis Slope = rate of change
Word Problem "per", "each" "total", "how much" What do you choose?

Question 1 of 5

A babysitter earns $12 per hour. In the equation \( p = 12h \), which is the independent variable?

Question 2 of 5

On a graph of \( d = 65t \), the independent variable \( t \) goes on which axis?

Question 3 of 5

A table shows: when \( x = 1 \), \( y = 4 \); when \( x = 2 \), \( y = 8 \); when \( x = 3 \), \( y = 12 \). What equation fits this pattern?

Question 4 of 5

A taxi charges a $3 base fee plus $2 per mile. Which equation represents the total fare \( f \) for \( m \) miles?

Question 5 of 5

Look at this scenario: "The more hours Mia studies, the higher her test score." Which statement is correct?