Relationships Between Variables

Interactive study guide — Reveal Math Unit 9 · Standard 6.EE.C.9

9.1 — Independent & Dependent Variables

Identify which variable depends on the other and write equations using two variables. The independent variable is the input; the dependent variable is the output.

Key Ideas

The independent variable is the value you choose (the input). The dependent variable changes based on it (the output).

To write an equation, figure out how the output depends on the input and express that relationship with a formula.

Example: You earn $12 per hour.
Independent variable: hours worked (h)
Dependent variable: money earned (m)
Equation: m = 12h
Tip: Ask yourself — "Which quantity do I control?" That is the independent variable. "Which quantity changes as a result?" That is the dependent variable.

Practice

A plant grows 3 cm each week. Which is the independent variable?

Height of the plant
Number of weeks
3 cm

In the equation y = 5x, which variable is the dependent variable?

y
x
5

A taxi charges $2 per mile. Which equation shows the total cost (c) for m miles?

m = 2c
c = m + 2
c = 2m
Score
0/3

9.2 — Equations & Tables

Represent relationships with equations and input-output tables. Find patterns in the table to write or verify the equation.

Key Ideas

An input-output table shows pairs of values that follow a rule. The rule is the equation.

Substitute each input (x) into the equation to find the corresponding output (y).

Example: y = 3x
x (Input) y (Output)
1 3
2 6
3 9
4 12
Finding the pattern: Each time x increases by 1, y increases by 3. The rule is "multiply by 3," so y = 3x.

Practice

Using the equation y = 4x, what is y when x = 7?

11
24
28

A table shows: x = 2 → y = 10, x = 3 → y = 15, x = 4 → y = 20. What is the equation?

y = x + 8
y = 5x
y = 2x + 6

For y = x + 6, what is the output when the input is 9?

15
54
3
Score
0/3

9.3 — Graphs of Relationships

Plot equations on a coordinate plane, analyze relationship graphs, and connect tables to graphs.

Key Ideas

Each row of an input-output table becomes an ordered pair (x, y) that you plot on the coordinate plane.

If the points form a straight line, the relationship is linear.

Example: y = 2x + 1
x = 0 → y = 1 → plot (0, 1)
x = 1 → y = 3 → plot (1, 3)
x = 2 → y = 5 → plot (2, 5)
x = 3 → y = 7 → plot (3, 7)
These points lie on a straight line.
Reading a graph: Pick a point on the line. Its x-coordinate is the input and its y-coordinate is the output.

Practice

For y = 2x + 1, what ordered pair do you get when x = 4?

(4, 9)
(4, 7)
(4, 8)

A graph shows a straight line passing through (0, 0) and (3, 12). What is the equation?

y = 3x
y = x + 4
y = 4x

Which point is NOT on the graph of y = 3x?

(2, 6)
(4, 15)
(5, 15)
Score
0/3

9.4 — Real-World Modeling

Use equations, tables, and graphs together to model and solve real-world problems.

Key Ideas

Real-world situations can be described with an equation, organized in a table, and visualized with a graph. All three representations show the same relationship.

Example: A car travels at 60 mph.
Equation: d = 60t (distance = 60 × time)
Independent variable: time (t) in hours
Dependent variable: distance (d) in miles
t (hours) d (miles)
1 60
2 120
3 180
4 240
Using the model: To find how far the car travels in 5 hours, substitute t = 5: d = 60(5) = 300 miles.

Practice

A pool fills at 15 gallons per minute. How many gallons after 8 minutes?

80 gallons
120 gallons
150 gallons

A store sells notebooks for $4 each. Which equation gives the total cost (c) for n notebooks?

c = 4n
c = n + 4
c = n / 4

Using d = 50t, how many hours does it take to travel 250 miles?

3 hours
4 hours
5 hours
Score
0/3

Vocabulary

Tap a card to reveal the definition.

Independent Variable
The variable you choose or control; the input of a relationship.
Dependent Variable
The variable that changes in response to the independent variable; the output.
Input
The value substituted into an equation or function; the independent variable's value.
Output
The result after applying the rule or equation to the input; the dependent variable's value.
Equation
A mathematical sentence with an equal sign showing two expressions have the same value. Example: y = 3x.
Table
An organized display of input-output pairs that shows the pattern in a relationship.
Coordinate Plane
A grid formed by a horizontal x-axis and vertical y-axis used to plot ordered pairs.
Ordered Pair
A pair of numbers (x, y) that identifies a point on the coordinate plane.
Relationship
A connection between two variables where a change in one causes a predictable change in the other.
Function
A rule that assigns exactly one output to each input.
Rate of Change
How much the dependent variable changes for each unit increase in the independent variable.
Linear
A relationship whose graph is a straight line. The rate of change is constant.