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Relationships Between Variables

Standard: 6.EE.C.9 (Independent & Dependent Variables, Tables, Graphs, Equations)

Directions: Read the vocabulary box and worked examples. Then answer each problem. Use tables and equations to organize your thinking. Show your work.

Vocabulary Helper

🎮 Independent Variable (x)
The variable YOU choose or change. Usually the input.
📈 Dependent Variable (y)
The variable that changes because of the other. Usually the output.
📋 Input-Output Table
A table showing x-values and the matching y-values.
📊 Coordinate Graph
Points (x, y) plotted to show a relationship.
📝 Equation (Rule)
A formula connecting x and y, like y = 2x + 1.
🔁 Rate / Pattern
How much y changes each time x goes up by 1.

Worked Examples

EXAMPLE A — Independent vs. Dependent

A baker earns $8 per hour.

  1. You choose the hours → hours (h) is independent.
  2. Pay depends on hours → total pay (p) is dependent.
  3. Equation: p = 8h.
EXAMPLE B — Make a Table from y = 2x + 1

Plug each x-value into the rule:

x y = 2x + 1 y (x, y)
0 2(0)+1 1 (0, 1)
1 2(1)+1 3 (1, 3)
2 2(2)+1 5 (2, 5)

Sentence Frames

The independent variable is because I choose it. The dependent variable is because it changes because of the other.
For the rule y = 4x, when x = 3, I replace x with 3 to get y = .
Each time x goes up by 1, y goes up by in the equation y = 5x.

Guided Practice

1. A taxi charges $3 per mile. Which is the independent variable: miles or total cost?

Hint: You choose how far to ride.

2. Write an equation: you earn $6 for each lawn you mow.

Hint: y = 6 × (number of lawns).

3. For y = 4x, find y when x = 3.

Hint: y = 4(3).

4. Complete the table for y = x + 5. x: 0, 1, 2, 3 → y: ?, ?, ?, ?

Hint: Add 5 to each x-value.

5. What is the ordered pair when x = 2 and y = 10?

Hint: Ordered pairs are written (x, y).

6. For y = 3x, find y when x = 5.

Hint: y = 3(5).

7. A movie streams 2 episodes per hour. Write a rule for episodes (y) after x hours.

Hint: y = 2x.

8. In a table, when x = 0 → y = 4, x = 1 → y = 5, x = 2 → y = 6. What is the rule?

Hint: y is always x plus the same number.

9. For the rule y = 2x, complete: x = 4 → y = ?

Hint: y = 2(4).

10. A plant grows 3 cm each week. Which variable depends on the other: weeks or height?

Hint: Height changes because of the weeks.

Answer Key

  1. Miles (independent); total cost is dependent
  2. y = 6x (x = lawns, y = money)
  3. y = 12
  4. y: 5, 6, 7, 8
  5. (2, 10)
  6. y = 15
  7. y = 2x
  8. y = x + 4
  9. y = 8
  10. Height depends on weeks (weeks independent, height dependent)