6th Grade Independent and Dependent Variables Worksheet Solutions (PDF)

6th Grade math worksheet solutions: Independent and Dependent Variables – Maths PDF

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This variable relationships answers for 6th Grade covers independent and dependent variables (Common Core 6.EE.C.9), with a free PDF to print.

Check your equations, tables, and graphs here. The solutions name the independent and dependent variables, show the pattern used to write each equation, and give final answers in bold. Tables and graphs are included for many problems.

Grade 6 · Common Core State Standards: 6.EE.C.9

Variable relationships answers: exercises 1 to 10

Exercise 1 – Name the variables

The number of miles x is chosen first, so it is the independent variable.

The cost y depends on x, so it is the dependent variable.

Equation: $y = 3x$.

independent x; dependent y; $y = 3x$

Exercise 2 – Equation from a table

Each y is 5 times x: $5 \cdot 1 = 5$, $5 \cdot 2 = 10$, $5 \cdot 3 = 15$.

$y = 5x$

Exercise 3 – Complete a table

Add 4 to each x: $1 + 4 = 5$, $2 + 4 = 6$, $3 + 4 = 7$, $5 + 4 = 9$.

y = 5, 6, 7, 9

Each y is x + 4.

Exercise 4 – Evaluate an equation

$y = 4 \cdot 7 = 28$.

$36 = 4x$, so $x = 36 \div 4 = 9$.

y = 28; x = 9

Exercise 5 – Graph y = 2x

Table: (0, 0), (1, 2), (2, 4), (3, 6).

See the graph: the points lie on a line through the origin.

Points on y = 2x.

Exercise 6 – Equation from a table with division

Each y is half of x: $2 \div 2 = 1$, $4 \div 2 = 2$.

$y = \frac{x}{2}$

Exercise 7 – Read a graph

y is 3 times x: $y = 3x$.

$y = 3 \cdot 5 = 15$.

$y = 3x$; y = 15

Exercise 8 – Graph y = x + 2

The line for $y = x + 2$ crosses the y-axis at 2, not at 0. It goes up 1 each time x goes up 1.

The line for $y = 2x$ goes through $(0, 0)$ and goes up 2 each time x goes up 1.

It crosses the y-axis at 2 and rises more slowly.

Exercise 9 – Walking distance

Distance = rate times time: $d = 4t$. The time t is independent.

$4 \cdot 1 = 4$, $4 \cdot 2 = 8$, $4 \cdot 3 = 12$, $4 \cdot 5 = 20$.

$d = 4t$; d = 4, 8, 12, 20

Distance walked.

Exercise 10 – Age difference

$j = p + 4$.

$j = 9 + 4 = 13$.

$j = p + 4$; Jamal is 13

Exercises 11 and up

Exercise 11 – Hourly pay

$\$9.00 \cdot 1 = \$9.00$, so the rate is $9 per hour: $p = 9h$.

$9 \cdot 8 = 72$.

$p = 9h$; $72.00

Exercise 12 – Perimeter of a square

A square has 4 equal sides.

$P = 4s$

The graph of P = 4s.

Exercise 13 – Which table matches?

Table A: $3 \cdot 1 = 3$, $3 \cdot 2 = 6$: $y = 3x$.

Table B: $1 + 3 = 4$, $2 + 3 = 5$: $y = x + 3$.

B matches $y = x + 3$; A matches $y = 3x$

Exercise 14 – Distance and time

$d = 55t$.

$275 = 55t$, so $t = 275 \div 55 = 5$.

$d = 55t$; 5 hours

Exercise 15 – Cookie trays

$c = 12t$.

$108 = 12t$, so $t = 9$.

$c = 12t$; 9 trays

Exercise 16 – Independent or dependent?

(a) Independent: hours studied. Dependent: test score.

(b) Independent: number of pizzas. Dependent: total cost.

The quantity that you choose or change is independent; the one that responds is dependent.

Exercise 17 – Water tank

$g = 6m$.

$90 = 6m$, so $m = 90 \div 6 = 15$.

$g = 6m$; 15 minutes

The line passes through (15, 90).

Exercise 18 – Match the situation

(a) The pay is 5 times the hours: $y = 5x$.

(b) Her age is 5 more than her brother’s age: $y = x + 5$.

(a) $y = 5x$; (b) $y = x + 5$

The two relationships grow differently.

Exercise 19 – Growth table

The graph starts at $(0, 3)$ and rises 1 each week: $y = x + 3$.

$10 + 3 = 13$.

$y = x + 3$; 13 inches

Exercise 20 – Multi-step relationship

$s = 7w$.

Table: 14, 28, 42, 56.

$84 = 7w$, so $w = 12$.

$s = 7w$; 12 weeks

Savings at $7 per week.

Key facts behind this variable relationships answers

Key facts

The independent variable is the input; the dependent variable is the output.
In y = kx, y is k times x and the graph passes through the origin.
In y = x + c, y is c more than x and the graph crosses the y-axis at c.
A table, a graph, and an equation can show the same relationship.
Use substitution to find y from x, and solve an equation to find x from y.

How to use this variable relationships answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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