Topics
6. Equations & Inequalities
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United States of America
Grade 6

6.05 Relationships between quantitites

Lesson
Practice

Independent and dependent variables

This time we will look at  data on a graph  and see how two pieces of information are related to each other. We do that by looking at the corresponding values on the vertical and horizontal axes. Let's compare your age to your sister's age and see what happens as your sister gets 1 year older each year. You don't have a sister? That's okay, just imagine someone else in the example.

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Examples

Example 1

A catering company uses the following table to work out how many sandwiches are required to feed a certain number of people.

Fill in the blanks:

Number of PeopleSandwiches
15
210
315
420
525
a

How many sandwiches the caterer needs for each person?

Worked Solution
Create a strategy

Refer to the table for the number of sandwich 1 person needs.

Apply the idea

For each person, the caterer needs to make 5 sandwiches.

b

How many sandwiches the caterer would need to make for 6 people?

Worked Solution
Create a strategy

Multiply the number of people by the number of sandwich 1 person needs.

Apply the idea
\displaystyle \text{No. of sandwich}\displaystyle =\displaystyle 6\times5Multiply the number of people by 5
\displaystyle =\displaystyle 30Evaluate

For 6 people, the caterer would need to make 30 sandwiches.

Idea summary

For a table of data, we can look for a relationship between the information in one column and the information in the other column.

For a graph of data we can look for a relationship by looking at the corresponding values on the vertical and horizontal axes.

Then we can use these relationships to find other values.

Write a rule

Now we can write a rule to describe one set of data to another. We identify the rule and then use variables, or letters, to write it in a shorter way. Let's write a rule using variables for our previous example that tells us how old you are based on your sister's age.

InformationVariable
\text{Your age}y
\text{Your sisters age}x

Let's see how we do this.

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We looked at how to calculate your age, once we knew your sister's age. What if we wanted to write the rule in a different way? This time we will work out your sister's age if we know your age. How do you think our rule, or equation, might change?

Let's have a look in our last video.

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Examples

Example 2

Consider the pattern shown on this line graph:

4
8
12
x
1
2
3
4
y
a

If the pattern continues on, the next point marked on the line will be?

A
(12,3)
B
(12,4)
C
(16,3)
Worked Solution
Create a strategy

Every time the x-values are increased by 4, the corresponding y-value goes up by 1.

Apply the idea
4
8
12
x
1
2
3
4
y

Considering the last point (8,2), we can now find the next point using the pattern.

Solving for the x-value:

\displaystyle x\text{-value}\displaystyle =\displaystyle 8+4Add 4 to the last x-value
\displaystyle =\displaystyle 12Evaluate

Solving for the y-value:

\displaystyle y\text{-value}\displaystyle =\displaystyle 2+1Add 1 to the last y-value
\displaystyle =\displaystyle 3Evaluate
4
8
12
x
1
2
3
4
y

The next point will be (12,3), option A.

b

Complete the table with the points from the graph, and the one you just found (the first one is filled in for you):

x\text{-value}y\text{-value}
00
1
8
3
Worked Solution
Create a strategy

Refer to the points on the line graph in part a.

Apply the idea
x\text{-value}y\text{-value}
00
41
82
123
c

Choose the two statements that correctly describe this pattern:

A
The rule is x=y\div4.
B
The rule is y=4\times{x}.
C
The rule is y=x\div4.
D
As x increases y increases.
Worked Solution
Create a strategy

Substitute one known point into each of the rules stated.

Apply the idea

Let us consider point (8,2).

For rule x=y\div4:

\displaystyle 8\displaystyle =\displaystyle 2\div4Substitute x and y values
\displaystyle 8\displaystyle \neq\displaystyle \dfrac{1}{2}The rule is incorrect

For rule y=4\times{x}:

\displaystyle 2\displaystyle =\displaystyle 4\times8Substitute x and y values
\displaystyle 2\displaystyle \neq\displaystyle 32The rule is incorrect

For rule y=x\div4:

\displaystyle 2\displaystyle =\displaystyle 8\div4Substitute x and y values
\displaystyle 2\displaystyle =\displaystyle 2The rule is correct

The statement "as x increases y increases" is correct since the pattern shows that every time the x-values increase by 4, the y-values increade by 1.

Option C and D are correct.

Idea summary

We can write a rule to describe the relationship between one set of data to another.

We write the rule using variables or letters. e.g. y=x+2

Outcomes

6.EE.C.9

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

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