Now that we know how to represent linear functions as equations, tables, and graphs we can put this knowledge to use to solve a variety of real-world problems.

Some examples will be the best way to show you how the mathematics we have learned can be applied to everyday situations.

Examples

Example 1

Buzz recorded his savings (in \text{dollars}) over a few months in the graph given.

\text{Months}

100

\text{Savings}

Complete the table of values.

\text{Months}	1	2	3	4
\text{Savings } \left(\$\right)

Worked Solution

Create a strategy

Find the corresponding y-coordinate (savings) of each x-coordinate (months) from the given graph and table.

Apply the idea

Based on the given graph, when x=1: \, y=20.

\text{Months}

100

\text{Savings}

Similarly, if we determine the remaining y-values for the remaining x-values: (x=2, x=3 and x=4) from the table of values, we get:

\text{Months}	1	2	3	4
\text{Savings } \left(\$\right)	20	40	60	80

Buzz estimates that he will have exactly \$60 in his savings by month 5. Is this true or false?

Worked Solution

Create a strategy

Plot (5,60) on the coordinate plane and check whether this point lies on the line.

Apply the idea

Based on the graph below, the point (5,60) does not lie on the line.

\text{Months}

100

\text{Savings}

This means that the statement is false, because Buzz will not have exactly \$60 in his savings by month 5.

Example 2

A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table below shows the depth of the diver, in yards, over 5 minutes:

\text{Number of minutes passed }\left(x\right)	0	1	2	3	4
\text{Depth of diver in yards }\left(y\right)	0	1.4	2.8	4.2	5.6

What is the increase in depth each minute?

Worked Solution

Create a strategy

Subtract the depth at one time from the depth one minute later.

Apply the idea

\displaystyle \text{Increase}	\displaystyle =	\displaystyle 1.4-0	Subtract 0 from 1.4
	\displaystyle =	\displaystyle 1.4\text{ meters/minute}	Evaluate

Write an equation for the relationship between the number of minutes passed (x) and the depth (y) of the diver.

Worked Solution

Create a strategy

We can use the linear relationship y=mx+b, where m is the change in depth per minute and b is the initial depth of the diver.

Apply the idea

\displaystyle y	\displaystyle =	\displaystyle 1.4x+0	Substitute the value of m and b
\displaystyle y	\displaystyle =	\displaystyle 1.4x	Evaluate

In the equation, y=1.4x, what does 1.4 represent?

The change in depth per minute

The diver's depth below the surface

The number of minutes passed

Worked Solution

Create a strategy

We can use the fact that y is the depth of the diver and x is the number of minutes that passed.

Apply the idea

From part (a), the 1.4 represents the change in depth per minute. This means that option A is the correct answer.

At what depth would the diver be after 45 minutes?

Worked Solution

Create a strategy

Multiply the change in depth per minute by 45.

Apply the idea

\displaystyle y	\displaystyle =	\displaystyle 1.4 \times 45	Substitute the value of the change in depth per minute and multiply it by 45
	\displaystyle =	\displaystyle 63 \text{ m}	Evaluate

We want to know how long the diver takes to reach 12.6 meters beneath the surface.

If we substitute 12.6 into the equation in part (b) we get 12.6=1.4x.

Solve this equation for x to find the time it takes.

Worked Solution

Create a strategy

Solve the equation for x.

Apply the idea

\displaystyle \dfrac{1.4x}{1.4}	\displaystyle =	\displaystyle \dfrac{12.6}{1.4}	Divide both sides of equation by 1.4
\displaystyle x	\displaystyle =	\displaystyle 9	Evaluate

It will take 9 minutes to reach 12.6\text{ m}.

Example 3

A carpenter charges a callout fee of \$150 plus \$45 per hour.

Write an equation to represent the total amount charged, y, by the carpenter as a function of the number of hours worked, x.

Worked Solution

Create a strategy

We can use the linear relationship y=mx+b, where m is the fee per additional hours of work and b is the initial callout fee.

Apply the idea

\displaystyle y

\displaystyle =

\displaystyle 45x+150

Substitute m and b

What is the slope of the function?

Worked Solution

Create a strategy

We can use the equation found from part (a), and use the fact that a function represented by an equation in the form y=mx+b, where m is the slope.

Apply the idea

The equation from part (a) was: y=45x+150. So m=45.

\text{Slope}=45

What does this slope represent?

The total amount charged increases by \$45 for each additional hour of work.

The minimum amount charged by the carpenter.

The total amount charged increases by \$1 for each additional 45 hours of work.

The total amount charged for 0 hours of work.

Worked Solution

Create a strategy

We can use the fact that the slope represents the rate of change of the total amount charged by the carpenter.

Apply the idea

Based on the equation found from part (a), the slope represents the total amount charged increases by \$45 for each additional hour of work.

This means option A is the correct answer.

What is the value of the y-intercept?

Worked Solution

Create a strategy

We can use the equation found in part (a), to find the value of y when x=0.

Apply the idea

\displaystyle y	\displaystyle =	\displaystyle 45 \times 0 +150	Substitute the value of x
	\displaystyle =	\displaystyle 0+150	Evaluate the multiplication
	\displaystyle =	\displaystyle 150	Evaluate

What does this y-intercept represent? Select all that apply.

The total amount charged increases by \$150 for each additional hour of work.

The maximum amount charged by the carpenter.

The callout fee.

The minimum amount charged by the carpenter.

Worked Solution

Create a strategy

We can use the fact that y-intercept represents the total amount charged for 0 hours of work.

Apply the idea

Based on the answer found from part (d), option C and option D represent the value of the y-intercept.

Find the total amount charged by the carpenter for 6 hours of work.

Worked Solution

Create a strategy

We can use the equation found from part (a): y=45x+150, where x=6.

Apply the idea

\displaystyle y	\displaystyle =	\displaystyle 45 \times 6 +150	Substitute the value of x
	\displaystyle =	\displaystyle 270+150	Evaluate the multiplication
	\displaystyle =	\displaystyle \$420	Evaluate

Idea summary

Linear relationships (functions) as equations, tables, and graphs can be used to solve a variety of real-world problems.

Outcomes

8.F.B.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.F.B.5

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g. Where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

4.08 Problem solving with tables, equations and graphs

Ideas

Problem solving with tables, equations and graphs of lines

Examples

Example 1

Example 2

Example 3

Outcomes

8.F.B.4

8.F.B.5

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