Lesson 4

Lesson 4: Applying Linear Relations and Solving Problems

Focus

Linear relations are common in your life; you just have to be aware of these relationships.

Can you think of a set of labels and a scenario to match the graph shown? How many scenarios can you think of that this graph could be used for? What is similar about all of the scenarios you came up with?

Lesson Outcome

At the end of this lesson you will be able to solve problems involving linear relations.

Lesson Questions

You will investigate the following questions:

• How do linear relations connect with the workplace?
• What applications involve linear relations?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 4 Assignment (Download the Lesson 4 Assignment and save it in your course folder now.)
  
• course folder submissions from Try This and Share activities
  

Materials and Equipment

You will need

• ruler or straight edge

• Grid Paper Template
• calculator

Discover

Try This 1

The fare passengers are charged for a taxi ride is based on a fixed cost plus a cost per kilometre.

Explore

In Discover you probably found that the independent variable is the distance travelled by a cab, and the dependent variable is the cost of the cab ride. Your graph is a linear relation and should have resembled the following graph:

The slope of your graph represents the cost per kilometre and the y-intercept represents the fixed cost. Finding the y-intercept is an example of extrapolation. It would be an example of interpolation to estimate the cost of a 3.5-km trip from the graph.

Try This 2

You have seen that the line of best fit identifies a trend in data but the line does not necessarily include all points. The situation in Self-Check 1 gives a more exact line of best fit. Now check your skills in Self-Check 1.

Self-Check 1

The amount of money spent on gas is related to the number of litres purchased. The following chart shows amounts of gas purchased.

1. Is this a direct linear relation or a partial linear relation? Explain. Answer
2. Should the points be connected? Answer
3. What is the cost per litre? Answer
4. Determine the equation that relates the data in the relation above. Answer

From your work and checked answers in Self-Check 1, you should feel comfortable in attempting Try This 4.

Try This 4

A professional photographer charges $75 for a sitting fee and an additional $12.50 for each portrait taken.

1. Find an equation that relates the total cost to the sitting fee and to the number of portraits taken.
2. Using the equation, how much would it cost for 14 portraits?
3. How much would it cost to get an individual portrait for each student in a graduating class of 350 people? The students will share one sitting.
4. How many individual portraits are taken if the total cost is $2725 and only one sitting fee is paid?

Save your responses in your course folder.

Self-Check 2 and Self-Check 3 provide extra practice for applications of linear relations. Check your skills by doing the following self-checks.

Self-Check 2

The mass of a beaker and ethanol (M) is recorded when the beaker contains varying volumes of ethanol (v). The results of the experiment are recorded in the table. The beaker can hold a maximum of 1000 mL of ethanol.

Measurements may be assumed to be correct to the nearest millilitre (mL) and to the nearest gram (g).

Graph the relation, and respond to the following questions.

1. Assuming this pattern continues, determine the mass of the beaker and ethanol when 250 mL of ethanol is present. Answer
2. What volume of ethanol is in the beaker if the total mass is 450 g? Answer
3. When the volume of ethanol is 700 mL, determine the mass of the ethanol alone. Answer
4. Find an equation that relates the mass of the beaker and ethanol to the volume of ethanol. Answer

Self-Check 3

A small business makes ball caps and sells the caps to stores. Each ball cap costs $5 to make. The fixed monthly costs of the business, such as rent, salaries, and insurance, are $4000. The business sells the ball caps to the stores for $15 each. An equation to represent the cost (y) of making x ball caps is y = 5x + 4000. An equation to represent the revenue (y) of selling x ball caps is y = 15x.

1. Complete two tables of values similar to the tables shown. Answer
   
   

   COST: y = 5x + 4000
   Ball Caps (x)	Cost (y)
   0
   100
   200
   300

   
   

   REVENUE: y = 15x
   Ball Caps (x)	Revenue (y)
   0
   100
   200
   300

   
   profit = revenue − cost
2. Calculate the profit when 600 ball caps are sold. Answer
3. Calculate the profit when 150 ball caps are sold. Answer
4. The break-even point is when the revenue is equal to the cost, and thus the profit is 0. Calculate the break-even point using
   1. your tables from question 1 Answer
   2. the equations provided Answer

Connect

Lesson 4 Assignment

Lesson 4 Summary

You looked at examples and applications of linear relations. The ideas in this lesson can be applied to any scenario that involves a constant change.

When interpreting the graph of a linear relation, it is often useful to determine the slope and y-intercept. The graph of a relation and the equation itself are connected by the equation y = mx + b, where m represents the slope and b is the y-intercept.