Consider the equations y = 3 x-1 and \\ y = - 5 x + 23.
Complete the given table by finding the y-values for each of the given x-values:
Hence, find the values for x and y which satisfy both the equations.
| x | y=3x-1 | y=-5x+23 |
|---|---|---|
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 |
Consider the equations y = - 2 x - 8 and y = 5 x + 34.
Complete the given table by finding the y-values for each of the given x-values:
Hence, find the values for x and y which satisfy both the equations.
| x | y=-2x-8 | y=-5x+34 |
|---|---|---|
| -7 | ||
| -6 | ||
| -5 | ||
| -4 |
Consider the equations y = 5 x and y = 9 - 4 x.
Complete the given table by finding the y-values for each of the given x-values:
Hence, find the values for x and y which satisfy both the equations.
| x | y = 5 x | y = 9 - 4 x |
|---|---|---|
| 1 | ||
| 3 | ||
| 5 |
Describe the graphical solution of a system of two linear equations.
State the solution to the system of equations on each graph as an ordered pair.
For each of the following pairs of linear equations:
Sketch the lines of the two equations on the same number plane.
Hence, state the values of x and y which satisfy both equations.
y = x and y = - 1
y = x + 4 and y=2x-3
y = 2 x + 2 and y = - 2 x + 2
Solve the following pairs of equations by using your CAS calculator, or other technology, to graph the lines on the same number plane:
\begin{aligned} y &= 5 x - 7 \\ y &= - x + 5 \end{aligned}A yoga studio has two options for people to attend their classes:
One option is to purchase an annual membership that gives you unlimited access to classes for \$840.
The other option is to pay a 'drop in' fee of \$21 per class.
Complete the table of values:
| Number of classes attended in one year | Cost for the annual membership (in dollars) | Cost to drop in for every class (in dollars) |
|---|---|---|
| 0 | ||
| 10 | ||
| 20 | ||
| 30 | ||
| 40 | ||
| 50 |
How many times would you have to drop into a class in one year to be paying the same amount as an annual subscription?
Which of the options should you choose if you plan on attending a class once per week, for a whole year? Assume that there are 52 weeks in a year.
Which of the options should you choose if you plan on attending a class once every two weeks for a whole year?
Dave obtained quotes from two plumbers:
Plumber A charges \$92 for a callout fee plus \$17 per hour.
Plumber B charges \$20 for a callout fee plus \$35 per hour.
Complete the table of values:
| Hours of work | Amount charged by Plumber A | Amount charged by Plumber B |
|---|---|---|
| 4 | ||
| 5 | ||
| 6 | ||
| 7 |
How many hours would the job have to take for Plumber A and Plumber B to charge the same amount?
Consider the following phone plans:
GO SMALL plan: This plan has a \$20 monthly base charge and charges 90\text{c} per minute for all calls.
GO MEDIUM plan: This plan has a \$26 monthly base charge and then charges 70 \text{c} per minute for all calls.
Complete the table of values for various total monthly call times for the two plans:
| \text{Call time} \\ \text{(in minutes)} | \text{Total cost for} \\ \text{GO SMALL plan } (S) | \text{Total cost for} \\ \text{GO MEDIUM plan } (M) |
|---|---|---|
| 20 | ||
| 30 | ||
| 40 | ||
| 50 |
Sketch a graph for the two plans on the same set of axes.
Hence, find how many minutes of calls would need to be made so that the monthly bill costs the same on both plans.
The two equations y = 3 x + 35 and y = 4 x represent Laura’s living expenses and income from work respectively.
Find the point of intersection of the two equations.
Sketch both equations on the same number plane.
State the meaning of the point of intersection of the two lines.
Two twin soccer players are having a contest over who will score the most goals in the season. The twins have scored a combined total of 34 goals so far over the season and Twin A has scored 6 more goals than Twin B.
Let x and y be the number of goals scored by Twin A and Twin B respectively.
Use the fact that the twins have scored a combined total of 34 goals to set up an equation involving x and y.
Use the fact that Twin A has scored 6 more goals than Twin B to set up an equation involving x and y.
Graph the two equations on the same number plane.
Use the graph to find the number of goals Twin A has scored this season.
Use the graph to find the number of goals Twin B has scored this season.
Two siblings, Apollo and Artemis, leave home at different times. They are travelling to see their favourite sports team compete in the championship. The stadium is 30 \text{ km} away and the match starts at 9:00 pm.
At what time did Apollo leave home?
At what time did Artemis leave home?
How far did Apollo travel by 9:00 pm?
How far did Artemis travel by 9:00 pm?
Did Apollo make it to the stadium on time?
Did Artemis make it to the stadium on time?
Castor and Pollux are working on a bushfire prevention team. Over the course of their 10-hour shift, they move along a boundary and remove all flammable material they find.
The following travel graph shows their distance from the base throughout their shift:
How far from the base was Pollux when the shift began?
How far apart were Castor and Pollux after 2 hours?
When did Castor and Pollux meet up?
When was Pollux travelling the same speed as Castor was traveling at in his first three hours?
David decides to start his own yoga class. The cost and revenue functions of running the class have been graphed:
Determine the fixed cost.
Determine the amount of revenue David receives for each student.
Determine the number of students that must attend the class so that David can cover his costs.
Determine the cost of the break even point.
Determine the cost if there are 8 students.
Determine the revenue if there are 8 students.
Calculate the profit David makes if there are 8 students in his class.
The cost for a furniture manufacturer to make an armchair is \$500 per armchair plus a fixed setup cost of \$3500. The armchairs will sell for \$650 each.
Write an expression to represent the cost of manufacturing x armchairs.
Write an expression to represent the revenue generated from the sale of x armchairs.
Find the x-coordinate of the break even point.
A band plans to record a demo at a local studio. The cost of renting Studio A is \$250 plus \$50 per hour. The cost of renting Studio B is \$50 plus \$100 per hour.
The cost, y, in dollars of renting the studios for x hours can be modelled by the following linear system:
Studio A: y = 50 x + 250
Studio B: y = 100 x + 50
Graph the lines of both equations on the same graph.
State the values of x and y which satisfy both equations.
Interpret the solution of the equations in context.
Luke purchases 30\text{ L} of mixed leaded and unleaded petrol solution, which costs him a total of \$103.20. The price of leaded petrol is \$3.54\text{/L}, while the price of unleaded petrol is \$2.94\text{/L}.
Let x and y be the number of litres of leaded and unleaded petrol that make up the solution respectively.
Use the fact that Luke purchased a total of 30\text{ L} of petrol to write an equation involving x and y.
Use the fact that Luke purchased a total of \$103.20 worth of petrol to set up a second equation involving x and y.
Use the graphing function of your CAS calculator to sketch the graphs of the two equations on the same number plane.
Hence, determine the amount of leaded petrol that was used in the solution.
Determine the amount of unleaded petrol that was used in the solution.
A clothing manufacturer is deciding whether to employ people or to purchase machinery to manufacture their line of t-shirts. After conducting some research, they discover that the cost of employing people to make the clothing is y = 800 + 60 x, where y is the cost and x is the number of t-shirts to be made, while the cost of using machinery (which includes the cost of purchasing the machines) is y = 3200 + 20 x.
Sketch a graph that depicts the two cost functions.
State the value of x, the number of t-shirts to be produced, at which it will cost the same whether the t-shirts are made by people or by machines.
State the range of values of x, the number of t-shirts to be produced, for which it will be more cost efficient to use machines to manufacture the t-shirts.
State the range of values of x, the number of t-shirts to be produced, at which it will be more cost efficient to employ people to manufacture the t-shirts.
Frank makes ceramic bowls and sells them online. To use a kiln for making these bowls, there is a flat fee of \$400 per month plus \$25 per bowl. He then sells the bowls for \$50 each.
Complete the table of values below:
| Number of bowls made in one month | Cost to make (in dollars) | Revenue from sales (in dollars) |
|---|---|---|
| 0 | ||
| 4 | ||
| 8 | ||
| 12 | ||
| 16 | ||
| 20 |
How many bowls does Frank need to sell to exactly cover his costs?
Determine the total costs of making the bowls when Frank breaks even.
Would Frank make a profit or a loss in selling 20 bowls?
If Frank wanted to break even after making only 8 bowls, how much would he have to charge his customers for each bowl?
The two equations y = 4 x + 400 and y = 6 x represent a company's revenue and expenditure respectively.
Find the point of intersection of the two equations.
Sketch both equations on the same number plane.
Explain the meaning of the point of intersection of the two lines.
For each of the following cost revenue graphs:
State the coordinates of the break even point.
State the fixed cost.
State which is higher: the cost per unit or the revenue per unit.
The cost of manufacturing car parts \left(C\right) is related to the number of car parts produced \left(n\right) by the formula C = 6000 + 4 n.
The revenue \left(R\right) made from selling n car parts is R = 8 n - 2000.
Complete the following table:
| Number of car parts | Cost (in dollars) | Revenue (in dollars) |
|---|---|---|
| 1000 | ||
| 2000 | ||
| 3000 | ||
| 4000 |
Graph the cost and the revenue functions on the same set of axes.
Find the number of car parts that need to be produced for revenue to equal cost.
Explain the meaning of the y-coordinate of the break even point.
An electronics manufacturer has found the cost of creating circuits is represented by the equation C = 80 + 2 x, and the income received from selling them is represented by the equation I = 2.8 x, where x represents the number of circuits.
Complete the table of values for both the cost and income functions:
| x | 0 | 50 | 100 | 150 | 200 | 250 | 300 |
|---|---|---|---|---|---|---|---|
| \text{Cost in dollars } (C) | |||||||
| \text{Income in dollars } (I) |
Use the graphing functionality of your CAS calculator, or other technology, to display the cost and income functions on the same set of axes.
Hence, find the point that satisfies both equations simultaneously.
Interpret the meaning of the point of intersection in this context.
Kerry currently pays \$50 a month for her internet service. She is planning to switch to a fibre optic cable service.
Write an equation for the total cost T of Kerry's current internet service over a period of n months.
For the fibre optic cable service, Kerry pays a one-off amount of \$1200 for the installation costs and then a monthly fee of \$25. Write an equation of the total cost T of Kerry's new internet service over n months.
Complete the table of values for the total cost of the current internet service:
| n | 1 | 6 | 12 | 18 | 24 |
|---|---|---|---|---|---|
| T\text{ (dollars)} |
Complete the table of values for the total cost of the fibre optic cable service:
| n | 1 | 6 | 12 | 18 | 24 |
|---|---|---|---|---|---|
| T\text{ (dollars)} |
Using the same set of axes, sketch a graph that corresponds to the total cost of Kerry's current internet service and the total cost of her new internet service.
Hence, determine how many months it will take for Kerry to break even on her new internet service.
The cost of manufacturing toys, C, is related to the number of toys produced, n, by the formula C = 400 + 2 n. The revenue, R, made from selling n toys is given by R = 4 n.
Sketch the graph of the cost and the revenue on the same number plane.
How many toys need to be produced for the revenue to equal the cost?
Explain the meaning of the y-coordinate of the point of intersection.
The following graph displays a cost function C \left( x \right) = 0.4 x + 2015, and a revenue function R \left( x \right) = 3 x:
Find the coordinates of the break even point.
If x = 1000, is there a profit or a loss?
If x = 500, is there a profit or a loss?
The following graph shows the cost of producing drink bottles and the income from the sales of the drink bottles for a particular company:
For the line representing the cost of producing the drink bottles, find the:
y-intercept
Gradient
Hence, state the equation of the line representing the cost of producing the drink bottles.
State the equation of the line representing the income from the sales of the drink bottles.
Find the quantity of bottles that need to be sold to break even.
The following graph shows two lines that represent the revenue and cost from selling sandwiches at a local fair:
State the coordinates of the break even point.
Find the equation of the line representing the revenue, y, from the sale of x sandwiches.
Find the equation of the line representing the cost, y, of making x sandwiches.
Would the stand make a profit or a loss if they sold 70 sandwiches?
The graph shows the cost C \left(x\right) and the revenue R \left(x\right) from making and selling x units of a certain good:
State which line corresponds to the following functions:
C \left(x\right)
R \left(x\right)
How many units must be sold to reach the break even point?
A family owns two businesses that made a combined profit of \$6 million in the previous financial year, with Business B making 2 times as much profit as Business A.
Let x and y be the profits (in millions) of Business A and Business B respectively.
Use the fact that the two businesses made a combined profit of \$6 million to set up an equation involving x and y.
Use the fact that Business B made 2 times as much as Business A to set up another equation relating x and y.
Graph the two equations on the same number plane.
Use the graph to find Business A's profit.
Use the graph to find Business B's profit.
The cost for a furniture manufacturer to make an armchair is \$600 per armchair plus a fixed setup cost of \$8500. The armchairs will sell for \$850 each.
Write an expression to represent the cost of manufacturing x armchairs.
Write an expression to represent the revenue generated from the sale of x armchairs.
Find the break even point.
Explain the meaning of the break even point.
For each pair of cost, C \left( x \right), and revenue functions, R \left( x \right), find the number of units x that must be sold to break even:
\begin{aligned} C \left( x \right) & = 20 x + 8100 \\ R \left( x \right) & = 32 x \end{aligned}
\begin{aligned} C \left( x \right) & = 0.3 x + 1275 \\ R \left( x \right) & = 2 x \end{aligned}