Drawing two related linear functions on the same set of axes can give us useful comparisons and provided the graphs are not parallel they will have a point of intersection.
On a coordinate plane, the point of intersection of two graphs will represent the solution to a set of simultaneous equations. That is, the point will satisfy the equation of both lines. In this course we will find the point of intersection by graphing the two functions and reading the point of intersection from the graph.
For the graph below we have the intersection point of $\left(\frac{1}{2},2\frac{1}{2}\right)$(12,212).
Practice questions
Question 1
Consider the following linear equations:
$y=5x-7$y=5x−7 and $y=-x+5$y=−x+5
Plot the lines of the two equations on the same graph.
Loading Graph...State the values of $x$x and $y$y which satisfy both equations.
$x$x = $\editable{}$
$y$y = $\editable{}$
Question 2
Consider the following phone plans:
GO SMALL plan: This plan has a $\$20$$20 monthly base charge and charges $90$90 cents per minute for all calls.
GO MEDIUM plan: This plan has a $\$26$$26 monthly base charge and then charges $70$70 cents per minute for all calls.
Complete the following table of values for various total monthly call times for the two plans:
Call time (in minutes) Total cost for GO SMALL plan Total cost for GO MEDIUM plan $20$20 $\editable{}$ $\editable{}$ $30$30 $\editable{}$ $\editable{}$ $40$40 $\editable{}$ $\editable{}$ $50$50 $\editable{}$ $\editable{}$ Sketch the graph of the two plans.
Loading Graph...Using the graphs, determine how many minutes of calls would need to be made so that the monthly bill costs the same on both plans.
Break-even point
It's really important for businesses to make a profit! Otherwise, they won't be around for very long. So businesses should know their break-even point, the amount of money they need to take in to cover all their expenses. At this point there is no profit or loss, and their income is equal to their expenses.
To find this point, we use break-even analysis. Break-even analysis looks at the graphs of cost and revenue together to determine where they cross. This point of intersection is the break-even point where income equals expenses.
Here is an example of a break-even analysis for a single day of operation for the company Lovely Lemonade.
The horizontal axis is the number of drinks, or units, they sell. The expenses line (red) starts off higher than the income line (green). So for a low number of units sold, Lovely Lemonade will lose money. For example, if they only sell $1000$1000 units, the green line tells us they earn $\$200$$200, the red line tells us they spend $\$600$$600, and so overall they make $\$200-\$600=-\$400$$200−$600=−$400. The negative sign means overall they lose $\$400$$400.
But eventually, the income line becomes higher than the expenses line, so for a high number of units sold, Lovely Lemonade will make money. For example, if they sell $5000$5000 units, the graph tells us that they will earn $\$1100$$1100 but only spend $\$1000$$1000. Overall they make $\$1100-\$1000=\$100$$1100−$1000=$100, and the positive sign means a profit of $\$100$$100.
The important point is when the two lines meet - this is the break-even point. In this example we can see that selling $4000$4000 units means they don't make any profit, but they don't lose any money either - both their income and their expenses are $\$900$$900. Reaching this amount should be an important first goal for Lovely Lemonade every single day!
We can also use break-even analysis to compare two different pricing plans for the same product or service to see which one offers the best deal for a particular situation.
Here's an example comparison between two energy plans provided by rival companies Thorgate and Callisto:
In this scenario the break-even point at $\left(40,10\right)$(40,10) tells us that both plans charge $\$10$$10 for $40$40 kWh. For energy amounts less than the break-even point we can see that Thorgate is cheaper, and for energy amounts more than the break-even point Callisto is cheaper. Knowing how much energy someone plans on using can then determine the best plan for them.
Practice questions
Question 3
David decides to start his own yoga class. The cost and revenue of running the class have been graphed.
How much revenue does David make for each student?
How many students must attend his class so that David can cover the costs of running the class?
How much profit does David make if there are $8$8 students in his class?
Question 4
The cost of manufacturing toys ($C$C) is related to the number of toys produced ($n$n) by the formula $C=400+2n$C=400+2n. The revenue ($R$R) made from selling $n$n toys is $R=4n$R=4n.
Complete the following table:
Number of toys Cost (in dollars) Revenue (in dollars) $150$150 $\editable{}$ $\editable{}$ $200$200 $\editable{}$ $\editable{}$ $250$250 $\editable{}$ $\editable{}$ $300$300 $\editable{}$ $\editable{}$ Sketch the graph of the cost and the revenue on the same coordinate axes.
Loading Graph...How many toys need to be produced for revenue to equal cost?
What does the $y$y value of this point of intersection mean? Select both correct answers.
The number of toys the company needs to sell to break even.
AThe amount of profit the company will make.
BThe amount of revenue generated at the break-even point.
CThe total cost of producing $200$200 toys.
D