Linear modelling- Simultaneous Equations
The magic of simultaneous equations comes to life when we see how useful it is in real life applications. It is usually used when we have at least two unknown quantities and at least two bits of information involving both of these quantities. The first part is to use some variables to represent these quantities and then figuring out how to write the bits of information down as equations.
Once we have worked out all the equations we need (usually two), we can graph them on a number plane and find their point of intersection (ie. the point where the lines meet).
These can also be found algebraically using the substitution method or the elimination method, but for now we'll focus on solving them graphically by looking through some examples.
Worked Examples
Question 1
A family owns two businesses that made a combined profit of $\$6$$6 million in the previous financial year, with business B making $2$2 times as much profit as business A.
Let $x$x and $y$y be the profits ( in millions) of business A and business B respectively.
Use the fact that the two businesses made a combined profit of $\$6$$6 million to set up an equation involving $x$x and $y$y.
Use the fact that business B made $2$2 times as much as business A to set up another equation relating $x$x and $y$y.
Which of the following sets of graphs correctly depicts the two equations?
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DUse the graph to find business A's profit.
Use the graph to find business B's profit.
Question 2
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 $y=800+60x$y=800+60x, where $y$y is the cost and $x$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 $y=3200+20x$y=3200+20x.
Which of the following graphs correctly depicts the two cost functions?
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DState the value of $x$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 $x$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 $x$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.
Question 3
Consider the following graph.
How many units must be sold to achieve the break-even point?
What is the income per unit sold?
What is the cost per unit?
What is the fixed cost of producing any number of units?