4.05 Applications of linear systems
The magic of simultaneous equations comes to life when we see how useful it is in real-life applications. When we solve problems involving at least two unknown quantities and have some information relating them, we can set up simultaneous equations to solve for the unknowns.
Let's have a look at this scenario:
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A commercial plane can carry a total passenger and luggage load of $20000$20000 kg. The airline calculates that, on average, each passenger weighs $70$70 kg and each piece of luggage weighs $15$15 kg. It expects that each passenger checks in two pieces of luggage. How many passengers is the plane able to carry? |
When presented with a problem like this to solve, we can follow these general steps:
| Steps | Applying to the scenario | |
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1. Identify the unknown quantities in the problem. What are we trying to solve for? |
There are two unknowns:
We want to solve for the number of passengers. |
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2. Introduce some variables to represent the unknown quantities. |
Let $x$x represent the number of passengers, and let $y$y represent the number of pieces of luggage. |
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3. Figure out how the pieces of information are related to each other. |
The total weight of passengers $\left(70x\right)$(70x) and luggage $\left(15y\right)$(15y) is $20000$20000. The number of luggage pieces are twice the number of passengers. |
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4. Use the relationships to form equations and solve them. |
Using the total weight, we get the equation: $70x+15y=20000$70x+15y=20000 And since the number of luggage pieces is twice the number of passengers, so: $y=2x$y=2x These two equations can be solved simultaneously to solve for $x$x. Graphing the lines using technology, we find that the point of intersection is $\left(200,400\right)$(200,400). $x$x represents the number of passengers and $y$y represents the number of pieces of luggage, so the plane can carry $200$200 passengers and $400$400 pieces of luggage at the given average weights. |
Once we have worked out the equations we need, we can plot them on a number plane and find their point of intersection, which represents the solution to the system we just modelled. The values of $x$x and $y$y in the solution represent different things depending on the situation.
Practice questions
Question 1
A family owns two businesses, EcoLine and Helios. These businesses made a combined profit of $\$12$$12 million in the previous financial year, with Helios making $3$3 times as much profit as EcoLine.
Let $x$x represent the profit (in millions of dollars) of EcoLine, and $y$y be the profit (in millions of dollars) of Helios.
Use the fact that the two businesses made a combined profit of $\$12$$12 million to set up an equation involving $x$x and $y$y.
Use the fact that Helios made $3$3 times as much as EcoLine 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 EcoLine's profit.
Use the graph to find Helios' profit.
Question 2
Toby's piggy bank contains only $5$5c and $10$10c coins. He knows that there are $53$53 coins in the piggy bank, and that the total value is $\$3.50$$3.50.
Let $x$x and $y$y be the number of $5$5c and $10$10c coins respectively. We will create two equations then solve them simultaneously to find the number of coins of each type.
Write the fact that the total number of coins is $53$53 as an equation (call it Equation 1).
Write the equation in the form $ax+by=c$ax+by=c, where $a$a is positive.
Write the fact that the total amount of coins are worth $\$3.50$$3.50 as an equation (call it Equation 2).
Write the equation in the form $ax+by=c$ax+by=c, where $a$a is positive.
We now have the following equations:
Equation 1 $x+y=53$x+y=53 Equation 2 $5x+10y=350$5x+10y=350 Solve the system of linear equations by either graphing or by using technology, leaving your answer in coordinate form.
$\left(x,y\right)=\left(\editable{},\editable{}\right)$(x,y)=(,)
Question 3
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 $y=400+60x$y=400+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 $y=1000+30x$y=1000+30x.
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 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 for which it will be more cost efficient to use machines to manufacture the t-shirts.
State the range of values of $x$x at which it will be more cost efficient to employ people to manufacture the t-shirts.