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3.02 Problem solving with linear equations


Now that we know how to solve equations we are given, the next step is to create our own equations to solve a particular situation or problem we have been given.

Let's work through an example first and then reflect on the general approach to take.

Worked example

To make a particular pasta dish, a restaurant has a fixed cost of $\$70$$70 per night plus a variable cost of $\$8$$8 per dish made.

a) How much will it cost the restaurant to make $12$12 dishes in one night?

b) If the restaurant sells the dish for $\$15$$15, will they make a profit if they sell $12$12 dishes in one night? 

ThinkThe total cost of producing $n$n dishes is equal to the sum of the fixed cost and the variable cost. We want to first set up a linear equation representing the given information that will allow us to find the cost of producing $n$n dishes. Substituting in $n=12$n=12 will allow us to find the cost. We can then compare this to the income made by selling $12$12 dishes for $\$15$$15 each.

Do: If we let the total cost of n dishes equal $y$y, we can set up an equation as follows:


Notice that the fixed cost is our constant term, and the coefficient of $n$n is the cost of producing each dish. 
We can now substitute in $n=12$n=12 to find the cost of producing $12$12 dishes.

$y$y $=$= $8n+70$8n+70

Start by writing the general equation

  $=$= $8\times12+70$8×12+70

Substitute in $n=12$n=12

  $=$= $96+70$96+70

Evaluate the product

  $=$= $166$166

Evaluate the sum


So the cost of producing $12$12 dishes is $\$166$$166.

Now, we want to compare this cost to the income made by selling $12$12 dishes at $\$15$$15 each. We can find this easily  by calculating the product of $12$12 and $15$15:


So the restaurant will make $\$180$$180 if they sell $12$12 dishes at $\$15$$15 each.

We can now compare this to the cost of making the $12$12 dishes, which was $\$166$$166:


We can see that as $180$180 is greater than $166$166, the restaurant has made a profit of $\$180$$180$-$$\$166$$166 $=$= $\$14$$14


The general approach

The following general steps can be taken to solve a problem using equations:

  1. Identify the unknown value you are trying to solve for and let it be represented by a pronumeral (the question may already have given you the pronumeral to use).
  2. Identify any equations, concepts or formula that may be relevant to the problem. For example, if the question refers to averages, it may be useful to remember that:$\text{Average }=\frac{\text{sum of scores }}{\text{number of scores }}$Average =sum of scores number of scores
  3. Try to relate the unknown to the other values given in the problem (either using words or mathematical symbols) to form an equation.

    It may be useful to describe the relationships you can see in words before writing them out as mathematical equations, or even to form smaller and more obvious mathematical expressions and see how these expressions relate to one another.

  4. Solve the equation

  5. Test your solution by substituting in the value you have found into the original equation.


Practice questions

Question 1

A diver starts at the surface of the water and starts to descend below the surface at a constant rate. The table shows the depth of the diver over $5$5 minutes.

Number of minutes passed ($x$x) $0$0 $1$1 $2$2 $3$3 $4$4
Depth of diver in metres ($y$y) $0$0 $1.35$1.35 $2.7$2.7 $4.05$4.05 $5.4$5.4
  1. Write an equation for the relationship between the number of minutes passed ($x$x) and the depth ($y$y) of the diver.

  2. At what depth would the diver be after $59$59 minutes?

  3. In the equation, $y=1.35x$y=1.35x, what does $1.35$1.35 represent?

    The change in depth per minute.


    The diver’s depth below the surface.


    The number of minutes passed.


    The change in depth per minute.


    The diver’s depth below the surface.


    The number of minutes passed.


Question 2

Sisters Ursula and Eileen are training for a triathlon event. Ursula finds that her average cycling speed is $13$13 kph faster than Eileen's average running speed.

Ursula can cycle $46$46 kilometres in the same time that it takes Eileen to run $23$23 kilometres.

  1. If Eileen's running speed is $n$n kilometres per hour, solve for $n$n.

  2. Determine Ursula's average cycling speed.

Question 3

Find the perpendicular height, $h$h, of a parallelogram that has an area of $66$66 cm2 and a base of $6$6 cm.

  1. Start by substituting the given values into the formula for the area of a paralleogram.


    Enter each line of working as an equation.



Solve problems involving linear equations, including those derived from formulas.

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