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3 Linear relationships
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Stage 5.1-3

3.02 Problem solving with linear equations

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Introduction

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.

Problems with linear equations

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 formulas that may be relevant to the problem.
  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.

Examples

Example 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 minutes.

\text{Number of minutes passed} \left(x\right)01234
\text{Depth of diver in metres} \left(y\right)01.42.84.25.6
a

Write an equation for the relationship between the number of minutes passed \left(x\right) and the depth \left(y\right) of the diver.

Worked Solution
Create a strategy

Find the gradient, m, and the y-intercept, c, from the table and substitute them into the equation y=mx+c.

Apply the idea

We can find the gradient by using the formula: m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}. For this problem, we will be using the points \left(1,1.4\right) and \left(2,2.8\right).

\displaystyle m\displaystyle =\displaystyle \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}Write the equation
\displaystyle =\displaystyle \dfrac{2.8-1.4}{2-1}Substitute values
\displaystyle =\displaystyle 1.4Evaluate

We can find the y-intercept by determining the value of y when x=0. Looking at the table, when x=0, the value of y is equal to 0. So the y-intercept, c, is 0.

Substituting the values, our linear equation is y=1.4x.

b

At what depth would the diver be after 65 minutes?

Worked Solution
Create a strategy

Substitute the value of x into the equation found in part (a).

Apply the idea
\displaystyle \text{Depth}\displaystyle =\displaystyle 1.4 \times 65Substitute values
\displaystyle =\displaystyle 91\text{ metres}Evaluate
c

In the equation, y=1.4x, what does 1.4 represent?

A
The change in depth per minute.
B
The diver's depth below the surface
C
The number of minutes passed.
Worked Solution
Create a strategy

Use the given table to determine what the 1.4 value represents.

Apply the idea

It represents the rate of change of y. So the correct answer is option A.

Example 2

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

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

a

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

Worked Solution
Create a strategy

Equate the time of Eileen to the time of Ursula and solve for the value of n using the inverse operations.

Apply the idea

We need to first determine the equation of time by rearranging the equation of speed to isolate time on one side of the equation.

\displaystyle \text{speed}\displaystyle =\displaystyle \dfrac{\text{distance}}{\text{time}}Write the equation for speed
\displaystyle \text{time}\displaystyle =\displaystyle \dfrac{\text{distance}}{\text{speed}}Rearrange the equation

Using the equation of time, we can now calculate Eileen's speed. Eileen cycled at a speed of n, for a distance of 23 kilometres while Ursula ran at a speed of n+13, for a distance of 46 kilometres.

\displaystyle \text{Time of Eileen}\displaystyle =\displaystyle \text{Time of Ursula}Set up the equation
\displaystyle \dfrac{23}{n}\displaystyle =\displaystyle \dfrac{46}{n+13}Substitute data
\displaystyle 23\times \left(n+13\right)\displaystyle =\displaystyle 46\times nCross multiply
\displaystyle 23n + 299\displaystyle =\displaystyle 46nExpand the brackets
\displaystyle 23n-23n-299\displaystyle =\displaystyle 46n-23nSubtract 23n from both sides
\displaystyle 299\displaystyle =\displaystyle 23nEvaluate difference
\displaystyle \dfrac{299}{23}\displaystyle =\displaystyle \dfrac{23n}{23}Divide both sides by 23
\displaystyle n\displaystyle =\displaystyle 13 \text{ kph}Evaluate
b

Determine Ursula's average cycling speed.

Worked Solution
Create a strategy

Substitute the value of n found in part (a) into the expression for Ursula's speed.

Apply the idea
\displaystyle \text{Average speed}\displaystyle =\displaystyle n+13Use the expression for Ursula's speed
\displaystyle =\displaystyle 13+13Substitute the value
\displaystyle =\displaystyle 26 \text{ kph}Evaluate
Idea summary

The general approach 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.
  2. Identify any equations, concepts or formulas that may be relevant to the problem.
  3. Write an equation that relate the unknown to the other values given in the problem.
  4. Solve the equation.
  5. Test your solution by substituting the value you have found into the original equation.

Outcomes

MA5.1-6NA

determines the midpoint, gradient and length of an interval, and graphs linear relationships

MA5.2-5NA

recognises direct and indirect proportion, and solves problems involving direct proportion

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