3 Linear equations

Lesson

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.

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

- 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).
- Identify any equations, concepts or formulas that may be relevant to the problem.
- 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.
- Solve the equation.
- Test your solution by substituting in the value you have found into the original equation.

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) | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

\text{Depth of diver in metres} \left(y\right) | 0 | 1.4 | 2.8 | 4.2 | 5.6 |

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

b

At what depth would the diver be after 65 minutes?

Worked Solution

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

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

b

Determine Ursula's average cycling speed.

Worked Solution

Idea summary

The general approach to solve a problem using equations:

- Identify the unknown value you are trying to solve for and let it be represented by a pronumeral.
- Identify any equations, concepts or formulas that may be relevant to the problem.
- Write an equation that relate the unknown to the other values given in the problem.
- Solve the equation.
- Test your solution by substituting the value you have found into the original equation.

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