Constructing Linear Equations Revision Notes for VCE SSCE Mathematical Methods

Constructing Linear Equations

Many real-world problems can be solved by translating them into mathematical language. When you come across a word problem, you can represent the unknown quantity with a variable (like $x$ ) and build an equation from the information given. Once you have the equation, you can solve it to find the value of the unknown.

Setting up the equation

Before you construct an equation, there are some important steps to follow:

Define your variable clearly. State exactly what your variable represents, including the units. For example, "Let $x$ be the weight of the turkey in kilograms" or "Let $t$ represent the time in hours."

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Ensure unit consistency. All elements in your equation must use the same system of units. If the problem gives you information in different units (like metres and centimetres), convert everything to the same unit before setting up your equation. This is a critical step that students often overlook!

Express everything in terms of your variable. Once you've defined your variable, write expressions for other quantities in the problem using that variable.

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When defining your variable, being specific about units helps you avoid confusion later. If a problem mixes units (like metres and centimetres), choose one unit system and convert all measurements to that system before constructing your equation.

Steps for solving word problems

Follow these systematic steps when tackling word problems:

Read the question carefully and identify the known information
Identify the unknown quantity you need to find
Assign a variable to this unknown quantity (remember to state units)
Form an expression in terms of your variable using the given information
Construct an equation from this expression
Solve the equation
Write your answer as a complete sentence, answering the original question

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Worked Example: Cooking Time

Problem: A chef uses this rule for cooking a turkey: "Allow $30$ minutes for each kilogram weight of turkey and then add an extra $15$ minutes." If a turkey took $3$ hours to cook, how much did it weigh?

Solution:

Let $x$ be the weight of the turkey in kilograms.

The cooking time in minutes is $30x + 15$ .

We know the turkey cooked for $3$ hours, which equals $180$ minutes.

Setting up our equation:

30x + 15 = 180

30x = 165

x = 5.5

Answer: The turkey weighed 5.5 kilograms.

Understanding this problem: We translated the chef's cooking rule into a mathematical expression. The " $30$ minutes per kilogram" becomes $30x$ , and the "extra $15$ minutes" is added on. The key was recognising that we needed to convert hours to minutes to keep our units consistent.

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Worked Example: Rectangle Problem

Problem: Find the area of a rectangle whose perimeter is $1.08$ m, if it is $8$ cm longer than it is wide.

Solution:

Let $\ell$ be the length in centimetres.

Then the width is $(\ell - 8)$ cm.

We know that for a rectangle:

Perimeter $= 2 \times \text{length} + 2 \times \text{width}$

In terms of $\ell$ :

Perimeter $= 2\ell + 2(\ell - 8)$

Perimeter $= 4\ell - 16$ cm

The perimeter is $1.08$ m $= 108$ cm.

Therefore:

4\ell - 16 = 108

4\ell = 124

\ell = 31 \text{ cm}

So the length is $31$ cm and the width is $23$ cm.

Area $= 31 \times 23 = 713$ cm²

Answer: The area is 713 cm².

Understanding this problem: The relationship between length and width was expressed using one variable. We converted the perimeter from metres to centimetres to match our other measurements. After finding the length, we could calculate the width and then find the area.

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Worked Example: Speed and Distance

Problem: Adam normally takes $5$ hours to travel between Higett and Logett. One day he increases his speed by $4$ km/h and the journey takes half an hour less. Find his normal speed.

Solution:

Let $x$ be Adam's normal speed in km/h.

Using the relationship: distance $= \text{speed} \times \text{time}$

The distance from Higett to Logett is:

x \times 5 = 5x \text{ kilometres}

Adam's new speed is $(x + 4)$ km/h.

His new time is $4.5$ hours (which is $\frac{9}{2}$ hours).

Since the distance is the same:

(x + 4) \times \frac{9}{2} = 5x

Multiplying both sides by $2$ :

9(x + 4) = 10x

9x + 36 = 10x

36 = x

Answer: His normal speed is 36 km/h.

Understanding this problem: The key formula here is distance $= \text{speed} \times \text{time}$ . Because the distance between the two places doesn't change, we could create an equation by setting up two different expressions for the same distance. Remember that in problems like this, we're working with average speed.

Key formulas

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When solving word problems, you'll often need these fundamental relationships:

Distance $= \text{speed} \times \text{time}$
Perimeter of rectangle $= 2 \times \text{length} + 2 \times \text{width}$
Area of rectangle $= \text{length} \times \text{width}$

These formulas appear frequently in word problems, so it's worth memorising them!

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Key Points to Remember:

Always clearly define your variable and state what it represents, including units
Ensure all parts of your equation use the same units
Read the problem carefully to identify what you know and what you need to find
Translate the word problem into a mathematical equation step by step
After solving, write your answer as a complete sentence that answers the original question

Constructing Linear Equations (VCE SSCE Mathematical Methods): Revision Notes