Developing a Formula: Setting Up and Solving an Equation in Two Unknowns Revision Notes for VCE SSCE General Mathematics

Developing a Formula: Setting Up and Solving an Equation in Two Unknowns

What you need to know

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Learning Intention

By the end of this topic, you should be able to set up and solve linear equations involving two unknown variables.

Understanding formula development

In everyday situations, you often need to create formulas to solve practical problems. Building a formula from a word problem is similar to creating an equation from a description. This skill helps you tackle real-world scenarios like calculating costs, working out totals, or planning purchases.

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The key is to identify what varies in the problem (your variables) and how these variables relate to each other through costs, rates, or other fixed values.

The general process

When you're working with two unknowns (usually called $x$ and $y$ ), follow this systematic approach:

Step 1: Build an expression for the first variable

Look at the cost or rate for the first item or quantity. Multiply this cost by your first variable to create an expression. For example, if one item costs $2, then $x$ items cost $2x$ dollars.

Step 2: Build an expression for the second variable

Do the same for the second item or quantity. If another item costs $3, then $y$ of these items cost $3y$ dollars.

Step 3: Combine the expressions

Add both expressions together to create a single formula. This gives you the total cost or total amount: $C = 2x + 3y$ (using the example above).

Step 4: Substitute and evaluate

When you're given specific values for $x$ and $y$ , substitute them into your formula and calculate the answer.

Worked example: Party food costs

Let's work through a complete example to see how this process works in practice.

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Worked Example: Calculating Party Food Costs

Problem: Sausage rolls cost $1.30 each and party pies cost $0.75 each.

Part a: Create a formula for finding the cost, $C$ dollars, of buying $x$ sausage rolls and $y$ party pies.

Part b: Calculate the cost of 12 sausage rolls and 24 party pies.

Solution to part a

Building the sausage roll expression:

Let's think about the pattern:

One sausage roll costs $1.30
Two sausage rolls cost $2 \times 1.30 = 2.60$ dollars
Three sausage rolls cost $3 \times 1.30 = 3.90$ dollars

Following this pattern, $x$ sausage rolls cost:

$x \times 1.30 = 1.3x \text{ dollars}$

Building the party pie expression:

Now for the party pies:

One party pie costs $0.75
Two party pies cost $2 \times 0.75 = 1.50$ dollars
Three party pies cost $3 \times 0.75 = 2.25$ dollars

Following this pattern, $y$ party pies cost:

$y \times 0.75 = 0.75y \text{ dollars}$

Combining into a total cost formula:

The total cost $C$ is the cost of sausage rolls plus the cost of party pies:

$C = 1.3x + 0.75y$

Solution to part b

Now we need to find the specific cost when $x = 12$ and $y = 24$ .

Step 1: Write down the formula

$C = 1.3x + 0.75y$

Step 2: Substitute the values $x = 12$ and $y = 24$

$C = 1.3 \times 12 + 0.75 \times 24$

Step 3: Evaluate the expression

\begin{aligned} C &= 15.6 + 18 \\ C &= 33.6 \end{aligned}

Step 4: Express the answer properly in dollars and cents

The total cost for 12 sausage rolls and 24 party pies is $33.60.

Key steps summary

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Key Steps for Developing a Formula with Two Unknowns:

Identify what each variable represents (e.g., $x$ = number of first item, $y$ = number of second item)
Create an expression for the first variable by multiplying its unit cost by the variable
Create an expression for the second variable in the same way
Add the two expressions together to form your complete formula
When needed, substitute specific values and calculate your answer
Don't forget to include appropriate units (like dollars and cents) in your final answer

Exam tips

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Common Mistakes to Avoid and Key Tips:

Always check your formula makes sense: if you buy zero items ( $x = 0$ and $y = 0$ ), the cost should be zero
When converting cents to dollars, remember that 75 cents = $0.75
Write out your substitution step clearly before calculating
Express money answers to two decimal places (dollars and cents)
Label your final answer with units

Remember!

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Key Takeaways:

Formula development helps solve practical problems involving two variables
The general form for cost problems is $C = ax + by$ , where $a$ and $b$ are the unit costs
Build each part separately before combining: work out the cost for $x$ items, then for $y$ items, then add them
Substitution means replacing variables with actual numbers
Always check your answer is reasonable and expressed with the correct units

Developing a Formula: Setting Up and Solving an Equation in Two Unknowns (VCE SSCE General Mathematics): Revision Notes