Substitution of Values Into a Formula and Constructing a Table of Values Revision Notes for VCE SSCE General Mathematics

Substitution of Values Into a Formula and Constructing a Table of Values

What is a formula?

A formula is a mathematical relationship that connects two or more variables. Variables are letters that represent values that can change or are unknown.

Common examples of formulas include:

$C = 45t + 150$ relates the cost ( $C$ dollars) of hiring a plumber for $t$ hours
$P = 4L$ finds the perimeter of a square, where $P$ is the perimeter and $L$ is the side length

Understanding formulas is essential because they allow us to calculate unknown values when we know some of the other variables in the relationship.

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Formulas are powerful tools in mathematics and real-world applications. They provide a systematic way to calculate values based on known information, making them invaluable in fields ranging from engineering to finance to everyday problem-solving.

Substituting values into a formula

When we substitute values into a formula, we replace the known variables with their numerical values to find the unknown variable. This process involves careful calculation and attention to the order of operations.

Steps for substitution

Follow these steps when working with formulas:

Write down the formula you are using
Substitute the known values into the formula
Evaluate the expression using the correct order of operations
Write your answer with appropriate units

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Common Mistake to Avoid: Many students forget to use the proper order of operations (BODMAS/PEMDAS) when evaluating formulas. Always remember:

Brackets/Parentheses
Orders/Exponents
Division and Multiplication (left to right)
Addition and Subtraction (left to right)

Worked example: Windsurfer hire

A windsurfer rental company charges according to the formula:

$C = 10 + 40t$

where $C$ is the cost in dollars and $t$ is the time in hours. Let's calculate the cost of hiring a windsurfer for $2$ hours.

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Worked Example: Calculating Windsurfer Hire Cost

Step 1: Write the formula

$C = 10 + 40t$

Step 2: Substitute $t = 2$ into the formula

$C = 10 + 40 \times 2$

Step 3: Evaluate

$C = 10 + 80$

$C = 90$

Step 4: Write the answer

It will cost $90 for a 2-hour hire.

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Exam Tip: Always check that you've substituted into the correct variable and used the proper order of operations (BODMAS/PEMDAS).

Worked example: Perimeter with π

The perimeter of this compound shape can be calculated using the formula:

$P = 2L + H\left(1 + \frac{\pi}{2}\right)$

where $L$ is the length of the rectangle and $H$ is the height. Find the perimeter to one decimal place if $L = 16.1$ cm and $H = 3.2$ cm.

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About π (pi): $\pi$ (pi) is the ratio of the circumference of any circle to its diameter. It is an irrational number approximately equal to $3.14159$ . Calculators have a special π key for accurate calculations.

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Worked Example: Finding the Perimeter with π

Step 1: Write the formula

$P = 2L + H\left(1 + \frac{\pi}{2}\right)$

Step 2: Substitute $L = 16.1$ and $H = 3.2$ into the formula

$P = 2 \times 16.1 + 3.2\left(1 + \frac{\pi}{2}\right)$

Step 3: Evaluate

$P = 40.4 \text{ (to one decimal place)}$

Step 4: Give the answer with correct units

The perimeter is $40.4$ cm.

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Exam Tip: When working with π, use the calculator's π button rather than typing 3.14 to ensure accuracy in your final answer.

Constructing a table of values

A table of values displays the relationship between variables in an organized format. We can create these tables by repeatedly substituting different values into a formula.

Creating tables by hand

To construct a table of values manually:

Set up a table with rows for each variable
Choose appropriate values for the independent variable
Substitute each value into the formula
Calculate and record the corresponding output values

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Creating tables by hand helps you understand the relationship between variables and develops your calculation skills. While technology can automate this process, understanding the manual method is crucial for developing mathematical intuition.

Worked example: Temperature conversion

The formula for converting degrees Celsius to degrees Fahrenheit is:

$F = \frac{9}{5}C + 32$

Let's construct a table of values for $F$ using values of $C$ in intervals of $10$ between $C = 0$ and $C = 100$ .

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Worked Example: Creating a Temperature Conversion Table

Method:

We substitute each value of $C$ into the formula to find the corresponding $F$ value:

If $C = 0$ : $F = \frac{9}{5}(0) + 32 = 32$

If $C = 10$ : $F = \frac{9}{5}(10) + 32 = 18 + 32 = 50$

Continue this process for all values.

Complete table:

C	0	10	20	30	40	50	60	70	80	90	100
F	32	50	68	86	104	122	140	158	176	194	212

Using technology to create tables

Modern calculators can automate the process of creating tables of values, saving time and reducing calculation errors.

Using the TI-Nspire calculator

To create a table of values on the TI-Nspire:

Start a new document and select Add Lists & Spreadsheet
Name your columns (lists) appropriately - for example, 'c' for Celsius and 'f' for Fahrenheit
Enter your input values in the first column

In the formula cell of the second column, type your formula (e.g., = 9 ÷ 5 × c + 32)
Press enter to calculate all values automatically

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Helpful Hint: When typing variable names like 'c', you may need to select 'Variable Reference' when prompted, or press the var key to avoid confusion with column names.

Using the ClassPad calculator

To create a table of values on the ClassPad:

Open the Graph & Table application from the menu
Enter your formula next to y1= (for example: $\frac{9}{5}x + 32$ )
Tap the Table Input icon to set your starting value, ending value, and step size
Tap the table icon to display the complete table of values

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In the Graph & Table application, y1 represents your output variable (like $F$ ) and x represents your input variable (like $C$ ).

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Key Points to Remember:

A formula is a mathematical relationship connecting two or more variables
Substitution means replacing variables with known numerical values
Always follow the correct order of operations (BODMAS/PEMDAS) when evaluating formulas
Include appropriate units in your final answer
A table of values organizes the relationship between variables systematically
Technology can automate table creation, but understanding the manual process is essential for learning

Substitution of Values Into a Formula and Constructing a Table of Values (VCE SSCE General Mathematics): Revision Notes