Introduction to Formulas

🌟 Definition of A Formula:

A formula is a mathematical equation that shows the relationship between two or more quantities.

It helps us calculate values by substituting different numbers.

🌟 Example of A Formula:

$$ y = 2x + 3 $$

• When $ x = 1 $, $ \ y = 2 \times 1 + 3 = 5 $
• When $ x = 2 $, $ \ y = 2 \times 2 + 3 = 7 $
• When $ x = 3 $, $ \ y = 2 \times 3 + 3 = 9 $

Formulas are used in many areas of math and science, such as calculating speed, area, and financial interest.

A formula can be represented in different ways to show the relationship between variables. Two common methods are using a table and a graph.

🌟 Using a Table to Visualise a Formula:

A table helps organize values by substituting different values of $x$ into the formula to calculate $y$.
$$
\begin{array}{c|@{\quad}l}
\ \textbf{x} & \ \textbf{y = 2x + 3} \\
\hline
1 & \ 2 \times 1 + 3 = 5 \\
2 & \ 2 \times 2 + 3 = 7 \\
3 & \ 2 \times 3 + 3 = 9 \\
4 & \ 2 \times 4 + 3 = 11 \\
5 & \ 2 \times 5 + 3 = 13 \\
6 & \ 2 \times 6 + 3 = 15 \\
\end{array}
$$

🌟 Using a Graph to Visualise a Formula:

A graph plots the values $(x, y)$ as points on a coordinate grid.

• The x-axis represents the input values ($x$).
• The y-axis represents the output values ($y$).
• The plotted points form a straight line, showing a linear relationship.

🌟 Comparing Tables and Graphs:

• A table makes it easy to calculate and organize values step by step.
• A graph provides a visual way to understand how the values change.

🚀 A good trick for exams: Start with a table to organize values and see patterns in the numbers. Then, use a graph to visualize the relationship more clearly and identify trends at a glance.

In real-life situations, formulas help us solve problems efficiently.

Example: If a pair of shoes costs £80, you currently have £20, and you can save £12 each week. How many weeks will it take to afford the shoes?

• Step 1: You can form a formula to calculate how long it will take. $$ y = 20 + 12x $$ where $y$ represents the total money saved, and $x$ represents the number of weeks.
• Step 2: You substitute different values of $x$ into the formula to find $y$. 
  \begin{array}{c|c}
  \mathbf{x} \text{ (weeks)} & \mathbf{y = 20 + 12x} \\
  \hline
  1 & \ 20 + 12 \times 1 = 32 \\
  2 & \ 20 + 12 \times 2 = 44 \\
  3 & \ 20 + 12 \times 3 = 56 \\
  4 & \ 20 + 12 \times 4 = 68 \\
  5 & \ 20 + 12 \times 5 = 80 \\
  \end{array}
• Step 3: After 5 weeks, the total money saved reaches £80, now you can afford the shoes. You don’t need to continue the calculation:)