Introduction to Formulas

🎬 Video: What Are Formulas and How to Use Them

What is a Formula in Math? (0:01)

🌟 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.

How to Visualising Formulas? (1:05)

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.

How to Use Formulas to Solve Real-life Problems? (1:32)

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:)

📂 Flashcards: Formula Definition, Visualisation and Examples

🍪 Quiz: Practice Solving Problems With Formulas

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Introduction to Formulas

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Q: If $y = 2x + 3$, what is $y$ when $x = 3$?

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Q: The formula for the distance traveled by a car is $y = 60x + 120$, where $y$ is the distance in miles and $x$ is the number of hours driven. How far will it travel after 3 hours?

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Q: If $y = 4x - 7$, what is $x$ when $y = 13$?

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Q: A company’s total revenue $y$ is represented by $y = 500x - 200$, where $x$ is the number of products sold. How many products must be sold to reach a revenue of £4,800?

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Q: The total cost of a gym membership is calculated by $y = 30x + 50$, where $x$ is the number of months and $y$ is the total cost in pounds. If the total cost is £200, how many months did the member sign up for?

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Q: A company charges a \$200 setup fee and \$50 for printing each book. If you print more than 10 books, you get a 10% discount on the total cost. How many books can be printed for \$720?

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