Mastering Maths: Your Guide to Formulae!
Hey everyone! Welcome to your study notes for the chapter on Formulae. Think of a formula as a secret code or a recipe in mathematics. It's a special rule that helps us solve problems quickly and easily. In this chapter, we're going to learn how to use these "recipes", how to change them to find different "ingredients", and even how to work with them when they look like fractions.
This is a super important skill in Maths, Science, and even in everyday life, like when you're calculating the cost of your mobile data or figuring out how long a trip will take. So, let's get started and unlock the power of formulae together!
Part 1: Using Formulae by Substitution
The easiest way to use a formula is by a method called substitution. It's a fancy word for a simple idea: swapping letters for numbers.
What is Substitution?
In a formula, letters (called variables) are used to represent values that can change. Substitution is just the act of replacing these letters with the actual numbers you've been given in a problem.
Imagine you're making a smoothie. The recipe says: "Blend 1 banana (b) and 2 apples (a)". If you have the banana and apples, you just put them in! Substitution is the same - you just put the numbers where the letters are.
How to Substitute: A Step-by-Step Guide
- Write down the formula. This helps you see exactly what you need.
- Identify the values you are given for each variable.
- Replace (substitute) each letter in the formula with its number. It's a great idea to put the numbers inside brackets `()` to avoid mistakes.
- Calculate the answer. Remember to follow the order of operations (BODMAS/PEMDAS)!
Example 1: Finding the Area of a Garden
Let's find the area (A) of a rectangular garden that is 10 metres long (l) and 5 metres wide (w). The formula is:
$$ A = l \times w $$
Step 1: The formula is $$A = lw$$. (Remember, `lw` means `l` times `w`).
Step 2: We are given $$l = 10$$ and $$w = 5$$.
Step 3: Substitute the numbers into the formula: $$A = (10) \times (5)$$.
Step 4: Calculate the answer: $$A = 50$$.
So, the area of the garden is 50 square metres. Easy, right?
Example 2: A Real-World Formula - Mobile Phone Plan
The total monthly cost (C) of a phone plan is $50 plus $10 for every gigabyte (G) of data used. The formula is:
$$ C = 50 + 10G $$
If you used 7 gigabytes of data this month, what is your bill?
Given: $$G = 7$$
Substitute: $$C = 50 + 10(7)$$
Calculate: $$C = 50 + 70$$
$$ C = 120 $$
Your monthly cost is $120.
Common Mistakes to Avoid!
Forgetting the order of operations! If the formula was $$A = 2(l+w)$$, and $$l=5, w=3$$, you must do the brackets first!
Correct: $$A = 2(5+3) = 2(8) = 16$$
Incorrect: $$A = 2 \times 5 + 3 = 10 + 3 = 13$$
Key Takeaway for Part 1
Substitution is just 'plugging in the numbers'. Write the formula, replace the letters with their given values, and carefully calculate the answer.
Part 2: Changing the Subject of a Formula
Sometimes, a formula is set up to find one specific thing, but we need to find something else. This is where changing the subject comes in. The subject of a formula is the variable that is all by itself on one side of the equals sign.
Think of it like this: In the formula $$A = lw$$, 'A' is the star of the show. But what if we already know the area and the width, and we want to find the length 'l'? We need to make 'l' the new star!
The Golden Rule: Keep it Balanced!
An equation is like a balanced seesaw. To keep it balanced, whatever you do to one side, you MUST do to the other side.
- To get rid of an added number, you subtract it from both sides.
- To get rid of a subtracted number, you add it to both sides.
- To get rid of a multiplied number, you divide both sides by it.
- To get rid of a divided number, you multiply both sides by it.
Quick Review: Inverse Operations
Inverse operations are opposites that 'undo' each other.
- Addition (+) undoes Subtraction (-)
- Multiplication (×) undoes Division (÷)
How to Change the Subject: A Step-by-Step Guide
- Identify the new subject - the letter you want to get by itself.
- See what is 'stuck' to it (what numbers or letters are on the same side).
- Undo the operations one by one, using inverse operations. Remember to do it to BOTH sides!
- Keep going until your new subject is all alone.
Example 1: Making 'l' the subject
Let's use the area formula again: $$A = lw$$
We want to make `l` the subject.
Step 1: The new subject is `l`.
Step 2: `l` is being multiplied by `w`.
Step 3: The inverse of multiplying by `w` is dividing by `w`. Let's divide both sides by `w`.
$$ \frac{A}{w} = \frac{lw}{w} $$
The `w` on the right side cancels out.
$$ \frac{A}{w} = l $$
Step 4: `l` is now the subject! We can write it as $$ l = \frac{A}{w} $$.
Example 2: A Two-Step Problem
A famous formula from physics is for velocity: $$v = u + at$$
Let's make `a` the subject.
Goal: Get `a` by itself.
First, `u` is added to `at`. Let's undo the addition by subtracting `u` from both sides.
$$ v - u = u + at - u $$
$$ v - u = at $$
Now, `a` is being multiplied by `t`. Let's undo this by dividing both sides by `t`.
$$ \frac{v-u}{t} = \frac{at}{t} $$
$$ \frac{v-u}{t} = a $$
We did it! The new formula is $$ a = \frac{v-u}{t} $$.
Key Takeaway for Part 2
Changing the subject is about isolating a variable. Use inverse operations on both sides of the equation to 'undo' everything around your target variable until it stands alone.
Part 3: Working with Algebraic Fractions
Don't panic! An algebraic fraction is just a fraction that has letters (variables) in it. The rules you learned for normal fractions still apply. Let's review them.
Don't worry if this seems tricky at first. It's like learning a new level in a game - it takes a bit of practice!
Multiplication and Division (The Easy Part!)
Multiplication: Just multiply the tops (numerators) together and the bottoms (denominators) together.
Example: $$ \frac{a}{2} \times \frac{b}{3} = \frac{a \times b}{2 \times 3} = \frac{ab}{6} $$
Division: Use the "Keep, Change, Flip" method. Keep the first fraction, change divide to multiply, and flip the second fraction.
Example: $$ \frac{x}{5} \div \frac{y}{2} = \frac{x}{5} \times \frac{2}{y} = \frac{2x}{5y} $$
Addition and Subtraction (The Tricky Part!)
The number one rule for adding or subtracting fractions is: You must have a common denominator!
A common denominator is a common multiple of the original denominators. The best one to use is the Lowest Common Multiple (LCM).
How to Add/Subtract Algebraic Fractions: Step-by-Step
- Find the Lowest Common Multiple (LCM) of the denominators. This will be your new common denominator. For variables, this often means just multiplying them.
- Rewrite each fraction so it has the new common denominator. Remember to multiply the top by whatever you multiplied the bottom by!
- Add or subtract the numerators (the tops). Keep the denominator the same.
- Simplify the final fraction if you can.
Example 1: Simple Addition
Let's calculate: $$ \frac{3}{x} + \frac{4}{y} $$
Step 1: The denominators are `x` and `y`. The LCM is just `xy`.
Step 2:
- For the first fraction ($$\frac{3}{x}$$), we multiplied the bottom (`x`) by `y`. So we must multiply the top by `y` too: $$ \frac{3 \times y}{x \times y} = \frac{3y}{xy} $$
- For the second fraction ($$\frac{4}{y}$$), we multiplied the bottom (`y`) by `x`. So we must multiply the top by `x` too: $$ \frac{4 \times x}{y \times x} = \frac{4x}{xy} $$
Step 3: Now add the numerators:
$$ \frac{3y}{xy} + \frac{4x}{xy} = \frac{3y + 4x}{xy} $$
Step 4: We can't simplify this any further. That's our final answer!
Example 2: Subtraction with Linear Factors
Let's calculate: $$ \frac{5}{x+1} - \frac{2}{x+2} $$
Step 1: The denominators are `(x+1)` and `(x+2)`. They are different, so the LCM is just their product: `(x+1)(x+2)`.
Step 2: Rewrite the fractions:
$$ \frac{5(x+2)}{(x+1)(x+2)} - \frac{2(x+1)}{(x+1)(x+2)} $$
Step 3: Subtract the numerators. Be very careful with the brackets!
$$ \frac{5(x+2) - 2(x+1)}{(x+1)(x+2)} $$
Now, expand the brackets on top:
$$ \frac{5x + 10 - 2x - 2}{(x+1)(x+2)} $$
Combine the like terms on top:
$$ \frac{3x + 8}{(x+1)(x+2)} $$
Step 4: This is the final simplified answer.
Common Mistakes to Avoid!
- Illegal Cancelling: You CANNOT cancel parts of an addition or subtraction. For example, in $$ \frac{x+5}{x} $$, you cannot cancel the `x`'s!
- Forgetting the Brackets: When subtracting, always put the second numerator in brackets, like in our example: $$... - 2(x+1)$$. This helps you remember to subtract the *entire* thing.
Did You Know?
The word "formula" comes from the Latin word which means "a small pattern or rule". Some of the most famous formulae, like Pythagoras' Theorem for triangles ($$a^2 + b^2 = c^2$$), have been used for thousands of years!
Key Takeaway for Part 3
Algebraic fractions follow the same rules as number fractions. For addition and subtraction, the golden rule is to find a common denominator before you do anything else!