Welcome to Formulae and Substitution!
Welcome to the world of Algebra! In this chapter, we are going to learn about Formulae and Substitution. Don't worry if those words sound a bit intimidating—you actually use these ideas every single day without realizing it.
Think of a formula like a "math recipe." Just as a recipe tells you exactly what to do to make a cake, a formula tells you exactly what to do to find an answer. Substitution is simply the act of "swapping" letters for numbers so we can get our final result. By the end of these notes, you'll be a pro at swapping and solving!
1. What is a Formula?
A formula is a rule that shows the relationship between different quantities. It usually has an equals sign (\( = \)) and uses letters to represent numbers that can change. We call these letters variables.
Real-World Example: Imagine you are a window cleaner. You charge £10 just to turn up, plus £5 for every window you clean. We can write this as a formula:
\( \text{Total Cost} = 10 + (5 \times \text{number of windows}) \)
In algebra, we use letters to make this shorter:
\( C = 10 + 5w \)
(Where \( C \) is the cost and \( w \) is the number of windows).
Did you know? The plural of formula can be either formulas or formulae. Both are correct, but formulae is the traditional mathematical way to say it!
Key Takeaway:
A formula is just a mathematical rule that uses letters (variables) to help us solve problems quickly.
2. The "Swap Shop": How to Substitute
Substitution is when we replace a letter in a formula with a specific number. It's like a "Swap Shop"—you take the letter out and put the number in its place.
Step-by-Step Substitution:
Let's look at the expression: \( 3a + 7 \)
Find the value when \( a = 4 \).
Step 1: Identify the letter. In this case, the letter is \( a \).
Step 2: Swap the letter for the number. We know \( a = 4 \), so we replace \( a \) with \( 4 \).
Step 3: Remember the hidden multiplication! In algebra, when a number and a letter are touching (like \( 3a \)), it means multiply.
\( 3 \times 4 + 7 \)
Step 4: Calculate the answer.
\( 12 + 7 = 19 \)
Quick Review:
- \( 2x \) means \( 2 \times x \)
- \( ab \) means \( a \times b \)
- \( \frac{x}{2} \) means \( x \div 2 \)
Key Takeaway:
Always remember that a number next to a letter means multiply!
3. Using BIDMAS in Substitution
When you have a long formula, you must follow the correct order of operations. We use the word BIDMAS to help us remember the order:
B - Brackets
I - Indices (Powers like \( ^2 \) or \( ^3 \))
D/M - Division and Multiplication
A/S - Addition and Subtraction
Example: Find the value of \( 2n^2 + 5 \) when \( n = 3 \).
1. Substitute the number: \( 2 \times (3^2) + 5 \)
2. Do the Indices first: \( 3^2 \) is \( 9 \).
3. Now we have: \( 2 \times 9 + 5 \)
4. Do the Multiplication: \( 18 + 5 \)
5. Finally, Addition: \( 23 \)
Common Mistake to Avoid: Don't multiply before doing the power! In the example above, if you did \( 2 \times 3 \) first to get \( 6 \), and then squared it, you would get \( 36 \). This is incorrect. Always square the number first!
4. Dealing with Negative Numbers
Substitution can get a little tricky when we use negative numbers. Don't worry if this seems tricky at first! A great trick is to always put the negative number in brackets when you swap it into the formula.
Example: Find the value of \( 5x - 2 \) when \( x = -4 \).
1. Substitute with brackets: \( 5(-4) - 2 \)
2. Multiply: \( 5 \times -4 = -20 \)
3. Subtract: \( -20 - 2 = -22 \)
Memory Aid for Signs:
- A positive and a negative make a negative: \( (+) \times (-) = (-) \)
- Two negatives make a positive: \( (-) \times (-) = (+) \)
Key Takeaway:
Use brackets around negative numbers to keep your calculation tidy and avoid sign errors.
5. Real-Life Formulae Examples
Let's see how we use these skills in "real life."
The Taxi Fare
A taxi company uses the formula: \( F = 2.50 + 1.2d \)
Where \( F \) is the fare in pounds and \( d \) is the distance in miles.
How much does a 5-mile trip cost?
1. Swap \( d \) for \( 5 \): \( F = 2.50 + 1.2(5) \)
2. Multiply first: \( 1.2 \times 5 = 6 \)
3. Add: \( 2.50 + 6 = 8.50 \)
The trip costs £8.50.
Area of a Triangle
The formula for the area of a triangle is: \( A = \frac{bh}{2} \)
Where \( b \) is the base and \( h \) is the height.
Find the area if \( b = 6\text{cm} \) and \( h = 4\text{cm} \).
1. Swap the letters: \( A = \frac{6 \times 4}{2} \)
2. Top part: \( 24 \)
3. Divide by 2: \( 24 \div 2 = 12\text{cm}^2 \)
6. Summary and Quick Tips
To wrap up this chapter, here are the most important things to remember:
- Formula: A mathematical rule using letters.
- Substitution: Swapping a letter for a number.
- Touching means multiply: \( 4y \) means \( 4 \times y \).
- Use BIDMAS: Always follow the correct order (Brackets and Powers first!).
- Negatives: Be extra careful with minus signs; use brackets to stay organized.
Top Tip: When you finish a calculation, read it back to yourself. Does the answer make sense? If you are calculating the price of a taxi ride and get £5,000, you might have accidentally multiplied where you should have added!
Final Takeaway:
Substitution is just like being a detective—you take the clues (the numbers) and put them in the right places to solve the mystery!