Welcome to the World of Algebraic Formulae!
In this chapter, we are going to learn how to speak "Maths." Think of Algebraic Formulae as shorthand recipes. Just like a recipe tells you how to turn flour and eggs into a cake, a formula tells you how to turn different pieces of information into an answer. Whether you're calculating the cost of a taxi ride or predicting how fast a rocket moves, you're using formulae!
Don't worry if algebra feels a bit like a foreign language right now. We’ll break it down step-by-step until you’re a pro.
1. Creating Your Own Formulae (Formulating)
Sometimes, a math problem is hidden inside a real-world sentence. Formulating is just the process of turning those words into an algebraic expression.
How to Build a Formula:
- Identify the variables: Pick letters to represent things that can change (like 'n' for number of miles).
- Find the "fixed" amount: This is a number that stays the same regardless.
- Find the "rate": This is the amount that changes depending on your variable.
A car hire company charges a fixed fee of £50 per day, plus 10p for every mile driven. Write a formula for the total cost (C) for one day.
1. The fixed fee is 50.
2. The miles driven can change, so let's call it m.
3. The rate is 10p (which is £0.10).
4. The Formula: \(C = 50 + 0.10m\)
Quick Tip: Always check your units! If the fixed fee is in pounds (£), make sure your "per mile" rate is also in pounds (£0.10 instead of 10p).
Key Takeaway: A formula is just a mathematical "sentence" where the equals sign (\(=\)) acts like the word "is".
2. Substitution: Plugging in the Numbers
Substitution is simply replacing the letters in a formula with actual numbers to find an answer. It’s like following the instructions on a LEGO set—you put the right piece in the right place.
Step-by-Step Substitution:
- Write down the formula.
- Replace the letters with the given numbers (use brackets for negative numbers!).
- Calculate the answer using the correct order of operations (BIDMAS/BODMAS).
Find the value of \(v\) in the formula \(v = u + at\) when \(u = 7\), \(a = 2\), and \(t = 1\).
Solution:
\(v = 7 + (2 \times 1)\)
\(v = 7 + 2\)
\(v = 9\)
Watch out for Negatives!
If you are substituting a negative number, like \(a = -9.8\), always put it in brackets: \(( -9.8 )\). This helps you remember that if you square it, the answer becomes positive!
Key Takeaway: Substitution is just "copy and paste" for math. Replace the letter with the number and solve.
3. Changing the Subject (Rearranging)
Usually, a formula starts with one letter on its own, like \(y = 3x - 2\). Here, \(y\) is the subject. Sometimes, we want a different letter to be the subject. This is like rearranging your furniture—the pieces are the same, they’re just in different spots.
The Golden Rule: The Balance Scale
Think of the equals sign as the center of a balance scale. Whatever you do to one side, you must do to the other to keep it level.
The Secret Trick: Inverse Operations
To move something, do the opposite:
- The opposite of Addition (+) is Subtraction (-)
- The opposite of Multiplication (\(\times\)) is Division (\(\div\))
- The opposite of Squaring (\(x^2\)) is Square Rooting (\(\sqrt{x}\))
1. We want \(x\) alone. First, get rid of the \(-2\) by adding 2 to both sides:
\(y + 2 = 3x\)
2. Now, get rid of the \(3\) (which is multiplying \(x\)) by dividing both sides by 3:
\(\frac{y + 2}{3} = x\)
Did you know?
Rearranging formulae is exactly the same as solving equations, except you are moving letters instead of just numbers!
Key Takeaway: Work backwards. Start with the thing furthest away from your target letter and move it to the other side using the opposite sign.
4. Standard Formulae You Need to Know
The GCSE syllabus expects you to recognize and use some "famous" formulae. You don't always have to create these; you just need to know how to use them.
Geometry Formulae:
- Circumference of a Circle: \(C = \pi d\) or \(C = 2\pi r\)
- Area of a Circle: \(A = \pi r^2\)
- Pythagoras’ Theorem: \(a^2 + b^2 = c^2\)
Kinematics (Motion) Formulae:
These tell us how things move. You will often see these in exams:
- \(v = u + at\) (Velocity)
- \(s = ut + \frac{1}{2}at^2\) (Displacement/Distance)
- \(v^2 = u^2 + 2as\)
Memory Aid: In these formulae, \(u\) is the starting speed and \(v\) is the final speed. Think: 'u' comes before 'v' in the alphabet, just like the start comes before the end!
Key Takeaway: You don't need to be a scientist to use these. Treat them like any other formula: identify your numbers and substitute them in.
5. Common Mistakes to Avoid
Don't worry if this seems tricky at first; even top mathematicians make these mistakes sometimes!
- Mixing up \(2r\) and \(r^2\): In the area of a circle formula (\(\pi r^2\)), remember to square the radius before multiplying by \(\pi\).
- Incorrect Order (BIDMAS): Always do brackets and powers before you add or subtract.
- Negative Numbers: When substituting a negative number into something like \(x^2\), remember that \((-3) \times (-3) = 9\). (A negative times a negative is a positive!)
- Forgetting the "Whole Side": When dividing to rearrange a formula, make sure you divide the entire other side, not just one part of it.
Quick Review Box
Formulating: Turning words into letters and numbers.
Substitution: Swapping letters for numbers.
Subject: The single letter on its own (usually on the left).
Inverse: The "opposite" math operation.
BIDMAS: Brackets, Indices, Division/Multiplication, Addition/Subtraction.