Using algebra (AQA GCSE Maths): Revision Notes
Using algebra
What is using algebra?
Using algebra means applying mathematical symbols and letters to solve real-world problems. When you encounter a question with unknown values, you can choose a letter to represent what you're trying to find, then build equations to solve the problem systematically.
This approach is particularly useful when guessing would take too long or when you need to show your working clearly.
Writing equations from given information
The key to using algebra effectively is translating word problems into mathematical expressions. You start by following a systematic approach that breaks down complex problems into manageable steps.
Problem-Solving Strategy:
- Choosing a variable - Pick a letter (usually A, x, or n) to represent the unknown quantity
- Building expressions - Use the given information to write mathematical statements
- Setting up an equation - Combine your expressions using the total or relationship given
- Solving systematically - Work through the algebra step by step
Worked example: Number cards problem
Worked Example: Three Number Cards Problem
Let's examine how this works with a practical problem involving three number cards A, B, and C.
Given information:
- Each card contains a whole number
- Card B's number is twice card A's number
- Card C's number is five more than card A's number
- The sum of all three numbers is 37
Solution approach:
- Let represent the number on card A
- Write expressions for the other cards:
- (twice card A)
- (five more than card A)
- Set up the equation:
- Substitute the expressions:
- Simplify:
- Solve: , so
Final answer: , ,
Always check your answer: ✓
Testing algebraic statements
Sometimes you need to determine whether an algebraic statement is true or false. The most effective method is to find a counter-example - just one example that proves the statement wrong.
Key concept: Counter-examples
To disprove a statement, you only need one counter-example. If you can find even a single case where the statement doesn't work, then the entire statement is false.
Worked example: Prime number claim
Worked Example: Testing a Prime Number Statement
Statement to test: " is always a prime number for any whole number n"
Method: Try different values of n until you find one that makes the expression composite (not prime).
Testing values:
- : (prime)
- : (prime)
- : (not prime!)
Conclusion: The statement is false because when , the result (21) is not prime.
Problem-solving tips
When working with algebra, these strategies will help you approach problems more effectively:
- Start with expressions - Before jumping into calculations, write out what each part represents algebraically
- Work systematically - Follow the order of operations and show each step clearly
- Check your solutions - Substitute your answers back into the original problem
- Look for patterns - In counter-example questions, try small whole numbers first
- Remember geometry - Algebra applies to shapes too, especially with perimeter and area problems
Key Points to Remember:
- Choose a letter to represent unknown quantities in word problems
- Build expressions step by step using the given information
- Set up equations by combining expressions with totals or relationships
- One counter-example is sufficient to prove an algebraic statement false
- Always check your final answers by substituting back into the original problem