Using algebra (Edexcel GCSE Maths): Revision Notes
Using algebra
Setting up equations from word problems
When solving word problems, you can use algebra to create equations that represent the given information. This approach helps you find unknown values systematically by transforming word descriptions into mathematical relationships.
Key steps for setting up equations:
- Choose a letter (variable) to represent an unknown quantity
- Write expressions for other unknown quantities using your chosen letter
- Set up an equation using the total or relationship given in the problem
Solving algebraic word problems
The systematic approach to solving algebraic word problems involves defining variables, expressing relationships, and solving step-by-step. This method ensures you don't miss important information and can verify your solution.
Worked Example: Number Relationships
Problem: Three cards A, B, and C contain whole numbers. Card B has twice the number on card A, and card C has five more than card A. The sum of all three numbers is 37.
Solution method:
- Define your variable: Let represent the number on card A
- Express other unknowns:
- (twice the number on A)
- (five more than A)
- Set up the equation:
- Substitute and solve:
- Find all values: , ,
- Check your answer: ✓
Testing algebraic statements with substitution
Sometimes you need to check whether an algebraic statement is always true or find examples that prove it false. Substitution means replacing variables with actual numbers to test whether a statement works.
To prove a statement is false:
- Try different values for the variable
- Look for a counter-example - a value that makes the statement incorrect
- You only need one counter-example to prove a statement is false
Finding counter-examples
When testing whether algebraic statements are always true, systematic substitution of different values can reveal when statements fail. A single counter-example is sufficient to disprove a general claim.
Worked Example: Testing an Algebraic Statement
Claim: Someone claims that is always a prime number for any whole number .
Testing different values:
- When : (prime ✓)
- When : (prime ✓)
- When : (not prime ✗)
Since 21 is not prime, provides a counter-example that proves the statement is false.
Working with perimeters algebraically
Algebra is particularly useful for solving problems involving shapes where some measurements are unknown. By expressing dimensions algebraically, you can set up equations to find missing values.
Key facts about rectangles:
- Perimeter =
- If you know expressions for length and width, substitute them into the perimeter formula
- When rectangles have the same perimeter, set their perimeter expressions equal to each other and solve
Problem-solving tip: Always define your variables clearly at the start of geometry problems. This prevents confusion when working with multiple unknown measurements.
Checking your work:
Always check your solutions by substituting your answers back into the original problem:
- Do the numbers satisfy all the given conditions?
- Does your arithmetic work out correctly?
- Have you answered the question that was actually asked?
This verification step catches calculation errors and ensures your solution makes sense in the context of the problem.
Key Points to Remember:
- Choose appropriate variables to represent unknown quantities in word problems
- Set up equations systematically using the relationships given in the problem
- Substitute values to test whether algebraic statements are true or false
- Find counter-examples to prove statements are false - you only need one!
- Always check your answers by substituting back into the original problem