IGCSE Class 9 > Class 9 Notes > Year 9 Mathematics (Cambridge) > Expressions and Formulae
Expressions and Formulae
Introduction to Algebraic Expressions
- Definition: An algebraic expression is a mathematical phrase that can include numbers, variables (letters that represent numbers), and operations (such as addition, subtraction, multiplication, and division).
- Example: The expression "three times a number increased by five" is written as 3x + 5.
Simplifying Expressions
- Combining Like Terms: To simplify an expression, combine like terms (terms with the same variable raised to the same power).
- Example: Simplify 2x + 3x - 4:
2x + 3x = 5x
So, 2x + 3x - 4 = 5x - 4.
Formulas and Perimeters
- Perimeter of a Rectangle: The perimeter (P) of a rectangle with length (l) and width (w) is given by: P = 2(l + w)
- Example: l = 5 and w = 3:
P = 2(5 + 3) = 2 × 8 = 16
Solving Equations
Steps to Solve Linear Equations:
- Isolate the variable on one side of the equation.
- Simplify both sides if necessary.
Example: Solve 3x - 7 = 14:
- Add 7 to both sides:
3x - 7 + 7 = 14 + 7
3x = 21 - Divide by 3:
x = 3/21
x = 7
Factoring Expressions
Factoring Difference of Squares: An expression of the form a2 - b2 can be factored as (a - b)(a + b).
Example:
Factor 4x2 - 9:
4x2 - 9 = (2x)2 - 32
= (2x - 3)(2x + 3)
Solving Quadratic Equations by Factoring
Steps:
- Set the equation to 0.
- Factor the quadratic expression.
- Solve for the variable by setting each factor equal to 0.
Example:
Solve x2 - 16 = 0:
Factor: x2 - 16 = (x - 4)(x + 4)
Set each factor to 0:
x - 4 = 0 or x + 4 = 0
x = 4 or x = -4
Distributive Property
- Definition: The distributive property states that a(b + c) = ab + ac.
- Example: Simplify 3(2x - 5) + 4(3x + 2):
Distribute: 3(2x) - 3(5) + 4(3x) + 4(2)
6x - 15 + 12x + 8
Combine like terms:
6x + 12x = 18x
-15 + 8 = -7
18x - 7
Expressing Verbal Phrases Algebraically
- Translating Phrases: To express "the sum of twice a number and seven" algebraically: 2x + 7
- Example: If the number is 4:
2(4) + 7 = 8 + 7 = 15
Solving Proportions
- Definition: A proportion is an equation that states that two ratios are equal.
- Example: Solve 2/x = 5
Cross-multiply:
2 = 5x
Solve for x:
x = 2/5 = 0.4
The document Expressions and Formulae is a part of the Class 9 Course Year 9 Mathematics IGCSE (Cambridge).
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FAQs on Expressions and Formulae
| 1. How do I simplify algebraic expressions with brackets and like terms? |  |
Ans. Simplifying algebraic expressions involves removing brackets using the distributive property, then combining like terms-variables with identical letters and powers. For example, 3(x + 2) + 2x becomes 3x + 6 + 2x, which simplifies to 5x + 6. Always expand brackets first, then collect like terms together to get your final simplified form.
| 2. What's the difference between an expression and a formula in IGCSE maths? | |
Ans. An expression is a mathematical phrase combining numbers, variables, and operations without an equals sign-like 2x + 5. A formula, however, shows a relationship between variables using an equals sign-like A = πr². Formulas are used to calculate specific values, whereas expressions are simplified or evaluated. Understanding this distinction helps when substituting values and solving problems accurately.
| 3. How do I substitute values into formulas correctly without making mistakes? | |
Ans. Substitution means replacing variables with given numerical values, following order of operations strictly. Write the formula, replace each variable carefully with its value in brackets, then calculate step-by-step using BODMAS rules. For instance, if v = u + at and u = 5, a = 2, t = 3, write v = 5 + (2)(3) = 11. Brackets around substituted numbers prevent operational errors.
| 4. Why do I get different answers when rearranging formulas to make a variable the subject? | |
Ans. Common mistakes occur when performing the same operation to both sides incorrectly or changing operation signs. To rearrange correctly, use inverse operations systematically-if a variable is multiplied, divide both sides; if added, subtract both sides. For example, rearranging A = bh to find h gives h = A/b. Always verify your rearranged formula by substituting values back into both versions.
| 5. How can I tell if I've expanded brackets correctly in complex expressions? | |
Ans. Check your expansion by multiplying each term inside brackets by the term outside-every term must be distributed. For (a + b)(c + d), multiply each of four terms: ac, ad, bc, bd. Then verify by reversing: factorise your answer back to brackets. Using mind maps and flashcards helps reinforce the distributive property and prevents skipping terms during expansion.
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