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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.
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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
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Expressions and Formulae
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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
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FAQs on Expressions and Formulae - Year 9 Mathematics (Cambridge)
| 1. What are algebraic expressions? |  |
| 2. How do you simplify algebraic expressions? | |
Ans. To simplify algebraic expressions, you combine like terms by adding or subtracting them, and you can also use the distributive property to factor out common terms.
| 3. What is the difference between an algebraic expression and an algebraic formula? | |
Ans. An algebraic expression is a mathematical phrase that may contain variables and constants, while an algebraic formula is a specific equation that relates variables to each other through mathematical operations.
| 4. How are algebraic expressions used in real-life situations? | |
Ans. Algebraic expressions are used in various real-life situations such as calculating costs, determining distances, and analyzing patterns in data.
| 5. Can you give an example of an algebraic formula used in a practical scenario? | |
Ans. An example of an algebraic formula used in a practical scenario is the formula for calculating the total cost of a purchase, which includes variables for price, quantity, and tax.
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Apr 05, 2025
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Document Description: Expressions and Formulae for Year 9 2025 is part of Year 9 Mathematics (Cambridge) preparation.
The notes and questions for Expressions and Formulae have been prepared according to the Year 9 exam syllabus. Information about Expressions and Formulae covers topics
like Introduction to Algebraic Expressions, Simplifying Expressions, Formulas and Perimeters, Solving Equations, Factoring Expressions, Distributive Property, Expressing Verbal Phrases Algebraically and Expressions and Formulae Example, for Year 9 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Expressions and Formulae.
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