Algebraic expressions are like little math puzzles where numbers and letters work together.
Imagine the letter "x" is like a mystery box, and you have to figure out what's inside. For example, in "2x + 3," the "x" can be any number, and the expression tells you what happens when you use different numbers for "x."
Think of it like this: if you want to buy some toys and each toy costs the same amount, the expression helps you figure out how much money you'll need. Algebraic expressions make solving problems fun and help you understand math in a new way!
Let's say you want to save money to buy a new bike that costs Rs 200. You decide to save a certain amount of money each week. For this, you can use an algebraic expression to figure out how many weeks it will take to save enough money.
If you save Rs10 every week, you can use the expression , where is the number of weeks.
By solving for , you can determine that it will take 20 weeks to save Rs 200.
How are Expressions Formed?
Expressions that contain only constants are called numeric or arithmetic expressions. Expressions that contain constants and variables, or just variables, are called algebraic expressions.
Variable: A variable does not have a fixed value. It can be varied. It is represented by letters like a, y, p m etc.
Constant: A constant has a fixed value. Any number without a variable is a constant. Examples of constants are 4, 100, etc
Terms of an Expression
In math, an algebraic expression consists of numbers, variables, and basic operations like addition, subtraction, multiplication, division, and exponentiation with rational exponents.
Terms : Parts of an expression that are formed separately and then added (or subtracted) are called terms.
Factors of a Term
Terms are often products of factors. For example, in the expression (4x + 5), the term 4x is a product of 4 and x.Algebraic Expression Tree Diagram
Examples:
For the expression (4x + 5):
Term 1: 4x (Factors: 4 and x)
Term 2: 5(Factor: 5)
The expression is formed by adding these terms.
For the expression (4x2 - 3xy):
Term 1: 4x2 (Factors: 4, x, and x)
Term 2: -3xy (Factors: -3, x, and y, where the minus sign is included in the term)
The expression is formed by subtracting these terms.
Addition of Terms: Terms are added to form expressions. For example, in (4x2- 3xy), 4x2 + (-3xy) = 4x2 - 3xy.
The minus sign is included in the term to avoid confusion. In (4x2 - 3xy), the term is (-3xy), not (3xy).
Question for Chapter Notes: Algebraic Expressions
Try yourself:How many terms are there in the expression 7x3+2xy+z−7y?
Explanation
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Coefficients
In a term, the numerical factor is referred to as the numerical coefficient. They indicate the scale or magnitude of the associated variables and help in simplifying expressions and solving equations efficiently.
Example: In the term 2x + 4y - 9, the coefficient of x is 2, and the coefficient of y is 4.
(i) When the Coefficient of a term is 1:
When the coefficient of a term is 1, it is usually omitted for simplicity.
Example: 1x is written as x; 1x2y2 is written as x2y2.
(ii) When the Coefficient of a term is -1:
The coefficient of -1 is indicated only by the minus sign.
Example: (-1)x is written as -x; (-1)x2y2 is written as -x2y2.
(iii) General Use of Coefficient:
The term 'coefficient' can be used in a more general sense. In this broader context, a coefficient can be either be a numerical factor or, an algebraic factor, or a product of two or more factors.
Example: In the term 5xy, 5 is the coefficient of xy, x is the coefficient of 5y, and y is the coefficient of 5x. Hence, coefficient is the factor by which the rest of the term is multiplied.
Similarly, in the term 10xy2, 10 is the coefficient of xy2, x is the coefficient of 10y2, and y2 is the coefficient of 10x.
(iv) Simplifying Expressions with Coefficients: Like terms are combined by addition or subtraction
When simplifying expressions, you can combine like terms by adding or subtracting their coefficients.
Example: To simplify 3x + 2x, you combine the coefficients of x (3 and 2) to get 5x.
Question for Chapter Notes: Algebraic Expressions
Try yourself:What is the coefficient of the term xyz in the expression 5xyz - 2xy + 3x?
Explanation
- The coefficient of a term is the numerical factor that appears in front of the variables.
- In the given expression, the term xyz has a coefficient of 5.
- Therefore, the correct answer is Option B: 5.
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Example: Identify the terms and their factors in the following expressions: Sol: (a) 5xy² + 7x²y Terms are 5xy² and 7x²y Factors are 5, x, y, y and 7, x, x, y.
(b) -ab + 2b² – 3a² Terms are -ab, 2b², -3a² Factors are -1, a, b, 2, b, b, -3, a, a.
(c) 1.2 ab – 2.4 b + 3.6 a Terms are 1.2 ab, -2.4 b, 3.6 a Factors are 1.2, a, b, -2.4, b, 3.6, a.
(d) 3/4 x + 1/4 Terms are 3/4 x and 1/4 Factors are 3/4, x and 1/4.
Example: Identify the numerical coefficients of terms (other than constants) in the following expressions: Sol: (a) 1.2a + 0.8b Coefficients are 1.2 for 1.2a, 0.8 for 0.8b.
(b) 3.14 r² Coefficient of 3.14 r² is 3.14.
(c) 2 (l + b) Coefficients are 2 for 2l and 2 for 2b.
Like and Unlike Terms
Terms that have the same algebraic factors are called like terms.
Terms that have different algebraic factors are called unlike terms.
Example: State whether a given pair of terms is of like or unlike terms.
Sol: (i) –7x, 5/2 x Like terms (both have the variable x)
(ii) 4m²p, 4mp² Unlike terms (different variable powers)
Question for Chapter Notes: Algebraic Expressions
Try yourself:Identify the factors of the terms in the following expression: 3xyz + 5xy - 2y
Explanation
- Terms in the expression are 3xyz, 5xy, and -2y.
- The factors of 3xyz are 3, x, y and z.
- The factors of 5xy are 5, x, and y.
- The factors of -2y are -2 and y.
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Monomials, Binomials, Trinomials and Polynomials
Algebraic expressions that contain only one term are called monomials.
Algebraic expressions that contain only two unlike terms are called binomials.
Algebraic expressions that contain only three unlike terms are called trinomials.
Algebraic expressions that have one or more terms are called polynomials. Therefore, binomials and trinomials are also polynomials.
Example: Classify into monomials, binomials, and trinomials. Sol: (i) 1 + x + x²: Trinomial Expression is Trinomial as it contains 3 terms: 1, x, and x²
(ii) a² + b²: Binomial Expression is Binomial as it contains 2 terms: a² and b²
(iii) 7mn: Monomial Expression is Monomial as it contains 1 term: 7mn
(iv) z² – 3z + 8: Trinomial Expression is Trinomial as it contains 3 terms: z², -3z and 8
Question for Chapter Notes: Algebraic Expressions
Try yourself:Which of the following is an example of a binomial?
Explanation
- A binomial is an algebraic expression that contains only two unlike terms.
- Option A is a trinomial since it has three unlike terms (3x2, 5x, and 2).
- Option B is also a trinomial since it has three unlike terms (4x3, -6x2, and 2x).
- Option C is a polynomial, but not a binomial or trinomial, as it contains five unlike terms.
- Option D is a binomial as it contains only two unlike terms (2x2 and -4x).
- Therefore, the correct answer is Option D.
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Do you know we can do operations on Algebraic Expressions:
To add similar terms in an algebraic expression, multiply the total of their numbers with their shared algebraic parts.
To take away similar terms in an algebraic expression, multiply the difference of their numbers with their shared algebraic parts.
For example,
Subtract 9ab from 15ab
15ab - 9ab = 15 x ab - 9 x ab
= (15 - 9) x ab
= 6ab
Add 4x + 12 and 8x - 9.
We know that the terms 4x and 8x are like terms and 12 and -9 are items.
Therefore,
= (4 + 8) x + 3
= 12x + 3
Thus, to add or subtract algebraic expressions, rearrange the terms in the sum of the given algebraic expressions, so that their like terms and constants are grouped. While rearranging the terms, move them with the correct '+' or '-'sign before them.
Unlike terms remain unchanged in the sum or difference of algebraic expressions.
Finding the Value of an Expression
Application of Algebraic Expressions Algebraic expressions can be used to represent number patterns. In the following example, we can find the relation between several cones and the number of ice cream scoops.
Number of cones (n)
Number of ice - cream scoops (2n)
1
2
2
4
3
6
4
8
15
2 x 15
Thus, we can find the value of an algebraic expression if the values of all the variables in the expression are known.
Similarly, we can write formulas for the perimeter and area for different geometrical figures using simple, easy-to-remember algebraic expressions.
Area and perimeter of different shapes
• If s represents the side of a square, then : =>The perimeter is 4s =>The area is s2
• If l represents the length and b represents the breadth of a rectangle, then : =>The perimeter is 2(l + b) =>The area is l x b
Example 1: If p = – 2, find the value of:
(i) 4p + 7
Sol: Putting p = -2, we get
4(-2) + 7 = -8 + 7 = – 1
(ii) -3p² + 4p + 7
Sol: Putting p = -2, we get -3(-2)² + 4(-2) + 7 = -3 × 4 – 8 + 7 = -12 – 8+ 7 = -13
FAQs on Algebraic Expressions Class 7 Notes Maths Chapter 10
1. How are algebraic expressions formed?
Ans. Algebraic expressions are formed by combining constants, variables, and mathematical operations such as addition, subtraction, multiplication, and division.
2. What are the terms of an expression?
Ans. The terms of an expression are the individual components separated by the plus or minus signs. For example, in the expression 3x + 2y - 5, the terms are 3x, 2y, and -5.
3. What are like and unlike terms in algebraic expressions?
Ans. Like terms in algebraic expressions are terms that have the same variables raised to the same powers. Unlike terms are terms that have different variables or different powers of the same variable.
4. What are monomials, binomials, trinomials, and polynomials in algebraic expressions?
Ans.
- Monomials are algebraic expressions with only one term, such as 3x or -5y.
- Binomials are algebraic expressions with two terms, such as 2x + 3y.
- Trinomials are algebraic expressions with three terms, such as 4x + 2y - 5z.
- Polynomials are algebraic expressions with any number of terms, such as x^2 + 3x - 5.
5. How do you find the value of an algebraic expression?
Ans. To find the value of an algebraic expression, substitute the given values of the variables into the expression and then simplify using the order of operations (PEMDAS - Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
