Chapter Notes: Algebraic Expressions

Algebraic Expressions Class 7 Notes Maths Chapter 10

 Table of contents How are Expressions Formed? Terms of an Expression Like and Unlike Terms Monomials, Binomials, Trinomials and Polynomials Finding the Value of an Expression

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 $10𝑥=200$, where $𝑥$ is the number of weeks.

By solving for $𝑥$, 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.

We use letters x, y, l, m, etc to denote variables. A variable can take various values. On the other hand constant has a fixed value. 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:

1. 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.
2. 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?

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?

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

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?

Operations on Algebraic Expressions

We can perform arithmetic operations on algebraic expressions. To add the like terms in an algebraic expression, multiply the sum of their coefficients with their common algebraic factors. To subtract the like terms in an algebraic expression, multiply the difference of their coefficients with their common algebraic factors. For example, let us subtract 9ab from 15ab

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.
Let us see this example.
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

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

(iii) -2p³ – 3p² + 4p + 7

Sol: Putting p = -2, we get
– 2(-2)³ – 3(-2)² + 4(-2) + 7
= -2 × (-8) – 3 × 4 – 8 + 7
= 16 – 12 – 8 + 7 = 3

Question for Chapter Notes: Algebraic Expressions
Try yourself:If the value of the expression x2 – 5x + k for x = 0 is 5, then the value of k is

Example 2: When a = 0, b = -1, find the value of the given expressions:

(i) 2a + 2b
Sol: = 2(0) + 2(-1)
= 0 – 2 = -2 which is required.

(ii) 2a² + b² + 1

Sol: = 2(0)² + (-1)² + 1 =0 + 1 + 1 = 2 which is required.

(iii) 2a²b + 2ab² + ab

Sol: = 2(0)² (-1) + 2(0)(-1)² + (0)(-1)
=0 + 0 + 0 = 0 which is required.

(iv) a² + ab + 2

Sol: = (0)² + (0)(-1) + 2
= 0 + 0 + 2 = 0 which is required.

Example 3: What should be the value of a if the value of 2x² + x – a equals to 5, when x = 0?

Sol: 2x² + x – a = 5
Putting x = 0, we get
2(0)² + (0) – a = 5
0 + 0 – a = 5
-a = 5
⇒ a = -5 which is required value.

Example 4: Simplify the following:(i) If z = 10, find the value of z² – 3(z – 10).(ii) If p = -10, find the value of p² -2p – 100.(iii)  If a = 5 and b = -3, find the value of 2(a² + ab) + 3 – ab.

Sol:(i) z² – 3(z – 10)
= z² – 3z + 30
Putting z = 10, we get
= (10)2 – 3(10) + 30
= 100 – 30 + 30 = 100 which is required.

(ii) p² – 2p – 100
Putting p = -10, we get
(-10)2 – 2(-10) – 100
= 100 + 20 – 100 = 20 which is required.

(iii) 2(a² + ab) + 3 – ab = 2a2 + 2ab + 3 – ab
= 2a² + 2ab – ab + 3
= 2a2 + ab + 3
Putting, a = 5 and b = -3, we get
= 2(5)² + (5)(-3) + 3
= 2 × 25 – 15 + 3
= 50 – 15 + 3
= 53 – 15 = 38
Hence, the required value = 38.

The document Algebraic Expressions Class 7 Notes Maths Chapter 10 is a part of the Class 7 Course Mathematics (Maths) Class 7.
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FAQs on Algebraic Expressions Class 7 Notes Maths Chapter 10

 1. What are the terms of an expression?
Ans. Terms of an expression are the individual parts that make up the expression, separated by addition or subtraction symbols. For example, in the expression 3x + 5y - 2, the terms are 3x, 5y, and -2.
 2. What is the difference between like and unlike terms in an expression?
Ans. Like terms in an expression have the same variables raised to the same powers, whereas unlike terms have different variables or different powers. For example, in the expression 2x + 3x - 5y, 2x and 3x are like terms, while 5y is an unlike term.
 3. What are monomials, binomials, trinomials, and polynomials?
Ans. Monomials are expressions with only one term, binomials have two terms, trinomials have three terms, and polynomials have more than three terms. For example, 4x, 2x + 3, 5x^2 - 2x + 1, and 3x^3 + 2x^2 - x + 5 are examples of monomials, binomials, trinomials, and polynomials, respectively.
 4. How can you find the value of an expression?
Ans. To find the value of an expression, substitute the given values for the variables in the expression and then simplify the expression using the order of operations (PEMDAS - Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
 5. How are expressions formed in algebra?
Ans. Expressions in algebra are formed by combining constants, variables, and mathematical operations such as addition, subtraction, multiplication, and division. By using these elements, algebraic expressions are created to represent mathematical relationships and situations.

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