10.Algebraic Expressions
Introduction
Expressions that contain only constants are called numeric or arithmetic expressions.
Expressions that contain only constants are called numeric or arithmetic expressions.
Expressions that contain constants and variables, or just variables, are called algebraic expressions.
While writing algebraic expressions, we do not write the sign of multiplication. An algebraic expression
containing only variables also has the constant 1 associated with it. The parts of an algebraic expression
joined together by plus (+) signs are called its terms.
A term that contains variables is called a variable term.
A term that contains only a number is called a constant term. The constants and the variables whose
product makes a term of an algebraic expression, are called the factors of the term. The factors of a
constant term in an algebraic expression are not considered. The numerical factor of a variable term
is called its coefficient. The variable factors of a term are called its algebraic factors.
Terms that have different algebraic factors are called unlike terms. Terms that have the same algebraic
factors are called like terms. 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. All algebraic expressions that have one or
more terms are called polynomials. Therefore, binomials and trinomials are also polynomials.
Question 1.
Find the rule which gives the number of matchsticks required to make the following patterns
(i) A pattern of letter ‘H’
(ii) A pattern of letter ‘V’
Solution:
i) A pattern of letter ‘H’
The letter ‘H’ is formed with 3 matchsticks.
∴ Its pattern should be 3, 6, 9, 12 (or)
3 × 1, 3 × 2, 3 × 3, 3 × 4 ……………….. 3 x n
∴ The required pattern is 3 × n = 3n
ii) A pattern of letter ‘V’
The letter ‘V’ is formed with 2 matchsticks.
∴ Its pattern should be 2, 4. 6. 8 (or)
2 × 1, 2 × 2, 2 × 3 …………………2 × n
∴ The required pattern is 2 × n = 2n
Question 3.
Write the following statements using variables, constants and arithmetic operations.
(i) 6 more than p
(ii) ‘x’ is reduced by 4
(iii) 8 subtracted from y
(iv) q multiplied by ‘-5’
(v) y divided by 4
(vi) One-fourth of the product of ’p’ and ‘q’
(vii) 5 added to the three times of ’z’
(viii) x multiplied by 5 and added to ‘10’
(ix) 5 subtracted from two times of ‘y’
(x) y multiplied by 10 and added to 13
Solution:
| Sentence | Algebraic expression |
| 1. 6 more than p | p + 6 |
| 2. x is reduced by 4 | x – 4 |
| 3. 8 subtracted from y | y – 8 |
| 4. q multiplied by ‘-5’ | -5q |
| 5. y divided by 4 | |
| 6. One-fourth of the product of ‘p’ and ‘q’ | |
| 7. 5 added to the three times of z | 3z + 5 |
| 8. ‘x’ multiplied by 5 and added to 10 | 5x + 10 |
| 9. 5 subtracted from two times of ‘y’ | 2y – 5 |
| 10. y multiplied by 10 and added to 13 | 10y |
Question 4.
Write the following expressions in Statements.
(i) x + 3
(ii) y – 7
(iii) 10l
(iv)
(v) 3m + 11
(vi) 2y – 5
Solution:
Expression Statements
| Expression | Statements |
| i) x + 3 | x is added to 3 |
| ii) y – 7 | 7 is subtracted from y. |
| iii) 10l | l is multiplied by 10. |
| iv) | x is divided by 5 |
| v) 3m + 11 | m is multiplied by 3 and added to 11 |
| vi) 2y – 5 | y is multiplied by 2 and 5 ¡s subtracted from the product |
Question 5.
Some situations are given below. State the number in situations is a variable or constant?
Example: Our age – its value keeps on changing so it is an example of a variable quantity.
(i) The number of days in the month of January
(ii) The temperature of a day
(iii) Length of your classroom
(iv) Height of the growing plant
Solution:
i) No. of days ¡n the month of January are fixed in every year. So, “Number of days” is a constant.
ii) The temperature of a day changes every minute. So (temperature) it ¡s a variable.
iii) The length of the classroom is fixed. So it is a constant.
iv) The height of a growing plant changes in every month. So it is a variable.
Exercise 2
Question 1.
Identify and write the like terms in each of the following groups.
(i) a2, b2, -2a2, c2, 4a
(ii) 3a, 4x, – yz, 2z
(iii) -2xy2, x2y, 5y2x, x2z
(iv) 7p, 8pq, -5pq, -2p, 3p
Solution:
i) Group of like terms : [a2, -2a2]
ii) Group of like terms: {-yz, 2zy}
iii) Group of like terms: {-2xy2, 5y2x}
iv) Group of like terms: {7p, -2p, 3p} , : {8pq,-5pq}
Question 2.
State whether the expression is a numerical expression or an algebraic expression.
(i) x + 1
(ii) 3m2
(iii) -30 + 16
(iv) 4p2 – 5q2
(v) 96
(vi) x2 – 5yz
(vii) 215x2yz
(viii) 95 ÷ 5 x 2
(ix) 2 + m + n
(x) 310 + 15 + 62
(xi) 11 a2 + 6b2 – 5
Solution:
Algebraic expression
(i) x + 1
(ii) 3m2
(iv) 4p2 – 5q2
(vi) x2 – 5yz
(vii) 215x2yz
(ix) 2 + m + n
(x) 310 + 15 + 62
(xi) 11 a2 + 6b2 – 5
Numerical expression
(iii) -30 + 16
(v) 96
(viii) 95 ÷ 5 x 2
(x) 310 + 15 + 62
Question 3.
State whether the algebraic expression given below is monomial, binomial, trinomial or multinornial.
(i) y2
(ii) 4y — 7z
(iii) 1 + x + x2
(iv) 7mn
(v) a2 + b2
(vi) 100 xyz
(vii) ax + 9
(viii) p2 – 3pq + r
(ix) 3y2 – x2y2 + 4x
(x) 7x2 – 2xy + 9y2 – 11
Solution:
| Algebraic Expression | Its name |
| (i) y2 | Monomial |
| (ii) 4y – 7z | Binomial |
| (iii) 1 + x + x2 | Trinomial |
| (iv) 7mn | Monomial |
| (v) a2 + b2 | Binomial |
| (vi) 100 xyz | Monomial |
| (vii) ax + 9 | Binomial |
| (viii) p2 – 3pq + r | Trinom ial |
| (ix) 3y2 – x2y2 + 4x | Trinomial |
Question 4.
What is the degree of each of the monomials.
(i) 7y
(ii) -xy2
(iii) xy2z2
(iv) -11y2z2
(v) 3mn
(vi) -5pq2
Solution:
| Monomial | Degree |
| (i) 7y | 1 |
| (ii) -xy2 | 1 + 2 = 3 |
| (iii) xy2z2 | 1 + 2 + 2 = 5 |
| (iv) -11y2z2 | 2 + 2 = 4 |
| (v) 3mn | 1 + 1 = 2 |
| (vi) -5pq2 | 1 + 2 = 3 |
Question 5.
Find the degree of each algebraic expression.
(i) 3x – 15
(ii) xy + yz
(iii) 2y2z + 9yz – 7z – 11x2y2
(iv) 2y2z + 10yz
(v) pq + p2q – p2q2
(vi) ax2 + bx + c
Solution:
| Algebraic Expression | Degree |
| (i) 3x – 15 | 1 |
| (ii) xy + yz | 2 |
| (iii) 2y2z + 9yz – 7z – 11x2y2 | 4 |
| (iv) 2y2z + 10yz | 3 |
| (v) pq + p2q – p2q2 | 4 |
| (vi) ax2 + bx + c | 2 |
Question 6.
Write any two Algebraic expressions with the same degree.
Solution:
Algebraic expression, with the same degree are
i) ax2 + bx + c .
ii) 4x2 – 5x – 1
Exercise 3
Question 1.
Find the length of the line segment PR in the following figure in terms of ’a’.
Solution:
The length of PR =
= 3a + 2a
= 5a units
Question 2.
(i) Find the perimeter of the following triangle.
(ii) Find the perimeter of the following rectangle.
Solution:
i) The perimeter of the triangle =
= 2x + 6x + 5x
= 13x units
ii) The perimeter of the rectangle ABCD =
or
= 2 (l + b)
= 2 (3x + 2x)
= 2 × 5x
= 10x units
Question 3.
Subtract the second terni from first term.
(i) 8x, 5x
(ii) 5p, 11p
(iii) 13m2, 2m2
Solution:
i) 8x – 5x = 3x
ii) 5p – 11p = -6p
iii) 13m2 – 2m2 = 11m2
Question 4.
Find the value of following monomials, if x =1.
(i) -x
(ii) 4x
(iii) -2x2
Solution:
1) If x = 1 ⇒ – x = – (1)= – 1
ii) If x = 1 ⇒ 4x = 4 x 1 = 4
iii) If x= 1 ⇒ – 2x2 = – 2(1)2 = – 2 × 1= – 2