Exercise 12.1 Algebraic-Expressions -NCERT Solutions Class 7

Question 1

Get the algebraic expressions in the following cases using variables, constants and arithmetic operations.

(i) Subtraction of \(z\) from \(y.\)

(ii) One-half of the sum of numbers \(x\) and \(y.\)

(iii) The number \(z\) multiplied by itself.

(iv) One-fourth of the product of numbers \(p\) and \(q.\)

(v) Numbers \(x\) and \(y\) both squared and added.

(vi) Number \(5\) added to three times the product of numbers \(m\) and \(n.\)

(vii) Product of numbers \(y\) and \(z\) subtracted from \(10.\)

(viii) Sum of numbers \(a\) and \(b\) subtracted from their product.

Reasoning:

Let us first understand the meaning or definition of terms variable, constants and arithmetic operations

Variables are the letters used in an algebraic expression that can take any value. For e.g. \(a, b, c\) or \(z\) etc. and it can take any value which can be either \(2\) or \(5\) or any other number. Constants always have fixed values in the algebraic expressions. They cannot be assumed or changed. Arithmetic Operations are Addition, subtraction, multiplication and division.

Steps:

(i) Subtraction of \(z\) from \(y.\)

\[y - z\]

(ii) One-half of the sum of numbers \(x\) and \(y.\)

\[\frac{1}{2}\left( {x + y} \right)\]

(iii) The number \(z\) multiplied by itself.

\[z \times z = {z^2}\]

(iv) One-fourth of the product of numbers \(p\) and \(q.\)

\[\frac{1}{4}pq\]

(v) Numbers \(x\) and \(y\) both squared and added.

\[\left( {x \times x} \right) + \left( {y \times y} \right) = {x^2} + {y^2}\]

(vi) Number \(5\) added to three times the product of numbers \(m\) and \(n.\)

\[5 + 3\left( {m \times n} \right) = 5 + 3mn\]

(vii) Product of numbers \(y\) and \(z\) subtracted from \(10.\)

\[10 - \left( {y \times z} \right) = 10 - yz\]

(viii) Sum of numbers \(a\) and \(b\) subtracted from their product.

\[\left( {a \times b} \right)-\left( {a + b} \right) = ab - \left( {a + b} \right)\]

Question 2

(i) Identify the terms and their factors in the following expressions. Show the terms and factors by tree diagrams.

(a) \(x – 3\)

(b) \(1 + x + x^2\)

(c) \(y – y^3\)

(d) \(5xy^2 + 7x^2y\)

(e) \(– ab + 2b^2 – 3a^2\)

(ii) Identify terms and factors in the expressions given below:

(f) \(–4x + 5\) (b) \(–4x + 5y\) (c) \(5y + 3y^2\)

(g) \(xy + 2x^2y^2\) (e) \(pq + q\) (f) \(1.2 ab – 2.4 b + 3.6 a\)

(h) \(\frac{3}{4}x + \frac{1}{4}\) (h) \(0.1 p^2 + 0.2 q^2\)

(i) Identify the terms and their factors in the following expressions. Show the terms and factors by tree diagrams.

(a) \(x – 3\)

Steps:

Term \(= x\) and Factor \(= 1\)

(b) \(1 + x + x^2\)

Steps:

Term \(= x\) and Factor \(= 1;\) Term \(= x^2\) and Factor \(= 1\)

(c) \(y – y^3\)

Steps:

Term \(= y\) and Factor \(= 1;\) Term \(= y^3\) and Factor \(= –1\) 

(d) \(5xy^2 + 7x^2y\)

Steps:

Term \(= xy^2\) and Factor \(= 5;\) Term \(= x^2y\) and Factor \(= 7\)

(e) \(– ab + 2b^2 – 3a^2\)

Steps:

Term \(= ab\) and Factor \(= –1; \)Term \(= b^2\) and Factor \(= 2;\) Term \(a^2\) and Factor \(= –3\)

(ii) Identify terms and factors in the expressions given below:

(f) \(–4x + 5\) (b) \(–4x + 5y\) (c) \(5y + 3y^2\)

(g) \(xy + 2x^2y^2\) (e) \(pq + q\) (f) \(1.2 ab – 2.4 b + 3.6 a\)

(h) \(\frac{3}{4}x + \frac{1}{4}\) (h) \(0.1 p^2 + 0.2 q^2\)

S.No.	Expression	Term	Factors
a)	\(–4x + 5\)	\(-4x\) and \(5\)	\(-4, x\)and \(5\)
b)	\(-4x + 5y\)	\(-4x\) and \(5y\)	\(-4, x\) and \(5, y\)
c)	\(5y + 3y^2\)	\(5y\) and \(3y^2\)	\(5, y \) and \(3, y, y\)
d)	\(xy + 2x^2y^2\)	\(xy\) and \(2x^2y^2\)	\(x, y\) and \(2, x, x,y, y\)
e)	\(pq + q\)	\(pq\) and \(q\)	\(p, q\) and \(q\)
f)	\(1.2ab - 2.4b + 3.6a\)	\(1.2ab, -2.4b\) and \(3.6a\)	\(1.2, a, b, -2.4, b\) and \(3.6, a\)
g)	\(\frac{{3}}{4} x + \frac{{1}}{4} \)	\(\frac{{3}}{4} x \) and \( \frac{{1}}{4}\)	\(\frac{{3}}{4}, x\) and \(\frac{{1}}{4} \)
h)	\(0.1 p^2 + 0.2 q^2\)	\(0.1p^2\) and \(0.2q^2\)	\(0.1, p, p\) and \(0.2, q, q\)

Question 3

Identify the numerical coefficients of terms (other than constants) in the following expressions:

(i) \(5 – 3t^2\)       

ii) \(1 + t + t^2 + t^3\)         

(iii) \(x + 2xy + 3y\)

(iv) \(100m + 1000n\)     

(v) \(– p^2q^2 + 7pq\)   

(vi) \(1.2a + 0.8b\)

(vii) \(3.14r^2\)         

(viii) \(2(l + b)\)                 

(ix) \(0.1y + 0.01y^2\)

 

 

S.No.	Expression	Term	Numerical Coefficient
(i)	\(5 - 3t^2\)	\(-3t^2\)	\(-3\)
(ii)	\(1 + t + t^2 + t^3\)	\(t, t^2 \) and \( t^3\)	\(1, 1 \) and \( 1\)
(iii)	\(x + 2xy + 3y\)	\(x , 2xy \) and \( 3y\)	\(1, 2 \) and \( 3\)
(iv)	\(100m + 1000n\)	\(100m \) and \( 1000n\)	\(100 \) and \( 1000\)
(v)	\(-p^2q^2 + 7pq\)	\(-p^2q^2 \) and \( 7pq\)	\(-1 \) and \( 7\)
(vi)	\(1.2 a + 0.8 b\)	\(1.2a \) and \( 0.8b\)	\(1.2 \) and \( 0.8\)
(vii)	\(3.14r^2\)	\(3.14r^2\)	\(3.14\)
(viii)	\(2(l + b)\)	\(2l \) and \( 2b\)	\(2 \) and \( 2\)
(ix)	\(0.1y + 0.01y^2\)	\(0.1y \) and \( 0.01 y^2\)	\(0.1 \) and \( 0.01\)

Question 4

(a) Identify terms which contain \(x\) and give the coefficient of \(x.\)

(i) \(y^2x + y\)

(ii) \(13y^2 – 8yx\)

(iii) \(x + y + 2\)

(iv) \(5 + z + zx\)

(v) \(1 + x + xy\)

(vi) \(12xy^2 + 25\)

(vii) \(7x + xy^2\)

(b) Identify terms which contain \(y^2\) and give the coefficient of \(y^2.\)

(i) \(8 – xy^2\)

(ii) \(5y^2 + 7x\)

(iii) \(2x^2y – 15xy^2\) + \(7y^2\)

(a) Identify terms which contain \(x\) and give the coefficient of \(x.\)

(i) \(y^2x + y\)

(ii) \(13y^2 – 8yx\)

(iii) \(x + y + 2\)

(iv) \(5 + z + zx\)

(v) \(1 + x + xy\)

(vi) \(12xy^2 + 25\)

(vii) \(7x + xy^2\)

S.No.	Expression	Term containing x	Coefficient of x
(i)	\(y^2x + y\)	\(y^2x\)	\(y^2\)
(ii)	\(13y^2 – 8yx\)	\(–8yx\)	\(–8y\)
(iii)	\(x + y + 2\)	\(x\)	\(1\)
(iv)	\(5 + z + zx\)	\(zx\)	\(z\)
(v)	\(1 + x + xy\)	\(x\) and \(xy\)	\(1\) and \(y\)
(vi)	\(12xy^2 + 25\)	\(12xy^2\)	\(12y^2\)
(vii)	\(7 x + xy^2\)	\(7 x\) and \(xy^2\)	\(7\) and \(y^2\)

 

(b) Identify terms which contain \(y^2\) and give the coefficient of \(y^2.\)

(i) \(8 – xy^2\)

(ii) \(5y^2 + 7x\)

(iii) \(2x^2y – 15xy^2\) + \(7y^2\)

S.No.	Expression	Term containing y2	Coefficient of y2
(i)	\(8 – xy^2\)	\(– xy^2\)	\(– x\)
(ii)	\(5y^2 + 7x\)	\(5y^2\)	\(5\)
(iii)	\(2x^2 y – 15xy^2 + 7y^2\)	\(– 15xy^2\) and \(7y^2\)	\(– 15x\) and \(7\)

Question 5

Classify into monomials, binomials and trinomials.

(i) \(4y – 7z\)

(ii) \(y^2\)

(iii) \(x + y – xy\)

(iv) \(100\)

(v) \(ab – a – b\)

(vi) \(5 – 3t\)

(vii) \(4p^2q – 4pq^2\)

(viii) \(7mn\)

(ix) \(z^2 – 3z + 8\)

(x) \(a^2 + b^2\)

(xi) \(z^2 + z\)

(xii) \(1 + x + x^2\)

Steps:

Monomial means expression having single term.

Binomials means expression having two terms.

Trinomials means expression having three terms.

S No.	Expression	No. of terms	Classification
(i)	\(4y – 7z\)	\(2\)	Binomial
(ii)	\(y^2\)	\(1\)	Monomial
(iii)	\(x + y – xy\)	\(3\)	Trinomial
(iv)	\(100\)	\(1\)	Monomial
(v)	\(ab – a – b\)	\(3\)	Trinomial
(vi)	\(5 – 3t\)	\(2\)	Binomial
(vii)	\(4p^2q – 4pq^ 2\)	\(2\)	Binomial
(viii)	\(7mn\)	\(1\)	Monomial
(ix)	\(z^2 – 3z + 8\)	\(3\)	Trinomial
(x)	\(a^2 + b^2\)	\(2\)	Binomial
(xi)	\(z^2 + z\)	\(2\)	Binomial
(xii)	\(1 + x + x^2\)	\(3\)	Trinomial

Question 6

State whether a given pair of terms is of like or unlike terms.

(i) \(1, 100\)

(ii) \(–7x, x\)

(iii) \(– 29x, – 29y\)

(iv) \(14xy, 42yx\)

(v) \(4m^2p, 4mp^2\)

(vi) \(12xz, 12x^2z^2\)

 

S.No.	Expression	Terms	Factors	Like/ Unlike	Reason
(i)	\(1, 100\)	\(1\) and \(100\)	\(1\) and \(100\)	Like	Bothe the terms has no variables
(ii)	\(–7x,  x\)	\(–7x\) and \( x\)	\(–7, x\) and \(x\)	Like	Both terms have same variable \(x\)
(iii)	\(– 29x, – 29y\)	\(– 29x\) and \( – 29y\)	\(– 29, x\) and \(– 29, y\)	Unlike	Both terms have different variables \(x\) & \(y\)
(iv)	\(14xy, 42yx\)	\(14xy\) and \(42yx\)	\(14, x,y\) and \(42, y,x\)	Like	Both terms have same variable \(xy\) & \(xy\)
(v)	\(4m^2p, 4mp^2\)	\(4m^2p\) and \(4mp^2\)	\(4, m^2, p\) and \(4, m, p^2\)	Unlike	Both terms have same variable but with different powers
(vi)	\(12xz, 12x^2z^2\)	\(12xz\) and \(12x^2z^2\)	\(12, x ,z\) and \(12,x^2 ,z^2 \)	Unlike	Both terms have same variable but with different powers

Question 7

Identify like terms in the following:

(a)

\(\begin{align}&–xy^2, –4yx^2, 8x^2, 2xy^2, 7y, –11x^2, \\&–100x, –11yx, 20x^2y, –6x^2, \\&y, 2xy, 3x\end{align}\)  

(b)

\(\begin{align}&10pq, 7p, 8q, –p^2q^2, –7qp, –100q,\\& –23, 12q^2p^2,–5p^2, 41,2405p, \\& 78qp, 13p^2q, qp^2, 701p^2 \end{align}\)

Reasoning:

This question is based on the concept of like terms. If there are same variable in all
the terms in the expression, then the expression has like terms. We have to ignore constants here.

Steps:

S.No.	Terms	Like terms
(i)	\(–xy^2, –4yx^2, 8x^2, 2xy^2, 7y, –11x^2, \\–100x, – 11yx, 20x^2y, –6x^2, y, 2xy, 3x\)	\(–xy^2, 2xy^2;\\ –4yx^2, 20x^2y;\\ 8x^2, –11x^2, –6x^2;\\ 7y, y;\\ –100x, 3 x;\\ –11yx, 2xy\\\)
(ii)	\(10pq, 7p, 8q, –p^2q^2, –7qp, –100q, –23, 12q^2p^2,\\ –5p^2,41, 2405p, 78qp, 13p^2q, qp^2, 701p^2\)	\(10pq, –7qp, 78qp;\\ 8q, –100q;\\ -5p^2, 701p^2;\\ 7p, 2405p;\\ –p^2q^2, 12q^2p^2;\\ -23, 41;\\ 13p^2q, qp^2\\\)