NCERT Solutions for Class 7 Maths Chapter 12 - Algebraic Expressions

Exercise 12.1

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

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

Ans: $y-z$

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

Ans: $\frac{x+y}{2}$

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

Ans: ${{z}^{2}}$

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

Ans: $\frac{pq}{4}$

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

Ans: ${{x}^{2}}+{{y}^{2}}$

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

Ans: $3mn+5$

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

Ans: $10-yz$

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

Ans: $ab-\left( a+b \right)$

 2.

(i) Identify the terms and their factors in the following expressions, show the term and factors by tree diagram:

(a) $x-3$ 

Ans: Terms: $x,-3$

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(b) $1+x+{{x}^{2}}$

Ans: Terms: $1,x,{{x}^{2}}$

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(c) $y-{{y}^{3}}$

Ans: Terms: $y,-{{y}^{3}}$

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(d) $5x{{y}^{2}}+7{{x}^{2}}y$

Ans: Terms: $5x{{y}^{2}},7{{x}^{2}}y$

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(e) $-ab+2{{b}^{2}}-3{{a}^{2}}$

Ans: Terms: $-ab,2{{b}^{2}},-3{{a}^{2}}$

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

(a) $-4x+5$

Ans: Terms: $-4x,5$ and factors: $-4,x;5$

(b) $-4x+5y$

Ans: Terms: $-4x,5y$ and factors: $-4,x;5,y$

(c) $5y+3{{y}^{2}}$

Ans: Terms: $5y,3{{y}^{2}}$ and factors: $5,y;3,y,y$

(d) $xy+2{{x}^{2}}{{y}^{2}}$

Ans: Terms: $xy,2{{x}^{2}}{{y}^{2}}$ and factors: $x,y;2,x,x,y,y$

(e) $pq+q$

Ans: Terms: $pq,q$ and factors: $p,q;q$

(f) $1.2ab-2.4b+3.6a$

Ans: Terms: $1,2ab,-2.4b,3.6a$ and factors: $1.2,a,b;-2.4,b;3.6,a$

(g) $\frac{3}{4}x+\frac{1}{4}$

Ans: Terms: $\frac{3}{4}x,\frac{1}{4}$ and factors: $\frac{3}{4},x;\frac{1}{4}$

(h) $0.1{{p}^{2}}+0.2{{q}^{2}}$

Ans: Terms: $0.1{{p}^{2}},0.2{{q}^{2}}$ and factors: $0.1,p,p;0.2,q,q$

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

(1) $5-3{{t}^{2}}$

Ans: Terms: $-3{{t}^{2}}$ , Numerical coefficients: $-3$

(2) $1+t+{{t}^{2}}+{{t}^{3}}$

Ans: Terms: $t,{{t}^{2}},{{t}^{3}}$ , Numerical coefficients: $1,1,1$

(3) $x+3xy+3y$

Ans: Terms: $x,2xy,3y$ , Numerical coefficients: $1,2,3$

(4) $100m+1000n$

Ans: Terms: $100m,1000n$ , Numerical coefficients: $100,1000$

(5) $-{{p}^{2}}{{q}^{2}}+7pq$

Ans: Terms: $-{{p}^{2}}{{q}^{2}},7pq$ , Numerical coefficients: $-1,7$

(6) $1.2a+0.8b$

Ans: Terms: $1.2a,0.8b$ , Numerical coefficients: $1.2,0.8$

(7) $3.14{{r}^{2}}$

Ans: Terms: $3.14{{r}^{2}}$ , Numerical coefficients: $3.14$

(8) $2\left( l+b \right)$

Ans: Terms: $2l,2b$ , Numerical coefficients: $2,2$

(9) $0.1y+0.01{{y}^{2}}$

Ans: Terms: $0.1y,0.01{{y}^{2}}$ , Numerical coefficients: $0.1,0.01$

4.

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

(1)  ${{y}^{2}}x+y$

Ans: Terms: ${{y}^{2}}x$ , coefficients: ${{y}^{2}}$

(2) $13{{y}^{2}}-8yx$

Ans: Terms: $-8yx$ , coefficients: $-8y$

(3) $x+y+2$

Ans: Terms: $x$ , coefficients: $1$

(4) $5+z+zx$

Ans: Terms: $zx$ , coefficients: $z$

(5) $1+x+xy$

Ans: Terms: $x,xy$ , coefficients: $1,y$

(6) $12x{{y}^{2}}+25$

Ans: Terms: $12x{{y}^{2}}$ , coefficients: $12{{y}^{2}}$

(7) $7x+x{{y}^{2}}$

Ans: Terms: $7x,x{{y}^{2}}$ , coefficients: $7,{{y}^{2}}$

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

(1) $8-x{{y}^{2}}$

Ans: Terms: $-x{{y}^{2}}$ , coefficients: $-x$

(2) $5{{y}^{2}}+7x$

Ans: Terms: $5{{y}^{2}}$ , coefficients: $5$

(3) $2{{x}^{2}}y-15x{{y}^{2}}+7{{y}^{2}}$

Ans: Terms: $-15x{{y}^{2}},7{{y}^{2}}$ , coefficients: $-15x,7$

5. Classify into the monomial, binomial and trinomials:

1. $4y-7x$

Ans: Binomial 

2. ${{y}^{2}}$

Ans: Monomial

3. $x+y-yx$

Ans: Trinomial 

4. $100$

Ans: Monomial

5. $ab-a-b$

Ans: Trinomial

6. $5-3t$

Ans: Binomial

7. $4{{p}^{2}}q-4p{{q}^{2}}$

Ans: Binomial

8. $7mn$

Ans: Monomial

9. ${{z}^{2}}-3z+8$

Ans: Trinomial

10. ${{a}^{2}}+{{b}^{2}}$

Ans: Binomial

11. ${{z}^{2}}+z$

Ans: Binomial

12. $1+x+{{x}^{2}}$

Ans: Trinomial

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

1. $1,100$

Ans: Like terms

2. $-7x,\frac{5}{2}x$

Ans: Like terms

3. $-29x,-29y$

Ans: Unlike terms

4. $14xy,42yx$

Ans: Like terms

5. $4{{m}^{2}}p,4m{{p}^{2}}$

Ans: Unlike terms

6. $12xz,12{{x}^{2}}{{z}^{2}}$

Ans: Unlike terms

7. Identify like terms in the following:

(1) $-x{{y}^{2}},-4y{{x}^{2}},8{{x}^{2}},2x{{y}^{2}},7y,-11{{x}^{2}},-100x,-11yx,20{{x}^{2}}y,-6{{x}^{2}},y,2xy,3x$

Ans: Like terms are:

$\left( -x{{y}^{2}},2x{{y}^{2}} \right),\left( -4y{{x}^{2}},20{{x}^{2}}y \right),\left( 8{{x}^{2}},-11{{x}^{2}},-6{{x}^{2}} \right),\left( 7y,y \right),\left( -110x,3x \right),\left( -11yx,2xy \right)$

(2)  $10pq,7p,8q,-{{p}^{2}}{{q}^{2}},-7qp,-100q,-23,12{{q}^{2}}{{p}^{2}},-5{{p}^{2}},41,2405p,78qp,13{{p}^{2}}q,q{{p}^{2}},701{{p}^{2}}$

Ans: Like terms are:

\[\left( 10pq,-7pq,78pq \right),\left( 7p,2405p \right),\left( 8q,-100q \right),\left( -{{p}^{2}}{{q}^{2}},12{{p}^{2}}{{q}^{2}} \right),\left( -12,41 \right),\left( -5{{p}^{2}},701{{p}^{2}} \right),\left( 13{{p}^{2}}q,q{{p}^{2}} \right)\]

Exercise 12.2

1. Simplify combining like terms:

1. $21b-32+7b-20b$

Ans:  $\Rightarrow 21b-32+7b-20b$

      $\Rightarrow 21b+7b-20b-32$

      $\Rightarrow 8b-32$

2. $-{{z}^{2}}+13{{z}^{2}}-5z+7{{z}^{3}}-15z$

Ans:  $\Rightarrow 7{{z}^{3}}+\left( -{{z}^{2}}+13{{z}^{2}} \right)-\left( 5z+15z \right)$

      $\Rightarrow 7{{z}^{3}}+12{{z}^{2}}-20z$

3. $p-\left( p-q \right)-q-\left( q-p \right)$

Ans: $\Rightarrow p-p+q-q-q+p$

    $\Rightarrow p-q$

4. $3a-2b-ab-\left( a-b+ab \right)+3ab+b-a$

Ans: \[\Rightarrow \left( 3a-a-a \right)-\left( 2b-b-b \right)-\left( ab+ab-3ab \right)\]

    $\Rightarrow a+ab$

5. $5{{x}^{2}}y-5{{x}^{2}}+3y{{x}^{2}}-3{{y}^{2}}+{{x}^{2}}-{{y}^{2}}+8x{{y}^{2}}-3{{y}^{2}}$

Ans: $\Rightarrow \left( 5{{x}^{2}}y+3{{x}^{2}}y \right)+8x{{y}^{2}}-\left( 5{{x}^{2}}-{{x}^{2}} \right)-\left( 3{{y}^{2}}+{{y}^{2}}+3{{y}^{2}} \right)$

    $\Rightarrow 8{{x}^{2}}y+8x{{y}^{2}}-4{{x}^{2}}-7{{y}^{2}}$

6. $\left( 3{{y}^{2}}+5y-4 \right)-\left( 8y-{{y}^{2}}-4 \right)$

Ans: $\Rightarrow \left( 3{{y}^{2}}+{{y}^{2}} \right)+\left( 5y-8y \right)-\left( 4-4 \right)$

    $\Rightarrow 4{{y}^{2}}-3y$

 2. Add:

1. $3mn,-5mn,8mn,-4mn$

Ans: $\Rightarrow 3mn+\left( -5mn \right)+8mn+\left( -4mn \right)$

    $\Rightarrow \left( 3-5+8-4 \right)mn$

    $\Rightarrow 2mn$

2. $t-8tz,3tz-z,z-t$

Ans: $\Rightarrow t-t-8tz+3tz-z+z$

     $\Rightarrow -5tz$

3. $-7mn+5,12mn+2,9mn-8,-2mn-3$

Ans: $\Rightarrow -7mn+12mn+9mn-2mn+5+2-8-3$

    $\Rightarrow 12mn-4$

4. $a+b-3,b-a+3,a-b+3$

Ans: $\Rightarrow \left( a-a+a \right)+\left( b+b-b \right)-3+3+3$

    $\Rightarrow a+b+3$

5. $14x+10y-12xy-13,18-7x-10y+8xy,4xy$

Ans: $\Rightarrow 14x-7x+10y-10y-12xy+8xy+4xy-13+18$

    $\Rightarrow 7x+5$

6. $5m-7n,3n-4m+2,2m-3mn-5$

Ans: $\Rightarrow 5m-4m+2m-7n+3n-3mn+2-5$

    $\Rightarrow 3m-4n+3mn-3$

7. $4{{x}^{2}}y,-3x{{y}^{2}},-5x{{y}^{2}},5{{x}^{2}}y$

Ans: $\Rightarrow 4{{x}^{2}}y+5{{x}^{2}}y-3x{{y}^{2}}-5x{{y}^{2}}$

    $\Rightarrow 9{{x}^{2}}y-8x{{y}^{2}}$

8. $3{{p}^{2}}{{q}^{2}}-4pq+5,-10{{p}^{2}}{{q}^{2}},15+9pq+7{{p}^{2}}{{q}^{2}}$

Ans: $\Rightarrow 3{{p}^{2}}{{q}^{2}}-10{{p}^{2}}{{q}^{2}}+7{{p}^{2}}{{q}^{2}}+4pq+9pq+5+15$

    $\Rightarrow 5pq+20$

9. $ab-4a,4b-ab,4a-4b$

Ans: $\Rightarrow -4a+4a+4b-4b+ab-ab$

    $\Rightarrow 0$

10. ${{x}^{2}}-{{y}^{2}}-1,{{y}^{2}}-1-{{x}^{2}},1-{{x}^{2}}-{{y}^{2}}$

Ans: $\Rightarrow {{x}^{2}}-{{x}^{2}}-{{x}^{2}}-{{y}^{2}}+{{y}^{2}}-{{y}^{2}}-1+1-1$

    $\Rightarrow -{{x}^{2}}-{{y}^{2}}-1$

3. Subtract:

1. $-5{{y}^{2}}$ from ${{y}^{2}}$

Ans: $\Rightarrow {{y}^{2}}-\left( -5{{y}^{2}} \right)$

    $\Rightarrow {{y}^{2}}+5{{y}^{2}}$

    $\Rightarrow 6{{y}^{2}}$

2. $6xy$ from $-12xy$

Ans: $\Rightarrow -12xy-\left( 6xy \right)$

    $\Rightarrow -12xy-6xy$

    $\Rightarrow -18xy$

3. $\left( a-b \right)$ from $\left( a+b \right)$

Ans: $\Rightarrow \left( a+b \right)-\left( a-b \right)$

    $\Rightarrow a-a+b+b$

    $\Rightarrow 2b$

4. $a\left( b-5 \right)$ from $b\left( 5-a \right)$$

Ans: $\Rightarrow b\left( 5-a \right)-a\left( b-5 \right)$

    $\Rightarrow 5b-ab-ab+5a$

    $\Rightarrow 5b-2ab+5a$$

5. $-{{m}^{2}}+5mn$ from $4{{m}^{2}}-3mn+8$

Ans: $\Rightarrow 4{{m}^{2}}-3mn+8-\left( -{{m}^{2}}+5mn \right)$

    \[\Rightarrow 4{{m}^{2}}+{{m}^{2}}-3mn-5mn+8\]

    $\Rightarrow 5{{m}^{2}}-8mn+8$

6. $-{{x}^{2}}+10x-5$ from $5x-10$

Ans: $\Rightarrow 5x-10-\left( -{{x}^{2}}+10x-5 \right)$

    $\Rightarrow {{x}^{2}}+5x-10x-10+5$

    \[\Rightarrow {{x}^{2}}-5x-5\]

7. $5{{a}^{2}}-7ab+5{{b}^{2}}$ from \[3ab-2{{a}^{2}}-2{{b}^{2}}\]

Ans: $\Rightarrow 3ab-2{{a}^{2}}-2{{b}^{2}}-\left( 5{{a}^{2}}-7ab+5{{b}^{2}} \right)$

    $\Rightarrow 3ab+7ab-2{{a}^{2}}-5{{a}^{2}}-2{{b}^{2}}-5{{b}^{2}}$

    $\Rightarrow 10ab-7{{a}^{2}}-7{{b}^{2}}$

8. $4pq-5{{q}^{2}}-3{{p}^{2}}$ from $5{{p}^{2}}+3{{q}^{2}}-pq$

Ans: $\Rightarrow 5{{p}^{2}}+3{{q}^{2}}-pq-\left( 4pq-5{{q}^{2}}-3{{p}^{2}} \right)$

    $\Rightarrow 5{{p}^{2}}+3{{p}^{2}}+3{{q}^{2}}+5{{q}^{2}}-pq-4pq$

    $\Rightarrow 8{{p}^{2}}+8{{q}^{2}}-5pq$

 4.

(a) What should be added to ${{x}^{2}}+xy+{{y}^{2}}$  to obtain $2{{x}^{2}}+3xy$?

Ans: Let we add $A$ in ${{x}^{2}}+xy+{{y}^{2}}$, then

${{x}^{2}}+xy+{{y}^{2}}+A=2{{x}^{2}}+3xy$

$A=2{{x}^{2}}+3xy-\left( {{x}^{2}}+xy+{{y}^{2}} \right)$

$A=2{{x}^{2}}+3xy-{{x}^{2}}-xy-{{y}^{2}}$

$A={{x}^{2}}+2xy-{{y}^{2}}$

Hence, ${{x}^{2}}+2xy-{{y}^{2}}$ should be added. 

(b) What should be subtracted from $2a+8b+10$ to get $-3a+7b+16$?

Ans: Let $B$ should be subtracted from $2a+8b+10$to get $-3a+7b+16$

$2a+8b+10-B=-3a+7b+16$

$B=2a+8b+10-\left( -3a+7b+16 \right)$

$B=2a+8b+10+3a-7b-16$

$B=5a+b-6$

Hence, $5a+b-6$ should be subtracted.

5. What should be taken away from $3{{x}^{2}}-4{{y}^{2}}+5xy+20$ to obtain $-{{x}^{2}}-{{y}^{2}}+6xy+20$ ?

Ans: Let $C$ should be subtracted from $3{{x}^{2}}-4{{y}^{2}}+5xy+20$ to get $-{{x}^{2}}-{{y}^{2}}+6xy+20$

$3{{x}^{2}}-4{{y}^{2}}+5xy+20-C=-{{x}^{2}}-{{y}^{2}}+6xy+20$

$C=3{{x}^{2}}-4{{y}^{2}}+5xy+20-\left( -{{x}^{2}}-{{y}^{2}}+6xy+20 \right)$

$C=3{{x}^{2}}-4{{y}^{2}}+5xy+20+{{x}^{2}}+{{y}^{2}}-6xy-20$

$C=4{{x}^{2}}-3{{y}^{2}}-xy$

Hence, $4{{x}^{2}}-3{{y}^{2}}-xy$ should be subtracted.

6. (a) From the sum of $3x-y+11$ and $-y-11$, subtract $3x-y-11$ .

Ans:  Sum of $3x-y+11$ and $-y-11$ is

$\Rightarrow 3x-y+11+\left( -y-11 \right)$

$\Rightarrow 3x-y+11-y-11$

$\Rightarrow 3x-2y$ ……eq(1)

Subtracting $3x-y-11$ from eq(1)

$\Rightarrow 3x-2y-\left( 3x-y-11 \right)$

$\Rightarrow 3x-3x-2y+y+11$

$\Rightarrow -y+11$

(b) From the sum of $4+3x$ and $5-4x+2{{x}^2}$, subtract the sum of $3{{x}^{2}}-5x$ and $-{{x}^{2}}+2x+5$.

Ans:  Sum of $4+3x$ and $5-4x+2{{x}^2}$ is

$\Rightarrow 2{{x}^2}+3x-4x+5+4$

$\Rightarrow 2{{x}^2}-x+9$ ……eq(1)

Sum of $3{{x}^{2}}-5x$ and $-{{x}^{2}}+2x+5$

$\Rightarrow 3{{x}^{2}}-5x+\left( -{{x}^{2}}+2x+5 \right)$

$\Rightarrow 3{{x}^{2}}-5x-{{x}^{2}}+2x+5$

$\Rightarrow 2{{x}^{2}}-3x+5$……eq(2)

Subtracting eq(2) from eq(1)

$\Rightarrow 2{{x}^2}-x+9 -\left( 2{{x}^{2}}-3x+5 \right)$

$\Rightarrow 2x+4$

Exercise 12.3

1. If $m=2$, find the value of :

(a) $m-2$

Ans: $\Rightarrow m-2$

$\Rightarrow 2-2$

$\Rightarrow 0$

(b) $3m-5$

Ans: $\Rightarrow 3m-5$

$\Rightarrow 6-5$

$\Rightarrow 1$

(c) $9-5m$

Ans: $\Rightarrow 9-5m$

$\Rightarrow 9-10$

$\Rightarrow -1$

(d) $3{{m}^{2}}-2m-7$

Ans: $\Rightarrow 3{{m}^{2}}-2m-7$

$\Rightarrow 12-4-7$

$\Rightarrow 1$

(e) $\frac{5}{2}m-4$

Ans: $\Rightarrow \frac{5}{2}m-4$

$\Rightarrow 5-4$

$\Rightarrow 1$

2. If $p=-2$, find the value of:

(a) $4p+7$

Ans: $\Rightarrow 4p+7$

$\Rightarrow -8+7$

$\Rightarrow -1$

(b) $-3{{p}^{2}}+4p+7$

Ans: $\Rightarrow -3{{p}^{2}}+4p+7$

$\Rightarrow -3\times 4+4\left( -2 \right)+7$

$\Rightarrow -12-8+7$

$\Rightarrow -13$

(c) $-2{{p}^{3}}-3{{p}^{2}}+4p+7$

Ans: $\Rightarrow -2{{p}^{3}}-3{{p}^{2}}+4p+7$

$\Rightarrow -2\left( -8 \right)-3\times 4-8+7$

$\Rightarrow 16-12-8+7$

$\Rightarrow 3$

 3. Find the value of the following expressions, when $x=-1$:

(a) $2x-7$

Ans:  $\Rightarrow 2x-7$

$\Rightarrow 2\left( -1 \right)-7$

$\Rightarrow -9$

(b) $-x+2$

Ans: $\Rightarrow -x+2$

$\Rightarrow -\left( -1 \right)+2$

$\Rightarrow 3$

(c) ${{x}^{2}}+2x+1$

Ans: $\Rightarrow {{x}^{2}}+2x+1$

$\Rightarrow {{\left( -1 \right)}^{2}}+2\left( -1 \right)+1$

$\Rightarrow 0$

(d) $2{{x}^{2}}-x-2$

Ans: $\Rightarrow 2{{x}^{2}}-x-2$

$\Rightarrow 2{{\left( -1 \right)}^{2}}-\left( -1 \right)-2$

$\Rightarrow 1$

4. If $a=2,b=-2$, find the value of:

(a) ${{a}^{2}}+{{b}^{2}}$

Ans: $\Rightarrow {{\left( -2 \right)}^{2}}+{{\left( -2 \right)}^{2}}$

$\Rightarrow 4+4$

$\Rightarrow 8$

(b) ${{a}^{2}}+ab+{{b}^{2}}$

Ans: \[\Rightarrow {{\left( -2 \right)}^{2}}+\left( -2 \right)\left( -2 \right)+{{\left( -2 \right)}^{2}}\]

$\Rightarrow 4-4+4$

$\Rightarrow 4$

(c) ${{a}^{2}}-{{b}^{2}}$

Ans: $\Rightarrow {{\left( -2 \right)}^{2}}-{{\left( -2 \right)}^{2}}$

$\Rightarrow 4-4$

$\Rightarrow 0$

5. If $a=0,b=-1$, find the value of given expression:

(a) $2a+2b$

Ans: $\Rightarrow 2\left( 0 \right)+2\left( -1 \right)$

$\Rightarrow 0-2$

$\Rightarrow -2$

(b) \[2{{a}^{2}}+{{b}^{2}}+1\]

Ans: $\Rightarrow 2{{\left( 0 \right)}^{2}}+{{\left( -1 \right)}^{2}}+1$

$\Rightarrow 0+1+1$

$\Rightarrow 2$

(c) \[2{{a}^{2}}b+2a{{b}^{2}}+ab\]

Ans: $\Rightarrow 2{{\left( 0 \right)}^{2}}\left( -1 \right)+2\left( 0 \right){{\left( -1 \right)}^{2}}+\left( 0 \right)\left( -1 \right)$

$\Rightarrow 0$

(d) \[{{a}^{2}}+ab+2\]

Ans: $\Rightarrow {{\left( 0 \right)}^{2}}+\left( 0 \right)\left( -1 \right)+2$

$\Rightarrow 0+0+2$

$\Rightarrow 2$

6. Simplify the expressions and find the value if $x$ is equal to $2$:

(a) $x+7+4\left( x-5 \right)$

Ans:

$\Rightarrow x+7+4x-20$

$\Rightarrow 5x-13$

$\Rightarrow 5\times 2-13$

$\Rightarrow 10-13$

$\Rightarrow -3$

(b) $3\left( x+2 \right)+5x-7$

Ans:

$\Rightarrow 3x+6+5x-7$

$\Rightarrow 8x-1$

$\Rightarrow 8\times 2-1$

$\Rightarrow 16-1$

$\Rightarrow 15$

(c) $6x+5\left( x-2 \right)$

Ans:

$\Rightarrow 6x+5x-10$

$\Rightarrow 11x-10$

$\Rightarrow 11\times 2-10$

$\Rightarrow 22-10$

$\Rightarrow 12$

(d) $4\left( 2x-1 \right)+3x+11$

Ans:

$\Rightarrow 8x-4+3x+11$

$\Rightarrow 11x+7$

$\Rightarrow 11\times 2+7$

$\Rightarrow 22+7$

$\Rightarrow 29$

7. Simplify these expressions and find their values if $x=3,a=-1,b=-2$:

(a) $3x-5-x+9$

Ans:

$\Rightarrow 2x+4$

$\Rightarrow 2\times 3+4$

$\Rightarrow 6+4$

$\Rightarrow 10$

(b) $2-8x+4x+4$

Ans:

$\Rightarrow 6-4x$

$\Rightarrow 6-4\times 3$

$\Rightarrow 6-12$

$\Rightarrow -6$

(c) $3a+5-8a+1$

Ans:

$\Rightarrow 6-5a$

$\Rightarrow 6-5\left( -1 \right)$

$\Rightarrow 6+5$

$\Rightarrow 11$

(d) $10-3b-4-5b$

Ans:

$\Rightarrow 6-8b$

$\Rightarrow 6-8\left( -2 \right)$

$\Rightarrow 6+16$

$\Rightarrow 22$

(e) $2a-2b-4-5+a$

Ans:

$\Rightarrow 3a-2b-9$

$\Rightarrow 3\left( -1 \right)-2\left( -2 \right)-9$

$\Rightarrow -3+4-9$

$\Rightarrow -8$

 8.

(a) If $z=10$, find the value of ${{z}^{3}}-3\left( Z-10 \right)$

Ans:

$\Rightarrow {{\left( 10 \right)}^{3}}-3\left( 10-10 \right)$

$\Rightarrow 1000-0$

$\Rightarrow 1000$

(b) If $p=-10$ , find the value of ${{p}^{2}}-2p-100$

Ans:

$\Rightarrow {{\left( -10 \right)}^{2}}-2\left( -10 \right)-100$

$\Rightarrow 100+20-100$

$\Rightarrow 20$

9. What should be the value of $a$ if the value of $2{{x}^{2}}+x-a$ equal to $5$, when $x=0$ ?

Ans:

Putting $x=0$ in $2{{x}^{2}}+x-a=5$, we get

$2{{\left( 0 \right)}^{2}}+0-a=5$

$0+0-a=5$

$a=-5$

Hence, the value of $a$ is $-5$ .

10. Simplify the expression and find its value when $a=5$ and $b=-3$: $2\left( {{a}^{2}}+ab \right)+3-ab$

Ans:

Simplifying the equation,

$\Rightarrow 2\left( {{a}^{2}}+ab \right)+3-ab$

$\Rightarrow 2{{a}^{2}}+2ab+3-ab$

$\Rightarrow 2{{a}^{2}}+ab+3$

Putting $a=5$ and $b=-3$ in the above equation

$\Rightarrow 2{{\left( 5 \right)}^{2}}+5\left( -3 \right)+3$

$\Rightarrow 2\times 25-15+3$

$\Rightarrow 50-15+3$

$\Rightarrow 38$

Value of the expression after simplifying and putting $a=5$ and $b=-3$ is $38$ .

Exercise 12.4

1. Observe the pattern of digits made from line segments of equal length. You will find such segmented digits on the display of electronic watches or calculators.

If the number of digits formed is taken to be $n$, the number of segments required to form $n$ digits is given by the algebraic expression appearing on the right of each pattern.

How many segments are required to form $5,10,100$ digits of the kind. 

(Images will be uploading soon)

    $6$       $11$ $16$   $21$       $\left( 5n+1 \right)$

Ans:

(Image will be uploading soon)

Symbol: 

Pattern Formulae: $5n+1$

Digit’s number: $5,10,100$

No. of segments in $n=5$,

$\Rightarrow 5\times 5+1$

$\Rightarrow 25+1$

$\Rightarrow 26$

No. of segments in $n=10$,

$\Rightarrow 5\times 10+1$

$\Rightarrow 50+1$

$\Rightarrow 51$

No. of segments in $n=100$,

$\Rightarrow 5\times 100+1$

$\Rightarrow 500+1$

$\Rightarrow 501$

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    $4$       $7$ $10$   $13$             $\left( 3n+1 \right)$

Ans:

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Symbol: 

Pattern Formulae: $3n+1$

Digit’s number: $5,10,100$

No. of segments in $n=5$,

$\Rightarrow 3\times 5+1$

$\Rightarrow 15+1$

$\Rightarrow 16$

No. of segments in $n=10$,

$\Rightarrow 3\times 10+1$

$\Rightarrow 30+1$

$\Rightarrow 31$

No. of segments in $n=100$,

$\Rightarrow 3\times 100+1$

$\Rightarrow 300+1$

$\Rightarrow 301$

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    $7$       $12$ $17$   $22$             $\left( 5n+2 \right)$

Ans:

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Symbol: 

Pattern Formulae: $5n+2$

Digit’s number: $5,10,100$

No. of segments in $n=5$,

$\Rightarrow 5\times 5+2$

$\Rightarrow 25+2$

$\Rightarrow 27$

No. of segments in $n=10$,

$\Rightarrow 5\times 10+2$

$\Rightarrow 50+2$

$\Rightarrow 52$

No. of segments in $n=100$,

$\Rightarrow 5\times 100+2$

$\Rightarrow 500+2$

$\Rightarrow 502$

 2.

Use the given algebraic expression to complete the table of number patterns:

Ans:

Calculating the blanks row-wise

Row $1$: $2n-1$

To calculate the $100th$ term, 

Putting $n=100$

$\Rightarrow 2\times 100-1$

$\Rightarrow 199$

Row $2$: $3n+2$

To calculate the $5th,10th,100th$ term, 

Putting $n=5$

$\Rightarrow 3\times 5+2$

$\Rightarrow 17$

Putting $n=10$

$\Rightarrow 3\times 10+2$

$\Rightarrow 32$

Putting $n=100$

$\Rightarrow 3\times 100+2$

$\Rightarrow 302$

Row $3$: $4n+1$

To calculate the $5th,10th,100th$ term, 

Putting $n=5$

$\Rightarrow 4\times 5+1$

$\Rightarrow 21$

Putting $n=10$

$\Rightarrow 4\times 10+1$

$\Rightarrow 41$

Putting $n=100$

$\Rightarrow 4\times 100+1$

$\Rightarrow 401$

Row $4$: $7n+20$

To calculate the $5th,10th,100th$ term, 

Putting $n=5$

$\Rightarrow 7\times 5+20$

$\Rightarrow 55$

Putting $n=10$

$\Rightarrow 7\times 10+20$

$\Rightarrow 90$

Putting $n=100$

$\Rightarrow 7\times 100+20$

$\Rightarrow 720$

Row $5$: ${{n}^{2}}+1$

To calculate the $5th,10th,100th$ term, 

Putting $n=5$

$\Rightarrow {{5}^{2}}+1$

$\Rightarrow 26$

Putting $n=10$

$\Rightarrow {{10}^{2}}+1$

$\Rightarrow 101$

Putting $n=100$

$\Rightarrow {{100}^{2}}+1$

$\Rightarrow 10001$

NCERT Solutions for Class 7 Maths Algebraic Expressions– Free PDF Download

Introduction

In Arithmetic, we deal with quantities, which are denoted by numbers having specific values but in Algebra we deal with quantities which are denoted by symbols (in general, letters from English Alphabets) that may have more than one value. Also, the values of these symbols vary based on the given conditions. These symbols are called variables.

Fundamental Concepts of Algebra

There are two types of terms in algebra:

1. Constant: A constant is a term that has a fixed numerical value.

Thus, 3, -2, 4/9, 0.2 are constant values.

2. Variable: A symbol, which takes on various numerical values, based on the given conditions, is called a variable or a literal.

Example: We know the formula of the perimeter of a square is: 

P = 4 x s

So, here 4 is constant.

When s = 6 then p = 4 x 6 = 24.

When s = 5 then p = 4 x 5 = 20.

This tells us that the values of p and s are not fixed. They vary. In the above example, s and p are variables.

Facts

• A variable can take on many values whereas a constant has a fixed value.

• Algebraic expressions can be formed by using basic operations on variables and constants.

• In general, any term of an algebraic expression has factors. Algebraic factors are variables and constant factors are numbers.

• Any factor in a term is called the coefficient of the remaining part of the term.

• The part of the algebraic expression that has only one term, is called a monomial.

• When a polynomial has two terms then it is called binomial whereas a trinomial has three terms.

• Terms that have the same algebraic factors are called like terms whereas terms that have different algebraic factors are called, unlike terms.

• The sum (or difference) of two like terms is like a term with a coefficient equal to the sum (or difference) of the coefficient of the two like terms.

• In mathematics, we can write rules and formulas in a concise and general form using algebraic expressions.

How are Algebraic Expressions Formed?

An algebraic expression is formed by a combination of constants and variables or only variables, connected by the symbols +, —, x, and ÷.

Example:

The expression x3 is obtained by multiplying the variable x by itself.

x *  x * x = x3

The expression 3x2 is obtained by multiplying the variable x by itself twice and multiplying the product x2 by the constant 3.

The expression ( 2x3 – 7 ) is obtained by multiplying the variable x by itself three times and multiplying the product x3 by the constant 2 to get 2x3. Next, we subtract 7 from the product to finally derive (2x3 – 7).

The expression pq is obtained by multiplying the variable p with another variable q. Thus, p x q = pq.

To get expression 5ab + 7, we first need to obtain ab, next we multiply it by 5 we get 5ab. Then, we add 7 to 5ab to get the expression 5ab + 7.

Terms of an Expression

To form an algebraic expression, we first its parts separately and then add them. Such parts of an expression that are formed separately first and then added are called terms.

Example:

The expression 4x + 9, has the terms ‘4x’ and ‘9’.

The expression 9y – 3p, has the terms ‘9y’ and ‘–3p’

Factors of a Term

When two or more quantities (numbers and literals) are multiplied, then each one of them is called a factor of the product.

Example:

In 9x2y, we have 9 as the numerical factor whereas x2 and y are the literal factors.

In ‘–pq ‘, the numerical factor is ‘–1’ whereas p and q are literal factors. 

Coefficients

Any of the factors of a term is called the coefficient of the product of all the remaining factors.

Example:

In 10pqr, the coefficient of pqr is 10.

In –6x2y, the coefficient of x2y is –6.

Like and Unlike Terms

The terms having the same literal factors are called like terms (or similar terms) whereas the terms that do not have the same literal factors are called unlike terms (or dissimilar terms).

Example: In the expression 5x2y + 3xy2 – xy -2xy2, we have 5x2y and –2yx2 as the like terms.

The terms 3xy2 and – xy are the, unlike terms.

Monomials, Binomials, Trinomials, and Polynomials

A monomial is an expression that has only one term.

When an expression has two terms then it is called a binomial.

An expression having three terms is called trinomial.

A polynomial is an algebraic expression that has two or more terms.

Example:

3x and – 2abc are monomials.

x2 – 3y2 and 6 – p are binomials.

x3 – y3 – z3 and xy + x2 – 5 are trinomials.

x – y, 3x2 – 2x + 5, 4x4 + 3x2 – 2x + 6 are polynomials.

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