Chapter 10 Algebraic Expressions

Solution:

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In an algebraic expression, terms are parts separated by addition or subtraction. A constant term is a term that does not contain any variables. Terms which are not constants contain variables. The numerical coefficient of a term is the number that multiplies the variable part of the term.

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Let's examine each expression:

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Expression: xy + 4

Terms are xy and 4.

The term '4' is a constant term.

The term 'xy' is not a constant term as it contains variables x and y.

The numerical coefficient of the term xy is 1 (since $xy = 1 \times xy$).

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Expression: 13 – y²

Terms are 13 and $-y^2$.

The term '13' is a constant term.

The term '$-y^2$' is not a constant term as it contains the variable y.

The numerical coefficient of the term $-y^2$ is -1 (since $-y^2 = -1 \times y^2$).

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Expression: 13 – y + 5y²

Terms are 13, -y, and $5y^2$.

The term '13' is a constant term.

The terms '-y' and '$5y^2$' are not constant terms as they contain the variable y.

The numerical coefficient of the term -y is -1 (since $-y = -1 \times y$).

The numerical coefficient of the term $5y^2$ is 5.

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Expression: 4p²q – 3pq² + 5

Terms are $4p^2q$, $-3pq^2$, and 5.

The term '5' is a constant term.

The terms '$4p^2q$' and '$-3pq^2$' are not constant terms as they contain variables p and q.

The numerical coefficient of the term $4p^2q$ is 4.

The numerical coefficient of the term $-3pq^2$ is -3.

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Summarizing the terms which are not constants and their numerical coefficients:

Expression	Term (not constant)	Numerical Coefficient
xy + 4	xy	1
13 – y2	-y2	-1
13 – y + 5y2	-y	-1
13 – y + 5y2	5y2	5
4p2q – 3pq2 + 5	4p2q	4
4p2q – 3pq2 + 5	-3pq2	-3

Solution:

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The coefficient of a variable in a term is the factor that multiplies that variable (including the sign and any other variables/numbers in the term, but excluding the variable itself for which we are finding the coefficient).

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(a) Coefficients of x:

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Expression: 4x – 3y

Terms containing x: 4x.

In the term 4x, the factor multiplying x is 4.

The coefficient of x is 4.

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Expression: 8 – x + y

Terms containing x: -x.

In the term -x, which can be written as $-1 \times x$, the factor multiplying x is -1.

The coefficient of x is -1.

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Expression: y²x – y

Terms containing x: $y^2x$.

In the term $y^2x$, the factor multiplying x is $y^2$.

The coefficient of x is $y^2$.

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Expression: 2z – 5xz

Terms containing x: -5xz.

In the term -5xz, the factors multiplying x are -5 and z. So, the factor multiplying x is -5z.

The coefficient of x is -5z.

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Summary of coefficients of x:

Expression	Term(s) containing x	Coefficient of x
4x – 3y	4x	4
8 – x + y	-x	-1
y2x – y	y2x	y2
2z – 5xz	-5xz	-5z

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(b) What are the coefficients of y in the following expressions?

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Expression: 4x – 3y

Terms containing y: -3y.

In the term -3y, the factor multiplying y is -3.

The coefficient of y is -3.

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Expression: 8 + yz

Terms containing y: yz.

In the term yz, the factor multiplying y is z.

The coefficient of y is z.

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Expression: yz² + 5

Terms containing y: $yz^2$.

In the term $yz^2$, the factor multiplying y is $z^2$.

The coefficient of y is $z^2$.

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Expression: my + m

Terms containing y: my.

In the term my, the factor multiplying y is m.

The coefficient of y is m.

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Summary of coefficients of y:

Expression	Term(s) containing y	Coefficient of y
4x – 3y	-3y	-3
8 + yz	yz	z
yz2 + 5	yz2	z2
my + m	my	m

Solution:

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Like terms are terms that have the same variables raised to the same powers. The numerical coefficients do not matter. Unlike terms are terms that do not have the same variables or the same powers of the variables.

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Let's examine each pair of terms:

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(i) 7x , 12y

The variables are x and y. The variables are different.

These are unlike terms.

Reason: The terms have different variables.

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(ii) 15x , –21x

The variable in both terms is x. The power of x in both terms is 1.

These are like terms.

Reason: The terms have the same variable raised to the same power (x¹).

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(iii) – 4ab , 7ba

The variables in the first term are a and b, both with power 1. The variables in the second term are b and a, both with power 1. Since multiplication is commutative ($ab = ba$), the variable parts are the same.

These are like terms.

Reason: The terms have the same variables (a and b) raised to the same powers (a¹ and b¹), although written in a different order.

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(iv) 3xy , 3x

The variables in the first term are x and y, both with power 1. The variable in the second term is x with power 1.

These are unlike terms.

Reason: The terms do not have the same variables (the first term has y, but the second term does not).

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(v) 6xy² , 9x²y

In the first term, the power of x is 1 and the power of y is 2. ($xy^2 = x^1y^2$)

In the second term, the power of x is 2 and the power of y is 1. ($x^2y = x^2y^1$)

The powers of the variables are not the same for both terms.

These are unlike terms.

Reason: The terms have the same variables (x and y), but the powers of the variables are different.

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(vi) pq² , – 4pq²

In the first term, the variable part is $pq^2$ (which is $p^1q^2$).

In the second term, the variable part is $pq^2$ (which is $p^1q^2$).

The variables and their powers are the same in both terms.

These are like terms.

Reason: The terms have the same variables (p and q) raised to the same powers (p¹ and q²).

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(vii) mn² , 10mn

In the first term, the variable part is $mn^2$ (which is $m^1n^2$).

In the second term, the variable part is $mn$ (which is $m^1n^1$).

The powers of the variable n are different ($n^2$ vs $n^1$).

These are unlike terms.

Reason: The terms have the same variables (m and n), but the powers of the variable n are different.

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Summary:

Pair of terms	Like/Unlike Terms	Reason
7x , 12y	Unlike terms	Different variables.
15x , –21x	Like terms	Same variable (x) with same power (1).
– 4ab , 7ba	Like terms	Same variables (a, b) with same powers (a1, b1).
3xy , 3x	Unlike terms	Different variables (first term has y, second does not).
6xy2 , 9x2y	Unlike terms	Same variables, but different powers (x1y2 vs x2y1).
pq2 , – 4pq2	Like terms	Same variables (p, q) with same powers (p1, q2).
mn2 , 10mn	Unlike terms	Same variables, but different powers (m1n2 vs m1n1).

Solution:

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We need to translate the given verbal statements into algebraic expressions using variables, constants, and arithmetic operations (+, -, $\times$, $\div$).

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(i) Subtraction of z from y.

This means y minus z.

The expression is $y - z$.

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

First, find the sum of x and y: $x + y$.

Then, take one-half of the sum: $\frac{1}{2} \times (x+y)$ or $\frac{x+y}{2}$.

The expression is $\frac{1}{2}(x+y)$ or $\frac{x+y}{2}$.

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(iii) The number z multiplied by itself.

This means z times z.

The expression is $z \times z$, which is written as $z^2$.

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

First, find the product of p and q: $p \times q$ or $pq$.

Then, take one-fourth of the product: $\frac{1}{4} \times pq$ or $\frac{pq}{4}$.

The expression is $\frac{1}{4}pq$ or $\frac{pq}{4}$.

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

First, square x: $x^2$.

Then, square y: $y^2$.

Finally, add the squared numbers: $x^2 + y^2$.

The expression is $x^2 + y^2$.

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

First, find the product of m and n: $mn$.

Then, find three times the product: $3 \times mn$ or $3mn$.

Finally, add 5 to this result: $3mn + 5$.

The expression is $3mn + 5$.

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

First, find the product of y and z: $yz$.

This product is subtracted from 10, meaning 10 minus the product.

The expression is $10 - yz$.

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

First, find the sum of a and b: $a + b$.

Then, find the product of a and b: $ab$.

The sum is subtracted from the product, meaning the product minus the sum.

The expression is $ab - (a+b)$.

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Summary of the algebraic expressions:

(i) $y - z$

(ii) $\frac{1}{2}(x+y)$ or $\frac{x+y}{2}$

(iii) $z^2$

(iv) $\frac{1}{4}pq$ or $\frac{pq}{4}$

(v) $x^2 + y^2$

(vi) $3mn + 5$

(vii) $10 - yz$

(viii) $ab - (a+b)$

Solution:

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(i) Identification of terms and factors using tree diagrams:

(a) $x - 3$

Expression: $x - 3$

Terms: $x$, $-3$

Factors of $x$: $x$

Factors of $-3$: $-3$

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

Expression: $1 + x + x^2$

Terms: $1$, $x$, $x^2$

Factors of $1$: $1$

Factors of $x$: $x$

Factors of $x^2$: $x, x$

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(c) $y - y^3$

Expression: $y - y^3$

Terms: $y$, $-y^3$

Factors of $y$: $y$

Factors of $-y^3$: $-1, y, y, y$

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

Expression: $5xy^2 + 7x^2y$

Terms: $5xy^2$, $7x^2y$

Factors of $5xy^2$: $5, x, y, y$

Factors of $7x^2y$: $7, x, x, y$

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

Expression: $- ab + 2b^2 - 3a^2$

Terms: $-ab$, $2b^2$, $-3a^2$

Factors of $-ab$: $-1, a, b$

Factors of $2b^2$: $2, b, b$

Factors of $-3a^2$: $-3, a, a$

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(ii) Identification of terms and factors:

(a) $-4x + 5$

Terms: $-4x$, $5$

Factors of $-4x$: $-4, x$

Factors of $5$: $5$

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(b) $-4x + 5y$

Terms: $-4x$, $5y$

Factors of $-4x$: $-4, x$

Factors of $5y$: $5, y$

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

Terms: $5y$, $3y^2$

Factors of $5y$: $5, y$

Factors of $3y^2$: $3, y, y$

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

Terms: $xy$, $2x^2y^2$

Factors of $xy$: $x, y$

Factors of $2x^2y^2$: $2, x, x, y, y$

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(e) $pq + q$

Terms: $pq$, $q$

Factors of $pq$: $p, q$

Factors of $q$: $q$

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(f) $1.2 ab – 2.4 b + 3.6 a$

Terms: $1.2ab$, $-2.4b$, $3.6a$

Factors of $1.2ab$: $1.2, a, b$

Factors of $-2.4b$: $-2.4, b$

Factors of $3.6a$: $3.6, a$

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

Terms: $\frac{3}{4}x$, $\frac{1}{4}$

Factors of $\frac{3}{4}x$: $\frac{3}{4}, x$

Factors of $\frac{1}{4}$: $\frac{1}{4}$

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

Terms: $0.1p^2$, $0.2q^2$

Factors of $0.1p^2$: $0.1, p, p$

Factors of $0.2q^2$: $0.2, q, q$

(i) $5 - 3t^2$

Terms (other than constant): $-3t^2$

Numerical coefficient of $-3t^2$ is $-3$.

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(ii) $1 + t + t^2 + t^3$

Terms (other than constant): $t$, $t^2$, $t^3$

Numerical coefficient of $t$ is $1$.

Numerical coefficient of $t^2$ is $1$.

Numerical coefficient of $t^3$ is $1$.

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

Terms (other than constant): $x$, $2xy$, $3y$

Numerical coefficient of $x$ is $1$.

Numerical coefficient of $2xy$ is $2$.

Numerical coefficient of $3y$ is $3$.

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(iv) $100m + 1000n$

Terms (other than constant): $100m$, $1000n$

Numerical coefficient of $100m$ is $100$.

Numerical coefficient of $1000n$ is $1000$.

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(v) $-p^2q^2 + 7pq$

Terms (other than constant): $-p^2q^2$, $7pq$

Numerical coefficient of $-p^2q^2$ is $-1$.

Numerical coefficient of $7pq$ is $7$.

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(vi) $1.2 a + 0.8 b$

Terms (other than constant): $1.2 a$, $0.8 b$

Numerical coefficient of $1.2 a$ is $1.2$.

Numerical coefficient of $0.8 b$ is $0.8$.

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(vii) $3.14 r^2$

Terms (other than constant): $3.14 r^2$

Numerical coefficient of $3.14 r^2$ is $3.14$.

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(viii) $2 (l + b) = 2l + 2b$

Terms (other than constant): $2l$, $2b$

Numerical coefficient of $2l$ is $2$.

Numerical coefficient of $2b$ is $2$.

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

Terms (other than constant): $0.1 y$, $0.01 y^2$

Numerical coefficient of $0.1 y$ is $0.1$.

Numerical coefficient of $0.01 y^2$ is $0.01$.

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

(i) $y^2x + y$

Term containing $x$: $y^2x$

Coefficient of $x$: $y^2$

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(ii) $13y^2 – 8yx$

Term containing $x$: $-8yx$

Coefficient of $x$: $-8y$

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(iii) $x + y + 2$

Term containing $x$: $x$

Coefficient of $x$: $1$

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(iv) $5 + z + zx$

Term containing $x$: $zx$

Coefficient of $x$: $z$

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(v) $1 + x + xy$

Terms containing $x$: $x$, $xy$

Coefficient of $x$ in $x$: $1$

Coefficient of $x$ in $xy$: $y$

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(vi) $12xy^2 + 25$

Term containing $x$: $12xy^2$

Coefficient of $x$: $12y^2$

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(vii) $7x + xy^2$

Terms containing $x$: $7x$, $xy^2$

Coefficient of $x$ in $7x$: $7$

Coefficient of $x$ in $xy^2$: $y^2$

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

(i) $8 – xy^2$

Term containing $y^2$: $-xy^2$

Coefficient of $y^2$: $-x$

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

Term containing $y^2$: $5y^2$

Coefficient of $y^2$: $5$

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(iii) $2x^2y – 15xy^2 + 7y^2$

Terms containing $y^2$: $-15xy^2$, $7y^2$

Coefficient of $y^2$ in $-15xy^2$: $-15x$

Coefficient of $y^2$ in $7y^2$: $7$

Classifying the given expressions:

(i) $4y – 7z$

Number of terms: 2

Classification: Binomial

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(ii) $y^2$

Number of terms: 1

Classification: Monomial

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(iii) $x + y – xy$

Number of terms: 3

Classification: Trinomial

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(iv) $100$

Number of terms: 1

Classification: Monomial

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(v) $ab – a – b$

Number of terms: 3

Classification: Trinomial

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(vi) $5 – 3t$

Number of terms: 2

Classification: Binomial

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(vii) $4p^2q – 4pq^2$

Number of terms: 2

Classification: Binomial

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(viii) $7mn$

Number of terms: 1

Classification: Monomial

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(ix) $z^2 – 3z + 8$

Number of terms: 3

Classification: Trinomial

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

Number of terms: 2

Classification: Binomial

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(xi) $z^2 + z$

Number of terms: 2

Classification: Binomial

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

Number of terms: 3

Classification: Trinomial

(i) $1$, $100$

These are constant terms.

Classification: Like terms

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(ii) $-7x$, $\frac{5}{2}x$

The variable part in both terms is $x$.

Classification: Like terms

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(iii) $-29x$, $-29y$

The variable parts are $x$ and $y$, which are different.

Classification: Unlike terms

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

The variable parts are $xy$ and $yx$. Since multiplication is commutative, $xy$ is the same as $yx$.

Classification: Like terms

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

The variable parts are $m^2p$ and $mp^2$. The powers of the variables are different.

Classification: Unlike terms

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

The variable parts are $xz$ and $x^2z^2$. The powers of the variables are different.

Classification: Unlike terms

(a) Identify like terms in the following:

Like terms with variable part $xy^2$: $-xy^2$, $2xy^2$

Like terms with variable part $x^2y$: $-4yx^2$, $20x^2y$

Like terms with variable part $x^2$: $8x^2$, $-11x^2$, $-6x^2$

Like terms with variable part $y$: $7y$, $y$

Like terms with variable part $x$: $-100x$, $3x$

Like terms with variable part $xy$: $-11yx$, $2xy$

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(b) Identify like terms in the following:

Like terms with variable part $pq$: $10pq$, $-7qp$, $78qp$

Like terms with variable part $p$: $7p$, $2405p$

Like terms with variable part $q$: $8q$, $-100q$

Like terms with variable part $p^2q^2$: $-p^2q^2$, $12q^2p^2$

Constant like terms: $-23$, $41$

Like terms with variable part $p^2$: $-5p^2$, $701p^2$

Like terms with variable part $p^2q$: $13p^2q$, $qp^2$

Given $x = 2$. We need to find the values of the following expressions:

(i) $x + 4$

Substitute $x=2$ into the expression:

$2 + 4$

Value of the expression is: $6$

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(ii) $4x – 3$

Substitute $x=2$ into the expression:

$4(2) - 3$

$8 - 3$

Value of the expression is: $5$

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(iii) $19 – 5x^2$

Substitute $x=2$ into the expression:

$19 - 5(2)^2$

$19 - 5(4)$

$19 - 20$

Value of the expression is: $-1$

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(iv) $100 – 10x^3$

Substitute $x=2$ into the expression:

$100 - 10(2)^3$

$100 - 10(8)$

$100 - 80$

Value of the expression is: $20$

Given $n = -2$. We need to find the value of the following expressions:

(i) $5n - 2$

Substitute $n = -2$ into the expression:

$5(-2) - 2$

$-10 - 2$

Value of the expression is: $-12$

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(ii) $5n^2 + 5n - 2$

Substitute $n = -2$ into the expression:

$5(-2)^2 + 5(-2) - 2$

$5(4) - 10 - 2$

$20 - 10 - 2$

$10 - 2$

Value of the expression is: $8$

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(iii) $n^3 + 5n^2 + 5n - 2$

Substitute $n = -2$ into the expression:

$(-2)^3 + 5(-2)^2 + 5(-2) - 2$

$-8 + 5(4) - 10 - 2$

$-8 + 20 - 10 - 2$

$12 - 10 - 2$

$2 - 2$

Value of the expression is: $0$

Given $a = 3$ and $b = 2$. We need to find the value of the following expressions:

(i) $a + b$

Substitute $a=3$ and $b=2$ into the expression:

$3 + 2$

Value of the expression is: $5$

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(ii) $7a – 4b$

Substitute $a=3$ and $b=2$ into the expression:

$7(3) - 4(2)$

$21 - 8$

Value of the expression is: $13$

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(iii) $a^2 + 2ab + b^2$

Substitute $a=3$ and $b=2$ into the expression:

$(3)^2 + 2(3)(2) + (2)^2$

$9 + 2(6) + 4$

$9 + 12 + 4$

$21 + 4$

Value of the expression is: $25$

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(iv) $a^3 – b^3$

Substitute $a=3$ and $b=2$ into the expression:

$(3)^3 - (2)^3$

$27 - 8$

Value of the expression is: $19$

Given $m = 2$. We need to find the value of the following expressions:

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(i) $m – 2$

Substitute $m = 2$ into the expression:

$2 - 2$

Value of the expression is: $0$

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(ii) $3m – 5$

Substitute $m = 2$ into the expression:

$3(2) - 5$

$6 - 5$

Value of the expression is: $1$

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(iii) $9 – 5m$

Substitute $m = 2$ into the expression:

$9 - 5(2)$

$9 - 10$

Value of the expression is: $-1$

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(iv) $3m^2 – 2m – 7$

Substitute $m = 2$ into the expression:

$3(2)^2 - 2(2) - 7$

$3(4) - 4 - 7$

$12 - 4 - 7$

$8 - 7$

Value of the expression is: $1$

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

Substitute $m = 2$ into the expression:

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

$\frac{10}{2} - 4$

$5 - 4$

Value of the expression is: $1$

Given $p = -2$. We need to find the value of the following expressions:

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(i) $4p + 7$

Substitute $p = -2$ into the expression:

$4(-2) + 7$

$-8 + 7$

Value of the expression is: $-1$

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(ii) $-3p^2 + 4p + 7$

Substitute $p = -2$ into the expression:

$-3(-2)^2 + 4(-2) + 7$

$-3(4) - 8 + 7$

$-12 - 8 + 7$

$-20 + 7$

Value of the expression is: $-13$

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(iii) $-2p^3 – 3p^2 + 4p + 7$

Substitute $p = -2$ into the expression:

$-2(-2)^3 - 3(-2)^2 + 4(-2) + 7$

$-2(-8) - 3(4) - 8 + 7$

$16 - 12 - 8 + 7$

$4 - 8 + 7$

$-4 + 7$

Value of the expression is: $3$

Given $x = -1$. We need to find the value of the following expressions:

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(i) $2x – 7$

Substitute $x = -1$ into the expression:

$2(-1) - 7$

$-2 - 7$

Value of the expression is: $-9$

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(ii) $- x + 2$

Substitute $x = -1$ into the expression:

$-(-1) + 2$

$1 + 2$

Value of the expression is: $3$

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(iii) $x^2 + 2x + 1$

Substitute $x = -1$ into the expression:

$(-1)^2 + 2(-1) + 1$

$1 - 2 + 1$

$-1 + 1$

Value of the expression is: $0$

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(iv) $2x^2 – x – 2$

Substitute $x = -1$ into the expression:

$2(-1)^2 - (-1) - 2$

$2(1) + 1 - 2$

$2 + 1 - 2$

$3 - 2$

Value of the expression is: $1$

Given $a = 2$ and $b = -2$. We need to find the value of the following expressions:

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(i) $a^2 + b^2$

Substitute $a = 2$ and $b = -2$ into the expression:

$(2)^2 + (-2)^2$

$4 + 4$

Value of the expression is: $8$

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(ii) $a^2 + ab + b^2$

Substitute $a = 2$ and $b = -2$ into the expression:

$(2)^2 + (2)(-2) + (-2)^2$

$4 + (-4) + 4$

$4 - 4 + 4$

$0 + 4$

Value of the expression is: $4$

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(iii) $a^2 – b^2$

Substitute $a = 2$ and $b = -2$ into the expression:

$(2)^2 - (-2)^2$

$4 - (4)$

$4 - 4$

Value of the expression is: $0$

Given $a = 0$ and $b = -1$. We need to find the value of the following expressions:

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(i) $2a + 2b$

Substitute $a = 0$ and $b = -1$ into the expression:

$2(0) + 2(-1)$

$0 - 2$

Value of the expression is: $-2$

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(ii) $2a^2 + b^2 + 1$

Substitute $a = 0$ and $b = -1$ into the expression:

$2(0)^2 + (-1)^2 + 1$

$2(0) + 1 + 1$

$0 + 1 + 1$

Value of the expression is: $2$

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(iii) $2a^2b + 2ab^2 + ab$

Substitute $a = 0$ and $b = -1$ into the expression:

$2(0)^2(-1) + 2(0)(-1)^2 + (0)(-1)$

$2(0)(-1) + 2(0)(1) + 0$

$0 + 0 + 0$

Value of the expression is: $0$

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(iv) $a^2 + ab + 2$

Substitute $a = 0$ and $b = -1$ into the expression:

$(0)^2 + (0)(-1) + 2$

$0 + 0 + 2$

Value of the expression is: $2$

Given $x = 2$. We need to simplify the expressions and find their values.

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(i) $x + 7 + 4 (x – 5)$

Simplify the expression:

$x + 7 + 4x - 4 \times 5$

$x + 7 + 4x - 20$

Combine like terms:

$(x + 4x) + (7 - 20)$

$5x - 13$

Now, substitute $x = 2$ into the simplified expression:

$5(2) - 13$

$10 - 13$

Value of the expression is: $-3$

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(ii) $3 (x + 2) + 5x – 7$

Simplify the expression:

$3x + 3 \times 2 + 5x - 7$

$3x + 6 + 5x - 7$

Combine like terms:

$(3x + 5x) + (6 - 7)$

$8x - 1$

Now, substitute $x = 2$ into the simplified expression:

$8(2) - 1$

$16 - 1$

Value of the expression is: $15$

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(iii) $6x + 5 (x – 2)$

Simplify the expression:

$6x + 5x - 5 \times 2$

$6x + 5x - 10$

Combine like terms:

$(6x + 5x) - 10$

$11x - 10$

Now, substitute $x = 2$ into the simplified expression:

$11(2) - 10$

$22 - 10$

Value of the expression is: $12$

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(iv) $4(2x – 1) + 3x + 11$

Simplify the expression:

$4 \times 2x - 4 \times 1 + 3x + 11$

$8x - 4 + 3x + 11$

Combine like terms:

$(8x + 3x) + (-4 + 11)$

$11x + 7$

Now, substitute $x = 2$ into the simplified expression:

$11(2) + 7$

$22 + 7$

Value of the expression is: $29$

Given $x = 3$, $a = -1$, and $b = -2$. We need to simplify the expressions and find their values.

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(i) $3x – 5 – x + 9$

Simplify the expression:

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

$2x + 4$

Now, substitute $x = 3$ into the simplified expression:

$2(3) + 4$

$6 + 4$

Value of the expression is: $10$

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(ii) $2 – 8x + 4x + 4$

Simplify the expression:

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

$-4x + 6$

Now, substitute $x = 3$ into the simplified expression:

$-4(3) + 6$

$-12 + 6$

Value of the expression is: $-6$

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(iii) $3a + 5 – 8a + 1$

Simplify the expression:

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

$-5a + 6$

Now, substitute $a = -1$ into the simplified expression:

$-5(-1) + 6$

$5 + 6$

Value of the expression is: $11$

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(iv) $10 – 3b – 4 – 5b$

Simplify the expression:

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

$-8b + 6$

Now, substitute $b = -2$ into the simplified expression:

$-8(-2) + 6$

$16 + 6$

Value of the expression is: $22$

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(v) $2a – 2b – 4 – 5 + a$

Simplify the expression:

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

$3a - 2b - 9$

Now, substitute $a = -1$ and $b = -2$ into the simplified expression:

$3(-1) - 2(-2) - 9$

$-3 + 4 - 9$

$1 - 9$

Value of the expression is: $-8$

(i) If $z = 10$, find the value of $z^3 – 3(z – 10)$.

Given expression: $z^3 – 3(z – 10)$

Given $z = 10$.

Substitute $z = 10$ into the expression:

$(10)^3 - 3(10 - 10)$

$1000 - 3(0)$

$1000 - 0$

Value of the expression is: $1000$

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(ii) If $p = – 10$, find the value of $p^2 – 2p – 100$.

Given expression: $p^2 – 2p – 100$

Given $p = -10$.

Substitute $p = -10$ into the expression:

$(-10)^2 - 2(-10) - 100$

$100 - (-20) - 100$

$100 + 20 - 100$

$120 - 100$

Value of the expression is: $20$

Given expression: $2x^2 + x – a$

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We are given that the value of the expression is $5$ when $x = 0$.

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Substitute $x = 0$ into the expression:

$2(0)^2 + (0) – a$

$2(0) + 0 – a$

$0 + 0 – a$

$-a$

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Now, we set this equal to the given value, $5$:

$-a = 5$

To find the value of $a$, multiply both sides of the equation by $-1$:

$a = -5$

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Therefore, the value of $a$ should be $-5$.

Given expression: $2(a^2 + ab) + 3 – ab$

Given values: $a = 5$ and $b = -3$.

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First, simplify the expression:

$2(a^2 + ab) + 3 – ab$

$2a^2 + 2ab + 3 – ab$

Combine like terms:

$2a^2 + (2ab – ab) + 3$

$2a^2 + ab + 3$

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Now, substitute the given values $a = 5$ and $b = -3$ into the simplified expression:

$2(5)^2 + (5)(-3) + 3$

$2(25) + (-15) + 3$

$50 - 15 + 3$

$35 + 3$

The value of the expression is: $38$