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Equivalent Expressions

Simplification of Expressions Involving Addition and Subtraction

  • Simplification means the process of making things easier.
  • We can simplify the expressions having various operations.
  • Here, we will learn to simplify the expressions involving two operations: addition and subtraction.

For example: 

(i) Simplify:

\(3x+5y+7z+2x+y+3z\)

First, arrange the like terms together.

\(3x+2x+5y+y+7z+3z\)

Now, take the common variables out.

\(x(3+2)+y(5+1)+z(7+3)\)

\(=x(5)+y(6)+z(10)\)

\(=5x+6y+10z\)

(ii) Simplify:

\(3x-7y-2z-x-3y-z\)

First, arrange the like terms together.

\(3x-x-7y-3y-2z-z\)

Now, take the common variables out.

\(x(3-1)+y(-7-3)+z(-2-1)\)

\(=x(2)+y(-10)+z(-3)\)

\(=2x-10y-3z\)

Illustration Questions

Simplify the given expression:   \(3s-2t-4s-7t+7\)

A \(s-9t+3\)

B \(-s+t+7\)

C \(-s-9t+7\)

D \(-s+9t-7\)

×

Given expression:

\(3s-2t-4s-7t+7\)

First, arrange the like terms together.

\(3s-4s-2t-7t+7\)

Now, take the common variables out.

\(s(3-4)+t(-2-7)+7\)

\(=s(-1)+t(-9)+7\)

\(=-s-9t+7\)

Hence, option (C) is correct.

Simplify the given expression:   \(3s-2t-4s-7t+7\)

A

\(s-9t+3\)

.

B

\(-s+t+7\)

C

\(-s-9t+7\)

D

\(-s+9t-7\)

Option C is Correct

Simplification of Expressions Involving Multiple Operations with Single Variable

  • Simplification means the process of making things easier.
  • We can simplify the expressions having various operations.
  • Here, we will learn to simplify the expression involving multiple operations i.e., addition, subtraction, multiplication and/or division.

Example:

(i) \(6b^3-6b^2\)

Expanded form of the given expression is: \(6×b×b×b-6×b×b\)

Here, \(6\) and two \(b\)'s are common in both the terms.

Thus, we can simplify the expression by taking the commons out i.e., \(6\) and two \(b\)'s.

\(6×b×b×b-6×b×b\)

\(=6×b×b(b-1)\)

\(=6b^2(b-1)\)

(ii) \(\dfrac{6d^2}{3d}+\dfrac{10d^3}{5d^2}\)

Expanded form of the given expression is: \(\dfrac{6×d×d}{3×d}+\dfrac{10×d×d×d}{5×d×d}\)

In expressions involving division, similar variables in numerator and denominator get canceled out by each other.

\(\dfrac{\not{6}^2×d×\not{d}}{\not{3}×\not{d}}+\dfrac{\not{10}^2×d×\not{d}×\not{d}}{\not{5}×\not{d}×\not{d}}\)

\(=2×d+2×d\)

\(=2d(1+1)\)

\(=2d(2)\)

\(=4d\)

Illustration Questions

Simplify the given expression:   \(\dfrac{4a^5+8a^4-4a^2}{4a^2}\)

A \(4(3a^2+4a-3)\)

B \(a(2a^2+4a-3)\)

C \(3(a^2+2a-3)\)

D \((a^3+2a^2-1)\)

×

Given expression:

\(\dfrac{4a^5+8a^4-4a^2}{4a^2}\)

Expanded form of the given expression is:

\(\dfrac{4×a×a×a×a×a+8×a×a×a×a-4×a×a}{4×a×a}\)

Here \(4\) and \(2a\)'s are common in all the terms of the numerator.

\(\dfrac{4×a×a[a×a×a+2×a×a-1]}{4×a×a}\)

Cancel out the same variables and numbers appearing in numerator and denominator.

\(\dfrac{\not{4}×\not{a}×\not{a}}{\not{4}×\not{a}×\not{a}}\;[a×a×a+2×a×a-1]=a^3+2a^2-1\)

Hence, option (D) is correct.

Simplify the given expression:   \(\dfrac{4a^5+8a^4-4a^2}{4a^2}\)

A

\(4(3a^2+4a-3)\)

.

B

\(a(2a^2+4a-3)\)

C

\(3(a^2+2a-3)\)

D

\((a^3+2a^2-1)\)

Option D is Correct

Simplification of Expressions Involving Multiple Operations with Two or more Variables

  • Simplification means the process of making things easier.
  • We can simplify the expressions having various operations.
  • Here, we will learn to simplify the expression involving multiple operations i.e., addition, subtraction, multiplication and/or division.

Example:

(i) \(2abc+5abc+7a^3\)

Expanded form of the given expression is:

\(2×a×b×c+5×a×b×c+7×a×a×a\)

Now, we take the common variables out.

\(2×a×b×c+5×a×b×c+7×a×a×a\)

\(=2abc+5abc+7a^3\)

\(=abc(2+5)+7a^3\)

\(=abc(7)+7a^3\)

\(=7abc+7a^3\)

Here, \(7\) and one \('a'\), are common in both the terms.

\(=7a(bc+a^2)\)

(ii) \(\dfrac{3ab^2c-6b^2c^2-3ab^2c^2}{3b^2c}\)

First, we take the common variables and numbers out and then we will divide.

\(=\dfrac{3ab^2c-6b^2c^2-3ab^2c^2}{3b^2c}\)

\(=\dfrac{3b^2c(a-2c-ac)}{3b^2c}\)

\(=a-2c-ac\)

\(=a-c(2+a)\)

Illustration Questions

Simplify the given expression:   \(\dfrac{8x^2y^2z-2xy^2zt+7xyz}{4xyz}\)

A \(\dfrac{3y(4x+7)+t}{4}\)

B \(\dfrac{7xy-4t}{8}\)

C \(\dfrac{8xy-2xy}{4}\)

D \(\dfrac{2y(4x-t)+7}{4}\)

×

Given expression:

\(\dfrac{8x^2y^2z-2xy^2zt+7xyz}{4xyz}\)

Take the common variables and numbers out.

\(\dfrac{xyz(8xy-2yt+7)}{4xyz}\)

Divide the terms of numerator by the denominator.

\(\dfrac{xyz(8xy-2yt+7)}{4xyz}\)

\(=\dfrac{8xy-2yt+7}{4}\)

Simplifying it further:

\(\dfrac{8xy-2yt+7}{4}=\dfrac{2y(4x-t)+7}{4}\)

Hence, option (D) is correct.

Simplify the given expression:   \(\dfrac{8x^2y^2z-2xy^2zt+7xyz}{4xyz}\)

A

\(\dfrac{3y(4x+7)+t}{4}\)

.

B

\(\dfrac{7xy-4t}{8}\)

C

\(\dfrac{8xy-2xy}{4}\)

D

\(\dfrac{2y(4x-t)+7}{4}\)

Option D is Correct

Finding Equivalent Expressions

Equivalent Expressions

  • If two expressions are equal, they are called equivalent expressions.
  • To find the equivalent expression, keep the following rules in mind:
  1. Commutative Property of Addition: On changing the order of numbers, their sum does not change. This is known as commutative property of addition, i.e.

 \(a+b=b+a\)

      2. Commutative Property of Multiplication: On changing the order of numbers, their product does not change. This is known as commutative property of multiplication, i.e.

 \(a×b=b×a\) 

      3. Associative Property of Addition: On changing the grouping of numbers, their sum does not change. This is known as associative property of addition, i.e.

\(a+(b+c)=(a+b)+c\)  

      4. Associative Property of Multiplication: On changing the grouping of numbers, their product does not change. This is known as associative property of multiplication, i.e.

\(a×(b×c)=(a×b)×c\)

      5. Distributive Property of Multiplication over Addition: The distributive property of multiplication over addition states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the product together, i.e.

\(a(b+c)=a×b+a×c\)

      6. Distributive Property of Multiplication over Subtraction: The distributive property of multiplication over subtraction is like the distributive property over addition. Either we find out the difference first and then multiply, or we first multiply with each number and then subtract, the result will always be same, i.e.

\(a(b-c)=ab-ac\)

      7. Like terms can be added and subtracted.

Illustration Questions

Which is the equivalent term of \(5(6+x)\)?

A \(5(6x)\)

B \(30+5x\)

C \(6(5+x)\)

D \(30x\)

×

Given: \(5(6+x)\)

 

Here, distributive property of multiplication over addition can be used.

\([a(b+c)=ab+ac]\)

Thus, the equivalent term of \(5(6+x)\) is

\(5(6+x)=5×6+5x\)

\(5(6+x)=30+5x\)

Hence, option (B) is correct.

Which is the equivalent term of \(5(6+x)\)?

A

\(5(6x)\)

.

B

\(30+5x\)

C

\(6(5+x)\)

D

\(30x\)

Option B is Correct

Factors of an Expression

  • If a term or an expression can completely divide a given algebraic expression, then that term or the expression is the factor of the given algebraic expression.
  • To determine factors of an algebraic expression, we need to follow the steps described in the given example.

Example: \(64\,abc-32\,ab+24\,abc\)

\(\Rightarrow\) By using the distributive property, take out the common factor.

Common factor is \(8\,ab\) .

\(\Rightarrow\) Rewrite the expression as:

\(8\,ab(8c-4+3c)\)

\(\Rightarrow\) The factors of above expression are the terms or expression which can divide the expression.

\(\Rightarrow\) \(8ab(8c-4+3c)\) can be written as:

 \(2×2×2×a×b(8c-4+3c)\) or \(2×4×a×b(8c-4+3c)\)

Thus, the factors are \(2,\;2a,\;2b,\;ab,\;a,\;b,\;(8c-4+3c),\;4,\;8,\;4a,\;4b,\;8a,\;8b,\;8ab\).

Illustration Questions

Which of the following options represents the factor of the expression \(81xyz-54xy+27yz\)?

A \(3y\)

B \(5y\)

C \(4y\)

D \(2y\)

×

Given: \(81xyz-54xy+27yz\)

By using the distributive property, take out the common factor which is \(27y\).

Rewriting the expression as:

 \(27y\,(3xz-2x+z)\)

It can be written as \(3×3×3y\,(3xz-2x+z)\) or \(3×9×y\,(3xz-2x+z)\)

We know that the factors of an expression are the terms or expression which can divide the given expression.

Therefore, the factors of \(81xyz-54xy+27yz\) will be:

\((3xz-2x+z),\;3,\;9,\;27,\;3y,\;9y,\;27y,\;y\)

Hence, option (A) is correct.

Which of the following options represents the factor of the expression \(81xyz-54xy+27yz\)?

A

\(3y\)

.

B

\(5y\)

C

\(4y\)

D

\(2y\)

Option A is Correct

Simplification of Expressions Involving Multiplication and Division

  • Simplification means the process of making things easier.
  • We can simplify the expressions having various operations.
  • Here, we will learn to simplify the expressions involving two operations: multiplication and division.

Example:

(i) \(4×a×a×b×2×c×b\)

Multiply the numbers and count the number of times each variable is multiplied by itself and write it as its power.

\(4×a×a×b×2×c×b\)

\(=4×2×a×a×b×b×c\)

\(=8a^2b^2c\)

 

(ii) \(\dfrac{16×a×b×a×b×c}{c×a×b×8}\)

In such type of expressions, we can cancel out the same variables and numbers appearing in both numerator and denominator.

\(\dfrac{16×a×\not{a}×\not{b}×b×\not{c}}{8×\not{a}×\not{b}×\not{c}}\)

\(=\dfrac{16×a×b}{8}\)

\(\because\) Factors of \(16=8,\;2,\;4\)

So, the expression can be written as,

\(\dfrac{8×2×a×b}{8}=2×a×b=2ab\)

Illustration Questions

Simplify the given expression:   \(\dfrac{27×x×y×z×x×x×y}{3×a×y×z}\)

A \(\dfrac{27x^2z}{ay}\)

B \(\dfrac{9x^3y}{a}\)

C \(\dfrac{3xyz}{a}\)

D \(\dfrac{2x^2y}{3a}\)

×

Given expression:

\(\dfrac{27×x×y×z×x×x×y}{3×a×y×z}\)

Arrange the same variables together.

\(\dfrac{27×x×x×x×y×y×z}{3×a×y×z}\)

Cancel out the same numbers and variables appearing in both numerator and denominator.

\(\dfrac{27×x×x×x×y×\not{y}×\not{z}}{3×a×\not{y}×\not{z}}\)

\(=\dfrac{27×x×x×x×y}{3×a}\)

\(\because\) Factors of \(27=3,\;9\)

So, the expression can be written as:

\(=\dfrac{9×\not{3}×x×x×x×y}{\not{3}×a}\)

\(=\dfrac{9×x××x×x×y}{a}\)

\(=\dfrac{9x^3y}{a}\)

Hence, option (B) is correct.

Simplify the given expression:   \(\dfrac{27×x×y×z×x×x×y}{3×a×y×z}\)

A

\(\dfrac{27x^2z}{ay}\)

.

B

\(\dfrac{9x^3y}{a}\)

C

\(\dfrac{3xyz}{a}\)

D

\(\dfrac{2x^2y}{3a}\)

Option B is Correct

Equivalent Expressions

  • When two expressions are equal, they are called equivalent expressions.
  • To find equivalent expressions, use the following properties:

(1) Commutative property of addition: On changing the order of numbers, their sum does not change. This is known as commutative property of addition.

i.e. \(a+b=b+a\)

(2) Commutative property of multiplication: On changing the order of numbers, their product does not change. This is known as commutative property of multiplication.

i.e. \(a×b=b×a\)

(3) Associative property of addition: On changing the grouping of numbers, their sum does not change. This is known as associative property of addition.

i.e. \(a+(b+c)=(a+b)+c\)

(4) Associative property of multiplication: On changing the grouping of numbers, their product does not change. This is known as associative property of multiplication.

i.e. \(a×(b×c)=(a×b)×c\)

(5) Distributive property of multiplication over addition: The distributive property of multiplication over addition states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.

i.e. \(a(b+c)=a×b+a×c\)

(6) Distributive property of multiplication over subtraction: The distributive property of multiplication over subtraction is like the distributive property over addition. Either we find out the difference first and then multiply or first we multiply with each number and then subtract, result will always be same.

i.e. \(a(b-c)=ab-ac\)

  • The algebraic expression can also be represented by tape diagrams.

For example: \(x+1\)

Here, \(x\) is an unknown quantity.

Now, if \((x+1)\) is multiplied with \(4,\) then, it can be represented as shown in figure.

This is just similar to the distributive property.

As, \(4(x+1)=4x+4\)

Thus, distributive property can be represented through diagrams.

Like terms can be added and subtracted.

Example:

  • Equivalent expression involving one variable

\(4a(a+2a)\)

\(\Rightarrow\;(4a×a)+(4a×2a)\)

\(\Rightarrow\;4a^2+8a^2\)

\(\Rightarrow\;a^2(4+8)\)

\(\Rightarrow\;a^2(12)\)

Thus, \(4a(a+2a)=12a^2\)

Both expressions are equivalent.

  • Equivalent expressions involving two variables

\(6(x+y)\)

\(\Rightarrow\;6x+6y\)

Thus, both expressions are equivalent.

  • Equivalent expression involving three variables

\(x(2x+3y+4z)\)

\(\Rightarrow\;(x×2x)+(x×3y)+(x×4z)\)

\(\Rightarrow\;2x^2+3xy+4xz\)

Thus, \(x(2x+3y+4z)=2x^2+3xy+4xz\)

Both expressions are equivalent.

Like terms can be added and subtracted.

Example:

  • Equivalent expression Involving one variable:

\(4a(a+2a)\)

\(\Rightarrow\;(4a×a)+(4a×2a)\)

\(\Rightarrow\;4a^2+8a^2\)

\(\Rightarrow\;a^2(4+8)\)

\(\Rightarrow\;a^2(12)\)

Thus, \(4a(a+2a)=12a^2\)

Both expression are equivalent.

  • Equivalent expression involving two variables:

\(6(x+y)\)

\(\Rightarrow\;6x+6y\)

Thus, both expression are equivalent.

Illustration Questions

Which one of the following options is equivalent to the given expression \(2(x+3)\;(y+7)+3(z+4)\)?

A \(2xy+14x+6y+3z+54\)

B \(4xy+7y\)

C \((x+3)\;(y+7)\;(z+4)\)

D \(2xy-14x-6y\)

×

Given:

\(2(x+3)\;(y+7)+3(z+4)\)

Here, distributive property of multiplication over addition is to be used.

\([a(b+c)=ab+ac]\)

Thus, the equivalent expression of \(2(x+3)\;(y+7)+3(z+4)\) is

\(2(x+3)\;(y+7)+3(z+4)\)

\(\Rightarrow\;[(2.x)+(2.3)](y+7)+(3.z)+(3.4)\)

\(\Rightarrow\;(2x+6)(y+7)+(3z+12)\)

\(\Rightarrow\;(2x.y)+(2x.7)+(6.y)+(6.7)+(3z+12)\)

\(\Rightarrow\;2xy+14x+6y+42+3z+12\)

\(\Rightarrow\;2xy+14x+6y+3z+54\)

Hence, option (A) is correct.

Which one of the following options is equivalent to the given expression \(2(x+3)\;(y+7)+3(z+4)\)?

A

\(2xy+14x+6y+3z+54\)

.

B

\(4xy+7y\)

C

\((x+3)\;(y+7)\;(z+4)\)

D

\(2xy-14x-6y\)

Option A is Correct

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