Algebraic Expressions and Solving Techniques
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Questions and Answers

What expression represents 'x increased by 12'?

  • x - 12
  • 12 - x
  • x + 12 (correct)
  • 1 + 12

    • The expression for 'y decreased by 7' is (7 - y).

    False (B)

    What is the sum of x and y represented as?

    x + y

    The product of x and y added to their sum is _____ + (x + y).

    <p>xy</p> Signup and view all the answers

    Which of the following represents '5 times x added to 7 times y'?

    <p>5x + 7y (D)</p> Signup and view all the answers

    X cubed less than y cubed is represented as y^2 - x^2.

    <p>False (B)</p> Signup and view all the answers

    What is the expression for 'one third of x multiplied by the sum of a and b'?

    <p>(1/3)x(a + b)</p> Signup and view all the answers

    Match the algebraic expressions to their descriptions:

    <p>5x + 7y = Five times x added to seven times y x - 2y = x minus twice y 80 + x = Total score including English and Hindi 3x + y^2 = Thrice x added to y squared</p> Signup and view all the answers

    What is the final simplified result of the expression $- a - [2b - a]$?

    <p>$-2b$ (C)</p> Signup and view all the answers

    The final result after simplifying the expression $5x - [4y - {7x - (3z - 2y) + 4z - 3(x + 3y - 2z)}]$ is $9x - 11y + 7z$.

    <p>True (A)</p> Signup and view all the answers

    What is the result of simplifying the expression $2a - [5b - 6a]$?

    <p>8a - 5b</p> Signup and view all the answers

    After removing the innermost grouping symbols from the expression $2a - [4b - {6a - b}]$, the next simplified result is __________.

    <p>5b - 6a</p> Signup and view all the answers

    Match the following expressions with their simplified results:

    <ul> <li>a - [2b - a] = -2b 2a - [5b - 6a] = 8a - 5b 5x - [4y - {7x - (3z - 2y) + 4z - 3(x + 3y - 2z)}] = 9x - 11y + 7z 2a - [4b - {4a - (b - 2a)}] = 5b - 6a</li> </ul> Signup and view all the answers

    What is the result of the expression $bbb*…$ repeated 15 times?

    <p>b^{15} (A)</p> Signup and view all the answers

    The expression $6 * x * x * y * y$ simplifies to $6x^2y^2$.

    <p>True (A)</p> Signup and view all the answers

    What is the solution for $(2×2)-(3×3)$ when $a=2$ and $b=3$?

    <p>-5</p> Signup and view all the answers

    The e______pression $3* z * z * z * y * y * $ can be simplified to $3z^3y^2$. Fill in the missing variable.

    <p>x</p> Signup and view all the answers

    Match the following substitutions with their results:

    <p>Substituting $x = 1$, $y = 2$, $z = 5$ in $3(1) - 2(2) + 4(5)$ = 19 Substituting $x = 1$, $y = 2$, $z = 5$ in $2 * 12 - 3 * 22 + 52$ = 15 Substituting $x = 1$, $y = 2$, $z = 5$ in $13 - 23 - 53$ = -132 Substituting $x = 1$, $y = 2$, $z = 5$ in $13 + 23 + 5$ = 134</p> Signup and view all the answers

    What is the final result when substituting $a = 2$ and $b = 3$ in $22 + (2 * 3)$?

    <p>10 (B)</p> Signup and view all the answers

    The result of the expression $23 - 33$ when $a = 2$ and $b = 3$ is $-19$.

    <p>True (A)</p> Signup and view all the answers

    What is the simplified form of $14 * a * a * a * a * b * b * b$?

    <p>14a^4b^3</p> Signup and view all the answers

    The result of $(−2)^2 + (−1)^2 − (3)^2$ is __________.

    <p>-4</p> Signup and view all the answers

    What is the simplified expression for the combination of terms $-3a - 3b + 8a - 12b - 2a + b$?

    <p>$3a - 14b$ (B)</p> Signup and view all the answers

    The expression $-4x^2 + (x^2 - 3) - (4 - 3x^2)$ simplifies to $x^2 - xy$.

    <p>False (B)</p> Signup and view all the answers

    What is the result after removing grouping symbols from the expression $a - [2b - {3a - (2b - 3c)}]$?

    <p>4a - 4b + 3c</p> Signup and view all the answers

    The expression $86 - [15x - 7(6x - 9) -2{10x - 5(2 - 3x)}]$ simplifies to _____ .

    <p>77x + 3</p> Signup and view all the answers

    Match the expressions with their simplified forms:

    <p>$-2x^2 + 2y^2 - 2xy - 3x^2 - 3y^2 + 3xy$ = $-5x^2 - y^2 + xy$ $-x + [10y - 3x]$ = $-4x + 10y$ $86 - [15x - 42x + 63 - 50x + 20]$ = $77x + 3$</p> Signup and view all the answers

    In the expression $-4x^2 + ext{{{group}}}$, which group should be removed first?

    <p>Group () (A)</p> Signup and view all the answers

    The expression $a - [4b - 3a - 3c]$ simplifies to $-4b + 4a + 3c$.

    <p>True (A)</p> Signup and view all the answers

    What is the simplified form of the expression $3a - [5b - (2a - 3b)]$?

    <p>5a - 3b</p> Signup and view all the answers

    What is the simplified result of the expression $3x + 7x$?

    <p>$10x$ (D)</p> Signup and view all the answers

    The expression $7y + (-9y)$ simplifies to $2y$.

    <p>False (B)</p> Signup and view all the answers

    What is the sum of the expressions $(3a - 2b + 5c)$ and $(2a + 5b - 7c)$, when also subtracting $(-a - b + c)$?

    <p>4a + 2b - c</p> Signup and view all the answers

    The result of $6a^3 - 4a^3 + 10a^3 - 8a^3$ equals _______.

    <p>4a^3</p> Signup and view all the answers

    Match the expressions with their simplified results:

    <p>$2x^3 - 5x^3 - x^3$ = $-4x^3$ $-4xy + 7y^2 + 4y^2$ = $10y^2$ $6x - 2x + 3$ = $4x + 3$ $-5 + 4x - 8$ = $4x - 13</p> Signup and view all the answers

    What is the simplified result of $2x^2 - 8xy + 7y^2 - 8xy^2 + 2xy^2 + 6xy - y^2 + 3x^2 + 4y^2 - xy - x^2 + xy^2$?

    <p>$4x^2 + 10y^2 - 3xy - 5xy^2$ (B)</p> Signup and view all the answers

    The expression $2 + x - x^2 + 6x^3$ can be rearranged to $6x^3 - x^2 + x + 2$.

    <p>True (A)</p> Signup and view all the answers

    The result of the expression $- x^3 + y^3 - z^3 + 3xyz$ when simplified is _______.

    <p>x^3 + y^3 - z^3 - 11xyz</p> Signup and view all the answers

    What is the result of the expression $5x + (-5x)$?

    <p>0</p> Signup and view all the answers

    What is the result of adding the following expressions: $(9x + 4y - 5z) + (3x + 3y - 4z)$?

    <p>$12x + 7y - 9z$ (A)</p> Signup and view all the answers

    It is necessary to change the sign of each term of the expression to be subtracted to compute the difference between two expressions.

    <p>True (A)</p> Signup and view all the answers

    What is the simplified form of $4x - (3y - x + 2z)$?

    <p>5x - 3y - 2z</p> Signup and view all the answers

    The simplified result of $2x - 3y + 4z - (2x + 5y - 6z + 2)$ is __________.

    <p>-8y + 10z - 2</p> Signup and view all the answers

    Match the following expressions with their simplified forms:

    <p>2x - 3y + 4z + (-2x - 5y + 6z - 2) = -8y + 10z - 2 4x - (3y - x + 2z) = 5x - 3y - 2z 1 - (2x - 3y - 4) = 5 - 2x + 3y a^2 + b^2 + 2ab - (a^2 + b^2 - 2ab) = 4ab</p> Signup and view all the answers

    What does the expression $1 - (2x - 3y - 4)$ simplify to?

    <p>$5 - 2x + 3y$ (C)</p> Signup and view all the answers

    Simplifying the expression $a^2 + b^2 + 2ab - a^2 - b^2 + 2ab$ yields $4ab$.

    <p>True (A)</p> Signup and view all the answers

    When calculating how much $2x - 3y + 4z$ is greater than $2x + 5y - 6z + 2$, the expression to be subtracted from the first is __________.

    <p>2x + 5y - 6z + 2</p> Signup and view all the answers

    Flashcards

    x increased by 12

    The result of adding 12 to the variable x.

    y decreased by 7

    The result of subtracting 7 from the variable y.

    x multiplied by itself

    The result of multiplying the variable x by itself.

    Thrice x added to y squared

    The result of multiplying the variable x by 3 and then adding the square of the variable y.

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    The quotient of x by 8 is multiplied by y

    The result of dividing the variable x by 8 and then multiplying the quotient by the variable y.

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    5 times x added to 7 times y

    The result of adding 5 times the variable x to 7 times the variable y.

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    x minus twice y

    The result of subtracting twice the variable y from the variable x.

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    The quotient of y by 5

    The result of dividing the variable y by 5.

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    Comparing expressions

    To find how much greater one expression is than another, subtract the smaller expression from the larger one.

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    Subtracting expressions

    When subtracting an expression, change the sign of each term in that expression and then add.

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    Simplifying expressions with parentheses

    To simplify an expression with parentheses preceded by a '-' sign, remove the parentheses and change the sign of each term within the parentheses.

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    Combining like terms

    Combining like terms involves adding or subtracting coefficients of terms with the same variables and exponents.

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    Simplifying algebraic expressions

    An algebraic expression can be simplified by combining like terms after removing parentheses.

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    Finding the excess

    To find how much one expression exceeds another, subtract the smaller expression from the larger one.

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    Simplifying with parentheses

    When simplifying algebraic expressions, change the signs of terms within parentheses preceded by a '-' sign.

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    Combining like terms in simplification

    Combining like terms involves grouping terms with the same variables and exponents and then adding or subtracting their coefficients.

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    What does a variable raised to a power indicate?

    A variable raised to a power indicates repeated multiplication of the variable by itself.

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    What is a coefficient?

    A coefficient is the number that multiplies a variable or a product of variables.

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    Explain power of a variable

    A power of a variable represents the number of times the variable is multiplied by itself. For example, x raised to the power of 3 (x^3) means x multiplied by itself three times: x * x * x.

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    How to evaluate an expression?

    To evaluate an expression, you substitute the given values for variables and then perform the calculations according to the order of operations (PEMDAS/BODMAS).

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    What is substitution ?

    Substituting values involves replacing variables in an expression with their numerical values. For example, if x = 2, then substituting x in "2x + 3" would give "2(2) + 3".

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    What is the order of operations?

    The order of operations (PEMDAS/BODMAS) governs the sequence in which operations are performed in an expression. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). BODMAS stands for Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

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    How to simplify an expression ?

    To simplify an expression, you combine like terms and perform calculations according to the order of operations.

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    What are like terms?

    Like terms are terms in an expression that have the same variable and exponent, allowing them to be combined. For example, 2x and 3x are like terms, but 2x and 3x^2 are not.

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    How to combine like terms?

    Combining like terms involves adding or subtracting coefficients of terms with the same variable and exponent.

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    What is an equation?

    An equation states that two expressions are equal. For example, 2x + 5 = 15.

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    Adding Algebraic Expressions

    Adding algebraic expressions involves combining like terms. Like terms have the same variables and exponents. Example: 3x and 7x are like terms, but 3x and 7y are not.

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    Adding Like Terms

    Combine the coefficients of like terms to find the sum. Example: 3x + 7x = (3 + 7)x = 10x

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    Subtracting Algebraic Expressions

    Subtracting algebraic expressions involves changing the sign of each term in the expression being subtracted and then adding the expressions. Example: 5x - 2y is the same as 5x + (-2y)

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    Subtracting Like Terms

    When subtracting like terms, combine the coefficients with the opposite sign of the term being subtracted. Example: 3x - 5x = 3x + (-5)x = -2x

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    Subtracting Expressions with Multiple Terms

    When subtracting algebraic expressions with multiple terms, distribute the negative sign to each term in the expression being subtracted - changing the sign before each term. Example: (3x - 2y) - (x + 4y) becomes 3x - 2y - x - 4y

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    Rearranging Terms

    Rearrange the terms of an expression so that like terms are grouped together. Then, combine the coefficients of like terms to simplify the expression. Example: (2a + 5b - c) + (3a - 2b + 4c) can be rearranged as 2a + 3a + 5b - 2b - c + 4c.

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    Combining Like Terms with Exponents

    When adding or subtracting algebraic expressions, we can only combine terms with the same variables and powers (exponents). Example: We can combine 3x^2 and 5x^2 but not 3x^2 and 5x.

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    Subtracting Expressions with Multiple Terms and Exponents

    Subtracting expressions with multiple terms involves changing the sign of each term in the expression being subtracted and then combining like terms. Example: (2x^2 - 3xy + 5y^2) - (-x^2 + xy - 2y^2) becomes 2x^2 - 3xy + 5y^2 + x^2 - xy + 2y^2

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    Removing Grouping Symbols

    Removing grouping symbols in expressions involves simplifying the expression by applying the order of operations and combining like terms.

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    Distributing a Minus Sign

    When a minus sign precedes parentheses, we distribute the minus sign to each term inside. This changes the sign of each term.

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    Nested Grouping Symbols

    To remove nested grouping symbols, we work from the innermost to the outermost.

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    Simplifying Expressions

    Combining like terms simplifies the expression by grouping similar terms together.

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    Simplifying Expressions (General)

    Expressions can be simplified using the order of operations, including removing grouping symbols, distributing, and combining like terms.

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    Grouping Terms

    In expressions involving variables, we group terms with the same variables and exponents together.

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    Simplifying Expressions (Recap)

    Simplifying expressions requires understanding the order of operations, distributing signs, and combining like terms.

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    Order of Grouping Symbol Removal

    Removing the innermost grouping symbols first, then proceeding outward to the outermost symbols. This method simplifies complex expressions by breaking them down systematically.

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    Distributive Property

    Multiplying a number or variable by the terms within parentheses. It's like distributing the number or variable to each term inside.

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    Study Notes

    Algebraic Expressions

    • Algebraic expressions combine constants and variables using mathematical operations.
    • Variables represent unknown values, and constants are fixed numerical values.
    • Exponents: Represent repeated multiplication of a variable.
    • A common mathematical operation is increasing or decreasing by a number, for instance multiplying by or dividing by, and taking a difference and sum.

    Solving Algebraic Expressions

    • Given different expressions, identify and perform calculations according to the question to arrive at the solution.
    • Evaluate the algebraic expressions when specific values are assigned to the variables.
    • Substitution: Replace variables with their assigned values.
    • Order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)

    Like Terms

    • Like terms have the same variables raised to the same power.
    • Combine like terms by adding or subtracting their coefficients.

    Simplifying Expressions

    • Simplify expressions by combining like terms and performing operations.
    • Apply order of operations when needed.

    Classifying Expressions

    • Monomial: A single term (e.g., 3x, 5y²).
    • Binomial: Two terms (e.g., 2x + 3y, 7x² - 2).
    • Trinomial: Three terms (e.g., 4x² - 5x + 7, -6 - 2y + 3z).
    • If an expression contains more than three terms, it does not fall into these categories.

    Constants in an Expression

    • The constant terms in an expression are the fixed values.
    • In the expression 3x² + 5x + 8, the constant term is 8.

    Coefficient of a Variable

    • The numerical factor in a term is called the coefficient.
    • In the term -7x, the coefficient is -7.

    Numerical Coefficients

    • The numerical factor in a term is termed as the numerical coefficient.
    • In the term 8xy, the numerical coefficient is 8.

    Examples of Algebraic Expressions with Solutions

    • Example 1: 5x + 2y = 100. Solve for x with y= 10.

      • Substituting y = 10: 5x + 2(10) = 100
      • Simplifying: 5x + 20 = 100
        • Isolate x: 5x = 80, and x = 16

    • Example 2: x² - 4x = 5x. Find x.

      • Rearrange to 𝑥²⁻9𝑥 = 0
      • Factorize to 𝑥(𝑥−9) = 0
      • So either x=0 or x = 9

    • Example 3: The term "twice x increased by y" is equivalent to 2𝑥+𝑦.

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    RS Aggarwal Solutions Class 6 Chapter-8 Algebraic Expressions (Ex 8A) Exercise 8.1 PDF

    Description

    This quiz focuses on algebraic expressions, their components, and solving techniques. Learn about variables, constants, exponents, and the importance of order of operations. Engage in identifying and evaluating expressions through substitution and combining like terms.

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