41. Simplification of expressions problems

41. Simplification of expressions problems - step by step solution procedure and detailed explanation for each problem.

Below are 10 simplification of expressions problems, each with a step-by-step solution procedure and a detailed explanation. These problems involve algebraic expressions and focus on simplifying them by combining like terms, applying the distributive property, handling exponents, and using the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction) where applicable. The problems range from basic to moderately complex to cover various simplification techniques.
Problem 1: Simplify
3x + 5x - 2x
Step-by-Step Solution:
  1. Identify like terms: The terms ( 3x ), ( 5x ), and
    -2x
    all have the same variable ( x ).
  2. Combine like terms: Add or subtract the coefficients of ( x ).
    • 3x + 5x = 8x
    • 8x - 2x = 6x
Final Answer: ( 6x )
Detailed Explanation:
  • Like Terms: Terms with the same variable and exponent (here,
    x^1
    ) can be combined by adding or subtracting their coefficients.
  • Common Mistake: Treating unlike terms (e.g., ( x ) and
    x^2
    ) as combinable.
  • Verification: Think of ( x ) as a placeholder (e.g., let
    x = 2
    ). Then,
    3(2) + 5(2) - 2(2) = 6 + 10 - 4 = 12
    . Simplified:
    6x = 6(2) = 12
    .
Problem 2: Simplify
4(y + 2) - 3y
Step-by-Step Solution:
  1. Apply distributive property: Distribute the 4 to terms inside the parentheses.
    • 4(y + 2) = 4y + 8
  2. Rewrite expression:
    4y + 8 - 3y
  3. Combine like terms:
    • For ( y )-terms:
      4y - 3y = y
    • Constant term: ( 8 )
  4. Write simplified expression:
    y + 8
Final Answer:
y + 8
Detailed Explanation:
  • Distributive Property:
    a(b + c) = ab + ac
    . Here,
    4(y + 2)
    becomes
    4y + 4 \cdot 2
    .
  • Like Terms: Combine ( 4y ) and
    -3y
    to get ( y ). The constant ( 8 ) has no like terms.
  • Common Mistake: Forgetting to distribute (e.g., treating
    4(y + 2)
    as
    4y + 2
    ).
  • Verification: Let
    y = 1
    . Original:
    4(1 + 2) - 3(1) = 4 \cdot 3 - 3 = 12 - 3 = 9
    . Simplified:
    1 + 8 = 9
    .
Problem 3: Simplify
2x^2 + 3x - x^2 + 4x
Step-by-Step Solution:
  1. Identify like terms:
    • Terms with
      x^2
      :
      2x^2
      ,
      -x^2
    • Terms with ( x ): ( 3x ), ( 4x )
  2. Combine like terms:
    • For
      x^2
      :
      2x^2 - x^2 = x^2
    • For ( x ):
      3x + 4x = 7x
  3. Write simplified expression:
    x^2 + 7x
Final Answer:
x^2 + 7x
Detailed Explanation:
  • Like Terms: Combine terms with the same variable and exponent.
    x^2
    and ( x ) are not like terms.
  • Common Mistake: Combining
    x^2
    and ( x ) (e.g.,
    2x^2 + 3x = 5x^3
    ), which is incorrect.
  • Verification: Let
    x = 2
    . Original:
    2(2^2) + 3(2) - (2^2) + 4(2) = 8 + 6 - 4 + 8 = 18
    . Simplified:
    2^2 + 7(2) = 4 + 14 = 18
    .
Problem 4: Simplify
5(2a - 3) + 4a
Step-by-Step Solution:
  1. Apply distributive property: Distribute the 5.
    • 5(2a - 3) = 10a - 15
  2. Rewrite expression:
    10a - 15 + 4a
  3. Combine like terms:
    • For ( a )-terms:
      10a + 4a = 14a
    • Constant:
      -15
  4. Write simplified expression:
    14a - 15
Final Answer:
14a - 15
Detailed Explanation:
  • Distributive Property: Ensures each term inside the parentheses is multiplied by 5.
  • Common Mistake: Incorrect distribution (e.g.,
    5(2a - 3) = 10a - 3
    ).
  • Verification: Let
    a = 1
    . Original:
    5(2 \cdot 1 - 3) + 4 \cdot 1 = 5(2 - 3) + 4 = 5(-1) + 4 = -5 + 4 = -1
    . Simplified:
    14(1) - 15 = 14 - 15 = -1
    .
Problem 5: Simplify
\frac{6x^2 + 9x}{3}
Step-by-Step Solution:
  1. Split the fraction: Divide each term in the numerator by the denominator.
    • \frac{6x^2 + 9x}{3} = \frac{6x^2}{3} + \frac{9x}{3}
  2. Simplify each term:
    • \frac{6x^2}{3} = 2x^2
    • \frac{9x}{3} = 3x
  3. Write simplified expression:
    2x^2 + 3x
Final Answer:
2x^2 + 3x
Detailed Explanation:
  • Fraction Simplification: The denominator is distributed to each term in the numerator.
  • Common Mistake: Dividing only one term (e.g.,
    \frac{6x^2}{3} + 9x = 2x^2 + 9x
    ).
  • Verification: Let
    x = 1
    . Original:
    \frac{6(1^2) + 9(1)}{3} = \frac{6 + 9}{3} = \frac{15}{3} = 5
    . Simplified:
    2(1^2) + 3(1) = 2 + 3 = 5
    .
Problem 6: Simplify
(2x + 3)(x - 1)
Step-by-Step Solution:
  1. Apply distributive property (FOIL method):
    • First:
      2x \cdot x = 2x^2
    • Outer:
      2x \cdot (-1) = -2x
    • Inner:
      3 \cdot x = 3x
    • Last:
      3 \cdot (-1) = -3
  2. Write expression:
    2x^2 - 2x + 3x - 3
  3. Combine like terms:
    • For ( x )-terms:
      -2x + 3x = x
  4. Write simplified expression:
    2x^2 + x - 3
Final Answer:
2x^2 + x - 3
Detailed Explanation:
  • FOIL Method: Ensures all combinations of terms are multiplied.
  • Common Mistake: Missing terms during distribution (e.g., forgetting
    -2x
    ).
  • Verification: Let
    x = 2
    . Original:
    (2 \cdot 2 + 3)(2 - 1) = (4 + 3)(1) = 7
    . Simplified:
    2(2^2) + 2 - 3 = 2 \cdot 4 + 2 - 3 = 8 + 2 - 3 = 7
    .
Problem 7: Simplify
3x - 2(x - 4) + 5
Step-by-Step Solution:
  1. Apply distributive property: Distribute the
    -2
    .
    • -2(x - 4) = -2x + 8
  2. Rewrite expression:
    3x - 2x + 8 + 5
  3. Combine like terms:
    • For ( x )-terms:
      3x - 2x = x
    • Constants:
      8 + 5 = 13
  4. Write simplified expression:
    x + 13
Final Answer:
x + 13
Detailed Explanation:
  • Distributive Property: The negative sign affects both terms inside the parentheses.
  • Common Mistake: Incorrectly distributing the negative (e.g.,
    -2(x - 4) = -2x - 8
    ).
  • Verification: Let
    x = 1
    . Original:
    3(1) - 2(1 - 4) + 5 = 3 - 2(-3) + 5 = 3 + 6 + 5 = 14
    . Simplified:
    1 + 13 = 14
    .
Problem 8: Simplify
\frac{12x^3 - 8x^2}{4x}
Step-by-Step Solution:
  1. Split the fraction: Divide each term by ( 4x ).
    • \frac{12x^3 - 8x^2}{4x} = \frac{12x^3}{4x} - \frac{8x^2}{4x}
  2. Simplify each term:
    • \frac{12x^3}{4x} = \frac{12}{4} \cdot \frac{x^3}{x} = 3x^2
    • \frac{8x^2}{4x} = \frac{8}{4} \cdot \frac{x^2}{x} = 2x
  3. Write simplified expression:
    3x^2 - 2x
Final Answer:
3x^2 - 2x
Detailed Explanation:
  • Exponent Rules: When dividing
    x^m / x^n
    , subtract exponents:
    x^{m-n}
    .
  • Common Mistake: Incorrectly canceling ( x ) (e.g.,
    \frac{12x^3}{4x} = 3x^3
    ).
  • Verification: Let
    x = 1
    . Original:
    \frac{12(1^3) - 8(1^2)}{4 \cdot 1} = \frac{12 - 8}{4} = \frac{4}{4} = 1
    . Simplified:
    3(1^2) - 2(1) = 3 - 2 = 1
    .
Problem 9: Simplify
(3x^2 - 2x + 1) + (x^2 + 4x - 5)
Step-by-Step Solution:
  1. Remove parentheses: Combine terms directly.
    • (3x^2 - 2x + 1) + (x^2 + 4x - 5) = 3x^2 - 2x + 1 + x^2 + 4x - 5
  2. Group like terms:
    • For
      x^2
      :
      3x^2 + x^2 = 4x^2
    • For ( x ):
      -2x + 4x = 2x
    • Constants:
      1 - 5 = -4
  3. Write simplified expression:
    4x^2 + 2x - 4
Final Answer:
4x^2 + 2x - 4
Detailed Explanation:
  • Combining Polynomials: Group and combine like terms based on their degree.
  • Common Mistake: Forgetting to combine constants or mismatching terms.
  • Verification: Let
    x = 1
    . Original:
    (3(1^2) - 2(1) + 1) + (1^2 + 4(1) - 5) = (3 - 2 + 1) + (1 + 4 - 5) = 2 + 0 = 2
    . Simplified:
    4(1^2) + 2(1) - 4 = 4 + 2 - 4 = 2
    .
Problem 10: Simplify
2x(3x - 4) - x(2x + 5)
Step-by-Step Solution:
  1. Apply distributive property:
    • 2x(3x - 4) = 6x^2 - 8x
    • -x(2x + 5) = -2x^2 - 5x
  2. Rewrite expression:
    6x^2 - 8x - 2x^2 - 5x
  3. Combine like terms:
    • For
      x^2
      :
      6x^2 - 2x^2 = 4x^2
    • For ( x ):
      -8x - 5x = -13x
  4. Write simplified expression:
    4x^2 - 13x
Final Answer:
4x^2 - 13x
Detailed Explanation:
  • Distributive Property: Each term in the parentheses is multiplied, and the negative sign in
    -x
    affects both terms.
  • Common Mistake: Incorrectly handling the negative (e.g.,
    -x(2x + 5) = -2x^2 + 5x
    ).
  • Verification: Let
    x = 1
    . Original:
    2(1)(3 \cdot 1 - 4) - 1(2 \cdot 1 + 5) = 2(3 - 4) - (2 + 5) = 2(-1) - 7 = -2 - 7 = -9
    . Simplified:
    4(1^2) - 13(1) = 4 - 13 = -9
    .
Summary of Key Simplification Techniques:
  • Like Terms: Combine terms with identical variables and exponents.
  • Distributive Property: Use
    a(b + c) = ab + ac
    to expand expressions.
  • Exponent Rules: Apply rules like
    x^m / x^n = x^{m-n}
    for fractions.
  • FOIL Method: For binomials, multiply First, Outer, Inner, Last terms.
  • Parentheses: Remove or distribute carefully, respecting signs.
  • Verification: Test with values to confirm equivalence.
These problems cover a range of simplification scenarios, from basic combining of terms to polynomial operations and fraction simplification. If you’d like more complex problems (e.g., with fractions, higher-degree polynomials, or nested expressions) or additional practice, let me know!

 

Comments

Post a Comment

|

Blog Archive

Show more