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Difficulty: MediumEvaluating Algebraic Expressions

If x=−2x = -2, y=13y = \frac{1}{3}, and z=−3z = -3, what is the value of the algebraic expression 3x2y−z2x+yz\frac{3x^2 y - z^2}{x + yz}?

  1. 53\frac{5}{3}Answer
  2. B
    −133-\frac{13}{3}
  3. C
    133\frac{13}{3}
  4. D
    −1-1
  5. E
    55

Answer

53\frac{5}{3}
Substituting the given values into the expression requires evaluating powers of negative numbers and performing multiplication prior to addition/subtraction. In the numerator, 3(−2)2(13)−(−3)2=3(4)(13)−9=4−9=−53(-2)^2(\frac{1}{3}) - (-3)^2 = 3(4)(\frac{1}{3}) - 9 = 4 - 9 = -5. In the denominator, −2+(13)(−3)=−2−1=−3-2 + (\frac{1}{3})(-3) = -2 - 1 = -3. Simplifying the fraction −5−3\frac{-5}{-3} gives 53\frac{5}{3}.

Step-by-Step Solution

1
Evaluate the numerator 3x2y−z23x^2 y - z^2 by substituting x=−2x = -2, y=13y = \frac{1}{3}, and z=−3z = -3.
3(−2)2(13)−(−3)2=3(4)(13)−9=4−9=−53(-2)^2\left(\frac{1}{3}\right) - (-3)^2 = 3(4)\left(\frac{1}{3}\right) - 9 = 4 - 9 = -5
Squaring negative numbers yields positive values: (−2)2=4(-2)^2 = 4 and (−3)2=9(-3)^2 = 9.
2
Evaluate the denominator x+yzx + yz using the same variable values.
−2+(13)(−3)=−2+(−1)=−3-2 + \left(\frac{1}{3}\right)(-3) = -2 + (-1) = -3
Perform the multiplication y⋅zy \cdot z before adding to xx according to the order of operations.
3
Divide the evaluated numerator by the evaluated denominator.
−5−3=53\frac{-5}{-3} = \frac{5}{3}
Dividing two negative numbers produces a positive quotient.

Key Concept

Evaluating Algebraic Expressions with Negative Values and Rational Numbers
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Topics

Evaluating Algebraic Expressions

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