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Elementary Algebra

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Question 301Question →

If x=−3x = -3 and y=2y = 2, what is the value of the algebraic expression x2y−(x−y)22x+y2\frac{x^2 y - (x - y)^2}{2x + y^2}?

Show answer & explanation

Answer: 72\frac{7}{2}

Answer

72\frac{7}{2}
Substituting x=−3x = -3 and y=2y = 2 gives a numerator of (−3)2(2)−(−3−2)2=18−25=−7(-3)^2(2) - (-3 - 2)^2 = 18 - 25 = -7 and a denominator of 2(−3)+22=−6+4=−22(-3) + 2^2 = -6 + 4 = -2. Dividing −7-7 by −2-2 gives 72\frac{7}{2}.

Step-by-Step Solution

1
Evaluate the terms in the numerator
x2y=(−3)2(2)=9⋅2=18x^2 y = (-3)^2(2) = 9 \cdot 2 = 18 and (x−y)2=(−3−2)2=(−5)2=25(x - y)^2 = (-3 - 2)^2 = (-5)^2 = 25
Apply order of operations by performing parenthetical subtraction before squaring negative numbers.
2
Subtract the terms to find the total numerator value
18−25=−718 - 25 = -7
Subtract the squared binomial result from the first product.
3
Evaluate the denominator
2x+y2=2(−3)+22=−6+4=−22x + y^2 = 2(-3) + 2^2 = -6 + 4 = -2
Multiply and square the variable values before adding.
4
Divide the numerator by the denominator and simplify
−7−2=72\frac{-7}{-2} = \frac{7}{2}
Dividing a negative number by a negative number yields a positive result.

Key Concept

Evaluating Algebraic Expressions with Negative Bases and Parentheses
Estimated Time:1m 0s
Question 302Question →

If x=−2x = -2, y=13y = \frac{1}{3}, and z=−4z = -4, what is the value of the algebraic expression x3y−1−(x−z)2x+yz\frac{x^3 y^{-1} - (x - z)^2}{x + y z}?

Show answer & explanation

Answer: 425\frac{42}{5}

Answer

425\frac{42}{5}
Substituting x=−2x = -2, y=13y = \frac{1}{3}, and z=−4z = -4 into the expression yields (−2)3=−8(-2)^3 = -8, (13)−1=3(\frac{1}{3})^{-1} = 3, and (−2−(−4))2=22=4(-2 - (-4))^2 = 2^2 = 4. Thus, the numerator equals −8×3−4=−28-8 \times 3 - 4 = -28. The denominator equals −2+(13)(−4)=−2−43=−103-2 + (\frac{1}{3})(-4) = -2 - \frac{4}{3} = -\frac{10}{3}. Dividing the numerator by the denominator gives −28−103=8410=425\frac{-28}{-\frac{10}{3}} = \frac{84}{10} = \frac{42}{5}.

Step-by-Step Solution

1
Substitute the given values into the numerator of the expression: x3y−1−(x−z)2x^3 y^{-1} - (x - z)^2.
Numerator = (−2)3(13)−1−(−2−(−4))2=(−8)(3)−(2)2=−24−4=−28(-2)^3 \left(\frac{1}{3}\right)^{-1} - (-2 - (-4))^2 = (-8)(3) - (2)^2 = -24 - 4 = -28.
First apply exponent rules and basic arithmetic inside the parentheses following standard order of operations.
2
Substitute the given values into the denominator of the expression: x+yzx + y z.
Denominator = −2+(13)(−4)=−2−43=−63−43=−103-2 + \left(\frac{1}{3}\right)(-4) = -2 - \frac{4}{3} = -\frac{6}{3} - \frac{4}{3} = -\frac{10}{3}.
Multiply yy and zz first, then find a common denominator to add the fraction to the integer.
3
Divide the simplified numerator by the simplified denominator.
\frac{-28}{-\frac{10}{3}} = -28 \times \left(-\frac{3}{10}\right) = \frac{84}{10} = \frac{42}{5}.
Dividing by a fraction is equivalent to multiplying by its reciprocal.

Key Concept

Evaluating Algebraic Expressions with Negative Numbers and Negative Exponents
Estimated Time:1m 15s
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