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Difficulty: MediumOrder of Operations and Number Properties

Given the algebraic expression A=x−y×z2A = x - y \times z^2, evaluate the expression for each of the following sets of values for (x,y,z)(x, y, z) and arrange the sets in order of their resulting values of AA from least to greatest.

  1. 1The set (x,y,z)=(−5,1,2)(x, y, z) = (-5, 1, 2)
  2. 2The set (x,y,z)=(−2,3,−1)(x, y, z) = (-2, 3, -1)
  3. 3The set (x,y,z)=(8,4,0)(x, y, z) = (8, 4, 0)
  4. 4The set (x,y,z)=(4,−2,3)(x, y, z) = (4, -2, 3)

Answer

The correct order of the sets from least to greatest is: (x,y,z)=(−5,1,2)(x, y, z) = (-5, 1, 2) resulting in −9-9, followed by (x,y,z)=(−2,3,−1)(x, y, z) = (-2, 3, -1) resulting in −5-5, then (x,y,z)=(8,4,0)(x, y, z) = (8, 4, 0) resulting in 88, and finally (x,y,z)=(4,−2,3)(x, y, z) = (4, -2, 3) resulting in 2222.
The correct order follows from evaluating the algebraic expression for each set of values according to the order of operations (PEMDAS): exponentiation first, then multiplication, and finally subtraction. This yields values of −9-9, −5-5, 88, and 2222, which are ordered from least to greatest.

Step-by-Step Solution

1
Evaluate the expression A=x−y×z2A = x - y \times z^2 for the set (x,y,z)=(−5,1,2)(x, y, z) = (-5, 1, 2).
A=−9A = -9
According to the order of operations, evaluate the exponent first: 22=42^2 = 4. Then perform multiplication: 1×4=41 \times 4 = 4. Finally, perform subtraction: −5−4=−9-5 - 4 = -9.
2
Evaluate the expression A=x−y×z2A = x - y \times z^2 for the set (x,y,z)=(−2,3,−1)(x, y, z) = (-2, 3, -1).
A=−5A = -5
Evaluate the exponent first: (−1)2=1(-1)^2 = 1. Next, multiply: 3×1=33 \times 1 = 3. Finally, subtract: −2−3=−5-2 - 3 = -5.
3
Evaluate the expression A=x−y×z2A = x - y \times z^2 for the set (x,y,z)=(8,4,0)(x, y, z) = (8, 4, 0).
A=8A = 8
Evaluate the exponent first: 02=00^2 = 0. Next, multiply: 4×0=04 \times 0 = 0. Finally, subtract: 8−0=88 - 0 = 8.
4
Evaluate the expression A=x−y×z2A = x - y \times z^2 for the set (x,y,z)=(4,−2,3)(x, y, z) = (4, -2, 3).
A=22A = 22
Evaluate the exponent first: 32=93^2 = 9. Next, multiply: −2×9=−18-2 \times 9 = -18. Finally, subtract: 4−(−18)=4+18=224 - (-18) = 4 + 18 = 22.
5
Arrange the resulting values from least to greatest.
−9<−5<8<22-9 < -5 < 8 < 22
Comparing the four calculated values shows that −9-9 is the least, followed by −5-5, then 88, and 2222 is the greatest.

Key Concept

Order of Operations (PEMDAS)

Alternative Method

To verify the calculations, substitute each set of coordinates carefully, using parentheses around negative values. Evaluate each term step-by-step: first compute z2z^2, then multiply by yy, and finally subtract this result from xx.
Estimated Time:1m 30s
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Topics

Order of Operations and Number Properties

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