Notes on Evaluating Algebraic Expressions

Evaluating Algebraic Expressions

To evaluate an algebraic expression:

  • Replace the variable with the given numeric values.
  • Perform the indicated operations, remembering the order of operations.
  • This is useful in statistics for plugging numbers into formulas.

Example 1: Evaluating 12+5x12 + 5x when x=8x = 8

  1. Replace xx with 8: 12+5(8)12 + 5(8)
  2. Multiplication before addition: 5×8=405 \times 8 = 40
  3. 12+40=5212 + 40 = 52

The value of the expression when x=8x = 8 is 52.

Using a Calculator

  • You can plug the expression directly into a calculator: 12+5×8=5212 + 5 \times 8 = 52

Example 2: Evaluating x2−y2x⋅y\frac{x^2 - y^2}{x \cdot y} when x=−2x = -2 and y=3y = 3

  1. Substitute the values of xx and yy: (−2)2−(3)2(−2)×(3)\frac{(-2)^2 - (3)^2}{(-2) \times (3)}
    • Use parentheses when raising a negative number to a power to ensure correct calculation.
  2. Evaluate the numerator:
    • (−2)2=4(-2)^2 = 4
    • 32=93^2 = 9
    • 4−9=−54 - 9 = -5
  3. Evaluate the denominator:
    • (−2)×(3)=−6(-2) \times (3) = -6
  4. The expression becomes: −5−6\frac{-5}{-6}
  5. Simplify the fraction:
    • A negative divided by a negative is positive, so −5−6=56\frac{-5}{-6} = \frac{5}{6}

The Concept of Variables

  • An expression's value changes depending on the numbers plugged in for its variables.

Example 3: Evaluating ab÷b2c\frac{a}{b} \div \frac{b^2}{c} where a=1a = 1, b=6b = 6, and c=−3c = -3

  1. Substitute the given values:
    16÷62−3\frac{1}{6} \div \frac{6^2}{-3}
  2. Evaluate the exponent:
    62=366^2 = 36
    So the expression becomes:
    16÷36−3\frac{1}{6} \div \frac{36}{-3}
  3. Divide the fractions using a calculator, with parentheses around each fraction:
    (16)÷(36−3)=−172(\frac{1}{6}) \div (\frac{36}{-3}) = -\frac{1}{72}

Example 4: Evaluating x−2ya+b\frac{x - 2y}{a + b} where x=10x = 10, y=−8y = -8, a=−4a = -4, and b=7b = 7

  1. Substitute the given values:
    10−2(−8)−4+7\frac{10 - 2(-8)}{-4 + 7}
  2. Evaluate the numerator (multiplication first):
    2×−8=−162 \times -8 = -16
    So the numerator becomes:
    10−(−16)10 - (-16)
  3. Evaluate the denominator:
    −4+7=3-4 + 7 = 3
  4. Simplify the numerator (subtracting a negative is addition):
    10−(−16)=10+16=2610 - (-16) = 10 + 16 = 26
  5. The expression becomes:
    263\frac{26}{3}
    This fraction does not reduce further.