Evaluating Algebraic Expressions - Sixth Grade Mathematics

Evaluating an algebraic expression determines its specific numerical value by substituting given numbers for variables and executing arithmetic using the order of operations.

The Substitution Principle

To evaluate an algebraic expression means to calculate its numerical value when each variable is replaced by a given number.

The Substitution Protocol:
Step 1: Replace every instance of the variable with the assigned number.
        (Always place parentheses around substituted numbers to protect signs!)
Step 2: Follow the standard Order of Operations (PEMDAS):
        - Parentheses / Grouping symbols
        - Exponents
        - Multiplication & Division (Left to Right)
        - Addition & Subtraction (Left to Right)

Worked Example: Evaluate 4x^2 - 3x + 7  when x = 3

Step 1: Substitute (3) in place of x:
        4(3)^2 - 3(3) + 7

Step 2: Evaluate exponent first:
        (3)^2 = 9
        Expression: 4(9) - 3(3) + 7

Step 3: Perform multiplications from left to right:
        4 * 9 = 36
        3 * 3 = 9
        Expression: 36 - 9 + 7

Step 4: Perform subtraction and addition from left to right:
        36 - 9 = 27
        27 + 7 = 34
Final Numerical Value: 34.

Evaluating Multi-Variable Expressions

Algebraic expressions often contain two or three distinct variables. Each variable is replaced by its own assigned numerical value.

Evaluate: 3a + 2b - c    when a = 4, b = 5, and c = 6
Step 1: Substitute all values:
        3(4) + 2(5) - (6)
Step 2: Multiply:
        12 + 10 - 6
Step 3: Add and subtract:
        22 - 6 = 16.

Managing Negative Numbers and Exponents during Substitution

Substituting negative values requires strict use of parentheses to ensure correct evaluation of powers and signs.

Evaluating with Negative Substitution: Evaluate x^2 and -x^2 when x = -4

Case 1: Evaluate x^2:
        Replace x with (-4):
        (-4)^2 = (-4) * (-4) = +16.

Case 2: Evaluate -x^2:
        Replace x with (-4):
        -(-4)^2 = -(16) = -16.

Evaluating Real-World Formulas

Evaluating expressions is the exact process scientists and engineers use when applying formulas: Area of a Triangle: A = 1/2 * b * h. When b = 8 cm and h = 5 cm: A = 1/2 * (8) * (5) = 4 * 5 = 20 square centimeters. Perimeter of a Rectangle: P = 2l + 2w. When l = 12 ft and w = 7 ft: P = 2(12) + 2(7) = 24 + 14 = 38 feet.

Exercises

Exercise 1. Evaluate each expression when x = 5: (a) 3x + 8; (b) 2x^2 - 15; (c) (40 / x) + 2x.

Exercise 2. Evaluate the multi-variable expression: 5a - 2b + 3c when a = 6, b = 4, and c = 2.

Exercise 3. Evaluate the fraction expression: (x^2 + 5) / (2y - 1) when x = 3 and y = 4.

Solutions

Solution to Exercise 1. For (a) 3x + 8 when x = 5: substitute 5: 3(5) + 8 = 15 + 8 = 23. For (b) 2x^2 - 15: substitute 5: 2(5)^2 - 15 = 2(25) - 15 = 50 - 15 = 35. For (c) (40 / x) + 2x: substitute 5: (40 / 5) + 2(5) = 8 + 10 = 18.

Solution to Exercise 2. Substitute a = 6, b = 4, and c = 2 into 5a - 2b + 3c: 5(6) - 2(4) + 3(2). Perform multiplications: 30 - 8 + 6. Perform subtraction and addition from left to right: 30 - 8 = 22; 22 + 6 = 28. The final value is 28.

Solution to Exercise 3. Evaluate numerator and denominator separately: Numerator: x^2 + 5 = (3)^2 + 5 = 9 + 5 = 14. Denominator: 2y - 1 = 2(4) - 1 = 8 - 1 = 7. Perform the final division: 14 / 7 = 2. The value of the expression is 2.