Evaluating Expressions

Now that you can write algebraic expressions, it's time to learn how to evaluate them! Evaluating means finding the numerical value of an expression by substituting numbers for variables.

What Does "Evaluate" Mean?

To evaluate an expression means to:

1. Replace the variable(s) with given number(s)
2. Perform the operations following the order of operations (PEMDAS)
3. Simplify to get a single numerical answer

Example: Evaluate 3x + 5 when x = 4

• Substitute: 3(4) + 5
• Multiply: 12 + 5
• Add: 17
• Answer: 17

Basic Evaluation with One Variable

Example 1: Simple Expression

Evaluate: 2n + 7 when n = 5

Step 1: Substitute 5 for n 2(5) + 7

Step 2: Follow PEMDAS - multiply first 10 + 7

Step 3: Add 17

Answer: 17

Example 2: Expression with Subtraction

Evaluate: 15 - 3x when x = 2

Step 1: Substitute 2 for x 15 - 3(2)

Step 2: Multiply first 15 - 6

Step 3: Subtract 9

Answer: 9

Example 3: Expression with Division

Evaluate: y/4 + 3 when y = 20

Step 1: Substitute 20 for y 20/4 + 3

Step 2: Divide first 5 + 3

Step 3: Add 8

Answer: 8

Evaluating with Parentheses

When an expression has parentheses, evaluate what's inside first!

Example 1: Parentheses First

Evaluate: 5(x + 3) when x = 6

Step 1: Substitute 6 for x 5(6 + 3)

Step 2: Parentheses first - add inside 5(9)

Step 3: Multiply 45

Answer: 45

Example 2: Multiple Operations

Evaluate: 4(a - 2) + 10 when a = 7

Step 1: Substitute 7 for a 4(7 - 2) + 10

Step 2: Parentheses first 4(5) + 10

Step 3: Multiply 20 + 10

Step 4: Add 30

Answer: 30

Evaluating with Two Variables

When expressions have multiple variables, substitute each one carefully.

Example 1: Addition with Two Variables

Evaluate: 2x + 3y when x = 4 and y = 5

Step 1: Substitute both values 2(4) + 3(5)

Step 2: Multiply each term 8 + 15

Step 3: Add 23

Answer: 23

Example 2: Mixed Operations

Evaluate: 5a - 2b when a = 6 and b = 4

Step 1: Substitute both values 5(6) - 2(4)

Step 2: Multiply each term 30 - 8

Step 3: Subtract 22

Answer: 22

Example 3: More Complex Expression

Evaluate: 3x + 2y - 5 when x = 3 and y = 4

Step 1: Substitute both values 3(3) + 2(4) - 5

Step 2: Multiply 9 + 8 - 5

Step 3: Add and subtract from left to right 9 + 8 = 17 17 - 5 = 12

Answer: 12

Evaluating with Exponents

Remember: Exponents come before multiplication and division in PEMDAS!

Example 1: Simple Exponent

Evaluate: x² + 3 when x = 4

Step 1: Substitute 4 for x 4² + 3

Step 2: Exponent first 16 + 3

Step 3: Add 19

Answer: 19

Example 2: Exponent with Coefficient

Evaluate: 2n² when n = 5

Step 1: Substitute 5 for n 2(5²)

Step 2: Exponent first (only the 5 is squared, not the 2!) 2(25)

Step 3: Multiply 50

Answer: 50

Important: In 2n², only n is squared. If it were (2n)², you would square the entire product.

Example 3: Multiple Terms with Exponents

Evaluate: a² + b² when a = 3 and b = 4

Step 1: Substitute both values 3² + 4²

Step 2: Calculate each exponent 9 + 16

Step 3: Add 25

Answer: 25

Evaluating Fractions and Decimals

Variables can represent any number, including fractions and decimals.

Example 1: Fraction Value

Evaluate: 6x + 4 when x = 1/2

Step 1: Substitute 1/2 for x 6(1/2) + 4

Step 2: Multiply 3 + 4

Step 3: Add 7

Answer: 7

Example 2: Decimal Value

Evaluate: 5y - 2 when y = 1.5

Step 1: Substitute 1.5 for y 5(1.5) - 2

Step 2: Multiply 7.5 - 2

Step 3: Subtract 5.5

Answer: 5.5

Negative Numbers in Expressions

When substituting negative numbers, use parentheses to avoid errors!

Example 1: Negative Substitution

Evaluate: 3x + 7 when x = -2

Step 1: Substitute -2 for x (use parentheses!) 3(-2) + 7

Step 2: Multiply -6 + 7

Step 3: Add 1

Answer: 1

Example 2: Subtraction with Negative

Evaluate: 10 - 2n when n = -3

Step 1: Substitute -3 for n 10 - 2(-3)

Step 2: Multiply (negative times negative = positive) 10 - (-6) 10 + 6

Step 3: Add 16

Answer: 16

Key Rule: Subtracting a negative is the same as adding a positive!

Example 3: Squaring a Negative

Evaluate: x² - 5 when x = -4

Step 1: Substitute -4 for x (-4)² - 5

Step 2: Square the negative (negative × negative = positive) 16 - 5

Step 3: Subtract 11

Answer: 11

Real-World Applications

Temperature Conversion

Formula: F = 9C/5 + 32 (converts Celsius to Fahrenheit)

Problem: If the temperature is 20°C, what is it in Fahrenheit?

Solution: Evaluate when C = 20 F = 9(20)/5 + 32 F = 180/5 + 32 F = 36 + 32 F = 68

Answer: 68°F

Distance Formula

Formula: d = rt (distance = rate × time)

Problem: If you travel at 55 mph for 3 hours, how far do you go?

Solution: Evaluate when r = 55 and t = 3 d = 55(3) d = 165

Answer: 165 miles

Area of Triangle

Formula: A = (1/2)bh (area = one-half × base × height)

Problem: Find the area when base = 8 and height = 5

Solution: Evaluate when b = 8 and h = 5 A = (1/2)(8)(5) A = (1/2)(40) A = 20

Answer: 20 square units

Common Mistakes to Avoid

Mistake 1: Forgetting parentheses with negative numbers Wrong: 3 × -2 (ambiguous) Right: 3(-2) = -6

Mistake 2: Not following PEMDAS Wrong: 2 + 3 × 4 = 5 × 4 = 20 Right: 2 + 3 × 4 = 2 + 12 = 14

Mistake 3: Squaring the coefficient too Wrong: 2x² when x = 3 → (2 × 3)² = 6² = 36 Right: 2x² when x = 3 → 2(3²) = 2(9) = 18

Mistake 4: Sign errors with negatives Wrong: 5 - 2(-3) = 5 - 6 = -1 Right: 5 - 2(-3) = 5 - (-6) = 5 + 6 = 11

Mistake 5: Forgetting to substitute all variables If an expression has x and y, you must substitute values for both!

Order of Operations Review (PEMDAS)

Parentheses - ( ) Exponents - powers and square roots Multiplication and Division - left to right Addition and Subtraction - left to right

Example: Evaluate 3(x + 2)² - 4x when x = 2

Step 1: Substitute 3(2 + 2)² - 4(2)

Step 2: Parentheses 3(4)² - 4(2)

Step 3: Exponents 3(16) - 4(2)

Step 4: Multiply (left to right) 48 - 8

Step 5: Subtract 40

Answer: 40

Evaluating vs. Simplifying

Evaluating: Substituting values and calculating a numerical answer

• Evaluate 2x + 3 when x = 5 → 2(5) + 3 = 13

Simplifying: Combining like terms without substituting

• Simplify 2x + 3x → 5x (no numerical answer, still has variables)

You evaluate when you have values to substitute. You simplify when you want to make an expression shorter.

Practice Strategy

Step 1: Write down the expression Step 2: Substitute each variable with its value (use parentheses!) Step 3: Follow PEMDAS strictly Step 4: Show all your work - don't skip steps Step 5: Check your answer - does it make sense?

Mental Check: If x = 0, most expressions become much simpler!

• 5x + 3 when x = 0 → 5(0) + 3 = 3

Creating a Substitution Chart

For complex problems with multiple values, make a chart:

Expression: 2a + 3b

| a | b | 2a + 3b | |---|---|---------| | 1 | 2 | 2(1) + 3(2) = 2 + 6 = 8 | | 3 | 1 | 2(3) + 3(1) = 6 + 3 = 9 | | 0 | 4 | 2(0) + 3(4) = 0 + 12 = 12 |

This helps you see patterns and practice evaluation!

Connection to Functions

Evaluating expressions is exactly what you do with functions!

Function notation: f(x) = 2x + 5 Find f(3): Evaluate 2x + 5 when x = 3

• f(3) = 2(3) + 5 = 6 + 5 = 11

You're already learning the foundation for algebra and functions!

Why Evaluation Matters

• Testing formulas: Science and math formulas need specific values
• Checking answers: Substitute your solution back to verify
• Real-world calculations: Recipes, budgets, distances all use evaluation
• Computer programming: Variables and evaluation are fundamental to coding
• Building algebra skills: Evaluation prepares you for solving equations

Master evaluation and you'll breeze through algebra, geometry, science, and beyond!