Evaluating Algebraic Expressions

An algebraic expression contains numbers, operations, and variables (letters that represent unknown numbers). To evaluate an algebraic expression means to find its exact numerical value when you are given specific numbers for those variables.

How to Evaluate an Expression

Evaluating an expression takes two simple steps:

1. Substitute: Replace the variables with their given numbers. It is a great habit to put parentheses around the numbers you plug in to keep your work organized.
2. Calculate: Use the Order of Operations (PEMDAS: Parentheses, Exponents, Multiplication & Division, Addition & Subtraction) to find the final answer.

Example 1: A Single Variable

Problem: Evaluate $3x + 5$ when $x = 4$.

• Step 1: Substitute $4$ for $x$. Remember that $3x$ means $3$ times $x$. $3(4) + 5$
• Step 2: Follow the order of operations. Multiply first: $12 + 5$
• Step 3: Add to get the final answer: $17$

Example 2: Multiple Variables and Exponents

Problem: Evaluate $2a^2 - b$ when $a = 3$ and $b = 7$.

• Step 1: Substitute $3$ for $a$ and $7$ for $b$. $2(3)^2 - 7$
• Step 2: Evaluate the exponent first ($3^2 = 9$). $2(9) - 7$
• Step 3: Multiply next. $18 - 7$
• Step 4: Subtract. $11$

Example 3: Fractions

Problem: Find the value of $\frac{x + y}{2}$ when $x = 8$ and $y = 6$.

• Step 1: Substitute the values into the fraction. $\frac{8 + 6}{2}$
• Step 2: A fraction bar acts like a grouping symbol. Evaluate the numerator (top) first. $\frac{14}{2}$
• Step 3: Divide the numerator by the denominator. $7$

Quick Tips

• Always rely on the Order of Operations after substituting.
• Use parentheses when plugging in numbers. This prevents confusion, especially when multiplication or exponents are involved.