Evaluating Algebraic Expressions and Combining Like Terms: A Step-by-Step Guide

 Evaluating Algebraic Expressions and Combining Like Terms: A Step-by-Step Guide

You’re a scout analyst for an NFL franchise. Someone hands you a formula: 3x + 2y − w + 5x, where x = vertical jump in inches, y = bench press reps, and w = pounds a player is overweight. Your job is to plug in real numbers for each variable and generate a composite score for every prospect in the upcoming draft. Simple in concept — but if you don’t know how to evaluate that expression or simplify it first, you’re going to get the wrong number every single time.

By the end, you’ll be able to evaluate any expression by substituting known values and applying PEMDAS, identify like terms, and combine them to simplify expressions into their cleanest form.

Quick Review: The Parts of an Expression

Every algebraic expression is made up of four components:

• • Terms: individual pieces separated by + or − signs. In 5x² − 2x + 7, there are three terms.
• • Coefficients: the numbers multiplied in front of a variable. In 5x², the coefficient is 5.
• • Variables: the letters representing unknowns — x, y, M, whatever fits the problem.
• • Constants: standalone numbers with no variable. In 5x² − 2x + 7, the constant is 7.
• • Exponents: the power a variable is raised to. In 5x², the exponent is 2.

Real-world example: Calories = 8M + 5S − 2R, where M = minutes running, S = sets lifted, R = rest minutes. Every term has a specific job.

Evaluating Expressions: Substituting and Solving

Evaluating an expression means replacing each variable with a given number, then computing the result using PEMDAS. Always three steps:

• Step 1: Write out the expression.

• Step 2: Substitute the given value(s) for each variable, using parentheses around every substitution.

• Step 3: Follow PEMDAS to compute the final answer.

The parentheses in Step 2 are not optional — they protect you from sign errors when substituting negative numbers.

Example: Evaluate 4x − 3 when x = 5.

4(5) − 3 = 20 − 3 = 17

Harder example: Evaluate 3x² − 2x + 1 when x = 3.

3(3)² − 2(3) + 1

= 3(9) − 6 + 1

= 27 − 6 + 1 = 22

What Are Like Terms?

Note: constants are always like terms with each other, regardless of their values.

Combining Like Terms: The Rules

Add or subtract the coefficients, and keep the variable exactly as it is. The variable and its exponent never change.

3x − 2x = (3−2)x = x

7x² + 2x² = (7+2)x² = 9x²

5 − 8 = −3

Tip: cross out terms as you combine them to prevent double-counting.

Practice: Simplify 7A + 3B − 2A + 5B.

A terms: 7A − 2A = 5A

B terms: 3B + 5B = 8B

Answer: 5A + 8B

Worked Examples

Example 1: Evaluate 2x + 3y when x = 4, y = 2.

2(4) + 3(2) = 8 + 6 = 14

Example 2: Simplify 7A + 3B − 2A + 5B.

= (7A − 2A) + (3B + 5B) = 5A + 8B

Example 3: Evaluate 3x² − 2x + 1 when x = 3.

3(3)² − 2(3) + 1 = 27 − 6 + 1 = 22

Example 4 (TRAP): Simplify 4x + 3x² − 2x + x².

The trap: combining 4x and 3x² because they both contain x. The exponents are different — x and x² are NOT like terms.

x terms: 4x − 2x = 2x

x² terms: 3x² + x² = 4x²

Answer: 4x² + 2x

5 Common Mistakes to Avoid

• Combining unlike terms because they share the same base. 5x + 3x² ≠ 8x³. Different exponents =
• different terms.
• Skipping parentheses when substituting. 3x where x = −2 should be written 3(−2), not 3−2.
• Adding exponents when combining like terms. 7x² + 2x² = 9x², NOT 9x⁴.
• Dropping negative signs mid-problem. Write each step fully.
• Treating x and x² as the same variable. x = x. x² = x × x. Never combine them.

Conclusion

Two skills, one lesson: evaluate and simplify. Evaluating an expression means substituting known values and

working through PEMDAS until you land on a single number. Combining like terms means identifying terms that

share the same variable and exponent, then adding or subtracting their coefficients.

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