11.2 Evaluating Algebraic Expressions
Key Takeaways
- Substitute carefully: replace each variable with parentheses, especially for negatives and fractions
- After substitution, simplify with PEMDAS—do not combine unlike terms across different variables until values are in
- Common sign errors: −x² when x is negative, and distributing a negative across a parenthesis
- Multi-variable expressions are evaluated the same way: plug in every letter, then compute
- On PERT, re-evaluate with the same values if two options look close—most wrong answers come from one missed sign
11.2 Evaluating Algebraic Expressions
Quick Answer: To evaluate an algebraic expression, replace every variable with the given number (use parentheses), then simplify with PEMDAS. Watch signs:
x = −3inx²is(−3)² = 9, while−x²is−9. Fractions and multi-variable expressions follow the same substitute-then-simplify process.
McCann’s PERT outline lists evaluating algebraic expressions as its own competency. These items check whether you can plug values in without rewriting the expression’s structure. The algebra is rarely deep—the danger is a single negative or fraction handled carelessly.
What “Evaluate” Means
An expression has no equals sign to “solve.” You are not finding x; you are computing a number when x (and other variables) are known.
| Expression | Given | Task |
|---|---|---|
3x + 7 | x = 4 | Compute the value → 19 |
2a − b | a = 5, b = −1 | Compute → 11 |
(x + y)/2 | x = 9, y = 3 | Compute → 6 |
Habit that saves points: Write the expression again with blanks or parentheses before inserting numbers:
3x + 7 with x = −2 → 3(−2) + 7, not 3 − 2 + 7.
Substituting Integers (Including Zero)
Worked Example 1 — One variable
Evaluate 5x − 3 when x = 6.
5(6) − 3 = 30 − 3 = 27.
Answer: 27.
Worked Example 2 — Zero
Evaluate 4x² + x − 8 when x = 0.
4(0)² + 0 − 8 = 0 + 0 − 8 = −8.
Zero in a product kills that term; it does not erase constant terms.
Negatives: The Highest-Yield Trap
When a variable is negative, always wrap the substituted value in parentheses.
Worked Example 3 — Square of a negative
Evaluate x² − 2x when x = −4.
x² = (−4)² = 16
−2x = −2(−4) = 8
16 + 8 = 24.
Answer: 24.
Wrong path A: −4² = −16 instead of (−4)² → −16 − 2(−4) = −16 + 8 = −8 (common distractor).
Wrong path B: −2x treated as −2 + (−4) instead of multiplication.
Worked Example 4 — Leading minus on a square
Evaluate −x² + 5 when x = −3.
First x² = (−3)² = 9, then apply the leading minus: −9 + 5 = −4.
Answer: −4.
Compare with (−x)² + 5: −x = −(−3) = 3, 3² + 5 = 14. Notation matters.
Worked Example 5 — Double negative in a product
Evaluate −3(x − 2) when x = −1.
Inside first after sub: −1 − 2 = −3, then −3(−3) = 9.
Alternatively distribute: −3x + 6 → −3(−1) + 6 = 3 + 6 = 9.
Answer: 9.
Fractions as Substitutions
Worked Example 6 — Fractional value
Evaluate 6x + 1 when x = 1/2.
6(1/2) + 1 = 3 + 1 = 4.
Worked Example 7 — Fraction with an exponent
Evaluate 8x² − 4x when x = 1/2.
x² = (1/2)² = 1/4
8(1/4) = 2
4x = 4(1/2) = 2
2 − 2 = 0.
Answer: 0.
Worked Example 8 — Negative fraction
Evaluate 2x + 3y when x = −3/4 and y = 1/2.
2(−3/4) + 3(1/2) = −6/4 + 3/2 = −3/2 + 3/2 = 0.
Answer: 0.
When denominators differ, convert to a common denominator before adding—another place PERT options hide “almost right” values.
Multi-Variable Expressions
Worked Example 9 — Two variables, mixed signs
Evaluate 3a − 2b + 4 when a = 5 and b = −3.
3(5) − 2(−3) + 4 = 15 − (−6) + 4 wait: −2(−3) = +6, so 15 + 6 + 4 = 25.
Answer: 25.
Trap: Reading −2b with b = −3 as −2 − 3 = −5 instead of multiplying.
Worked Example 10 — Product and quotient
Evaluate (2x − y)/(x + y) when x = 4 and y = −2.
Numerator: 2(4) − (−2) = 8 + 2 = 10
Denominator: 4 + (−2) = 2
10/2 = 5.
Answer: 5.
Denominator zero is undefined—if a choice set includes “undefined,” check whether x + y = 0.
Worked Example 11 — Three variables
Evaluate a² + 2ab + b² when a = 3, b = −1 (optional c unused).
Notice a² + 2ab + b² = (a + b)².
(3 + (−1))² = 2² = 4.
Directly: 9 + 2(3)(−1) + 1 = 9 − 6 + 1 = 4.
Answer: 4.
Recognizing structure can speed checks, but always be ready to expand term-by-term under PEMDAS.
Common Sign and Structure Errors
| Mistake | Example | Why it fails |
|---|---|---|
| Dropping parentheses on negative base | x = −2 in x² written as −2² | Gets −4 instead of 4 |
| Treating juxtaposition as addition | 2x with x = −5 as 2 − 5 | Product is −10, not −3 |
| Distributing only the first term | −(x − 4) as −x − 4 | Should be −x + 4 |
| Squaring term-by-term illegally | (x + 3)² = x² + 9 | Missing middle term 6x |
| Order after sub | Adding before multiplying | PEMDAS still applies |
PERT-Style Checklist
- List each variable’s value at the top of your scratch work.
- Rewrite with parentheses:
3x² − 5x→3( )² − 5( ). - Exponents before multiply on the substituted numbers.
- Simplify numerator and denominator separately in rational expressions.
- Re-run the arithmetic if your answer is not among the choices—usually a sign, not a brand-new method, is wrong.
Evaluating expressions is the bridge from arithmetic (Section 11.1) to solving equations (Section 11.3). If substitution is automatic, equation checking becomes automatic too.
Evaluate 4x − 9 when x = −2. What is the value?
Evaluate x² − 3x when x = −5.
Evaluate 2a + 3b when a = 1/2 and b = −2/3.
Evaluate (3x − y)/(x + 2) when x = 4 and y = −2.