SAT Math: Evaluating Linear Expressions

1. Sign Errors with Negative Values

When substituting negative numbers, always use parentheses. For example, if \( x = -3 \), write \( 2x \) as \( 2(-3) \), not \( 2 \times -3 \). This prevents sign errors.

2. Forgetting to Multiply the Coefficient

Remember that \( 5x \) means \( 5 \times x \). Many students write the substituted value but forget to multiply by the coefficient.

3. Order of Operations Mistakes

Always multiply before adding or subtracting. In \( 4x + 7 \), calculate \( 4x \) first, then add 7.

4. Fraction and Decimal Arithmetic

Take extra care with fraction multiplication and addition. Keep denominators in mind and simplify at the end.

5. Double-Check the Variable

Make sure you substitute for every occurrence of the variable, especially in longer expressions.

Question:

What is the value of the expression \( 5x - 8 \) when \( x = 4 \)?

A) 4
B) 12
C) 17
D) 28

Solution:

Step 1: Write the original expression:

\( 5x - 8 \)

Step 2: Substitute \( x = 4 \):

\( 5(4) - 8 \)

Step 3: Multiply first (order of operations):

\( 20 - 8 \)

Step 4: Subtract:

\( 12 \)

Answer: B) 12

💡 Tip: Always perform multiplication before addition or subtraction according to PEMDAS.

Question:

If \( n = -3 \), what is the value of \( -4n + 7 \)?

A) -19
B) -5
C) 5
D) 19

Solution:

Step 1: Write the expression:

\( -4n + 7 \)

Step 2: Substitute \( n = -3 \) (use parentheses!):

\( -4(-3) + 7 \)

Step 3: Multiply (negative × negative = positive):

\( 12 + 7 \)

Step 4: Add:

\( 19 \)

Answer: D) 19

💡 Tip: When substituting negative values, always use parentheses to avoid sign errors. Remember: \( -4 \times (-3) = +12 \).

Question:

What is the value of \( \frac{2}{3}x + 5 \) when \( x = 12 \)?

A) 9
B) 11
C) 13
D) 15

Solution:

Step 1: Write the expression:

\( \frac{2}{3}x + 5 \)

Step 2: Substitute \( x = 12 \):

\( \frac{2}{3}(12) + 5 \)

Step 3: Multiply the fraction by 12:

\( \frac{2 \times 12}{3} + 5 = \frac{24}{3} + 5 \)

Step 4: Simplify the fraction:

\( 8 + 5 \)

Step 5: Add:

\( 13 \)

Answer: C) 13

💡 Tip: When working with fractions, multiply first, then simplify. Notice that \( 12 \div 3 = 4 \), so \( \frac{2}{3} \times 12 = 2 \times 4 = 8 \).

Question:

A phone repair shop charges a flat fee of $25 plus $18 per hour of labor. The total cost \( C \) in dollars for \( h \) hours of labor is given by the expression \( C = 18h + 25 \). What is the total cost for a repair that takes 2.5 hours?

A) $43
B) $55
C) $68
D) $70

Solution:

Step 1: Identify the expression and value:

Expression: \( C = 18h + 25 \)

Value: \( h = 2.5 \) hours

Step 2: Substitute \( h = 2.5 \):

\( C = 18(2.5) + 25 \)

Step 3: Multiply:

\( C = 45 + 25 \)

(Since \( 18 \times 2.5 = 18 \times \frac{5}{2} = \frac{90}{2} = 45 \))

Step 4: Add:

\( C = 70 \)

Answer: D) $70

💡 Tip: In real-world problems, the expression represents a formula. The variable (here \( h \)) represents an input, and evaluating gives you the output (total cost). Always include units in your final answer when solving word problems.

Question:

What is the value of the expression \( 3(2x - 4) + 5 \) when \( x = 5 \)?

A) 13
B) 17
C) 23
D) 27

Solution:

Method 1: Substitute First

Step 1: Write the expression:

\( 3(2x - 4) + 5 \)

Step 2: Substitute \( x = 5 \):

\( 3(2(5) - 4) + 5 \)

Step 3: Work inside parentheses first:

\( 3(10 - 4) + 5 = 3(6) + 5 \)

Step 4: Multiply:

\( 18 + 5 \)

Step 5: Add:

\( 23 \)

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Method 2: Distribute First (Alternative)

Step 1: Distribute the 3:

\( 3(2x - 4) + 5 = 6x - 12 + 5 = 6x - 7 \)

Step 2: Substitute \( x = 5 \):

\( 6(5) - 7 = 30 - 7 = 23 \)

Answer: C) 23

💡 Tip: Both methods work! Substituting first is often faster on the SAT. However, if you need to simplify the expression for multiple values, distributing first creates a simpler form.

⏱️ Time Management

Evaluation questions should take 30-45 seconds. If you're stuck, skip and return later. Easy points add up!

🧮 Use Your Calculator Wisely

For SAT calculator-allowed sections, use your calculator to avoid arithmetic errors, especially with decimals and fractions.

✅ Show Your Work

Write each step in your test booklet. This reduces errors and helps you catch mistakes if you review your answer.

🎯 Eliminate Wrong Answers

If you're unsure, eliminate obviously wrong answers. Use estimation to narrow choices before calculating precisely.