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Evaluating
Expressions

Learn how to substitute a number for a variable and calculate the result: the core skill behind every algebra problem.

SSAT Middle Level: Quantitative

What Is an Expression?

An expression is a combination of numbers, variables, and operations. Unlike an equation, it has no equals sign: it simply represents a value that depends on whatever the variable stands for.

\( 3x + 5 \)
A variable \( x \), multiplied by 3, then 5 is added.
\( 2(n - 4) \)
Subtract 4 from \( n \), then multiply the result by 2.
\( \frac{y}{3} + 7 \)
Divide \( y \) by 3, then add 7 to the quotient.

Key Vocabulary

Variable
A letter (like \( x \), \( n \), \( y \)) that stands in for an unknown number.
Substitute
Replace the variable with a given number.
Evaluate
Perform all the operations after substituting to get a single number.
Coefficient
The number multiplied by a variable. In \( 5x \), the coefficient is 5.

Why Evaluating Matters

Every time you check a formula, verify a solution, or solve a word problem, you are evaluating an expression. It is the bridge between an abstract formula and a concrete answer.

The Substitution Rule

\(3x + 5\)
\(x = 4\)
\(3(4) + 5\)
\(17\)

Remember PEMDAS

After you substitute, you must simplify in the correct order. Use PEMDAS to avoid mistakes.

1
Parentheses: simplify inside grouping symbols first.
2
Exponents: evaluate powers next.
3
Multiplication and Division: work left to right.
4
Addition and Subtraction: work left to right.

Expression vs. Equation

Students often confuse these two. The key difference is the equals sign.

Expression Equation
Equals sign? No Yes
Example \( 2x + 7 \) \( 2x + 7 = 15 \)
Goal Evaluate (simplify) Solve for the variable
Result A single number A value that makes it true

Step-by-Step Method

1
Write the original expression.
2
Substitute: Replace every variable with the given number. Use parentheses around the number.
3
Simplify step by step using PEMDAS.
4
State the final numerical answer.

Example 1: Simple Substitution

Evaluate \( 4x - 3 \) when \( x = 6 \).

1
Write: \( 4x - 3 \)
2
Substitute \( x = 6 \):   \( 4(\color{#4ade80}{6}) - 3 \)
3
Multiply: \( 24 - 3 \)
4
Subtract: \( \color{#fbbf24}{21} \)

Example 2: Parentheses First

Evaluate \( 5(n + 2) \) when \( n = 3 \).

1
Write: \( 5(n + 2) \)
2
Substitute \( n = 3 \):   \( 5(\color{#fbbf24}{3} + 2) \)
3
Parentheses: \( 5(5) \)
4
Multiply: \( \color{#fbbf24}{25} \)

Example 3: Exponents

Evaluate \( x^2 + 3x - 1 \) when \( x = 5 \).

1
Substitute: \( (\color{#818cf8}{5})^2 + 3(\color{#818cf8}{5}) - 1 \)
2
Exponent: \( 25 + 3(5) - 1 \)
3
Multiply: \( 25 + 15 - 1 \)
4
Add, subtract: \( \color{#fbbf24}{39} \)

Example 4: Two Variables

Evaluate \( 2a + 3b \) when \( a = 4 \) and \( b = 5 \).

1
Substitute: \( 2(\color{#818cf8}{4}) + 3(\color{#4ade80}{5}) \)
2
Multiply each term: \( 8 + 15 \)
3
Add: \( \color{#fbbf24}{23} \)

Visualising the Process

EXPRESSION
\(2x^2 - x + 3\)
 GIVEN VALUE
\(x = 3\)
 SUBSTITUTED
\(2(3)^2 - (3) + 3\)
 EXPONENT
\(2(9) - 3 + 3\)
 MULTIPLY
\(18 - 3 + 3\)
 RESULT
\(18\)

Substituting Negative Values

When the given value is negative, wrap it in parentheses to keep the signs clear.

1
Evaluate \( x^2 - 2x \) when \( x = -3 \).
2
Substitute: \( (\color{#f87171}{-3})^2 - 2(\color{#f87171}{-3}) \)
3
Exponent: \( 9 - 2(-3) \)
4
Multiply: \( 9 - (-6) = 9 + 6 = \color{#fbbf24}{15} \)

Common Mistakes

✕ Forgetting Parentheses
Writing \( 34 \) instead of \( 3(4) \) when substituting \( x = 4 \) into \( 3x \).
✕ Wrong Order
Adding before multiplying. In \( 2 + 3x \) with \( x = 5 \), students compute \( (2 + 3) \times 5 = 25 \) instead of \( 2 + 15 = 17 \).
✕ Sign Errors
Treating \( (-3)^2 \) as \( -9 \). With parentheses, the answer is \( +9 \).

Quick Reference Table

Expression Given Value Substituted Result
\( 2x + 1 \) \( x = 5 \) \( 2(5) + 1 \) 11
\( n^2 - 4 \) \( n = 3 \) \( (3)^2 - 4 \) 5
\( 3(a + 7) \) \( a = 2 \) \( 3(2 + 7) \) 27
\( \frac{y}{4} + 6 \) \( y = 12 \) \( \frac{12}{4} + 6 \) 9
\( x^2 - 2x \) \( x = -3 \) \( (-3)^2 - 2(-3) \) 15

Question 1 of 5

Evaluate \( 3x + 7 \) when \( x = 4 \).

Question 2 of 5

Evaluate \( 2(n - 5) \) when \( n = 9 \).

Question 3 of 5

Evaluate \( x^2 + 4x \) when \( x = 3 \).

Question 4 of 5

Evaluate \( \frac{y}{2} + 8 \) when \( y = 10 \).

Question 5 of 5

Evaluate \( a^2 - 3a + 2 \) when \( a = -2 \).