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\(x\)
\( = 5 \)

Evaluate Expressions
With Variables

Learn to substitute values into algebraic expressions and calculate the result using the order of operations.

6th Grade Math

What Is a Variable?

A variable is a letter (like \( x \), \( y \), or \( n \)) that stands in for a number we don't know yet. Think of it as a blank box waiting to be filled with a specific value.

\(x\)
\(=\)
\(7\)
 The variable holds a value

What Is an Expression?

An algebraic expression is a combination of numbers, variables, and operations. It does not have an equals sign: that would make it an equation.

Expressions
\( 3x + 5 \)
\( 2n - 4 \)
\( y^2 + 7 \)
Not Expressions
\( 3x + 5 = 20 \)
\( 2n - 4 = 10 \)
These are equations

Why Evaluate Expressions?

In real life, we use expressions as formulas. When we know the actual value of a variable, we substitute it in and calculate to get a concrete answer.

Earning Money
You earn \( \$8 \) per hour. Total earnings: \( 8h \). If \( h = 5 \), you earn \( \$40 \).
Perimeter
A square's perimeter: \( 4s \). If \( s = 6 \), the perimeter is \( 24 \) units.
Temperature
Convert: \( \frac{9}{5}C + 32 \). If \( C = 20 \), that's \( 68^\circ\text{F} \).

The Substitution Rule

To evaluate an expression, replace every variable with its given value, then simplify using the order of operations.

Expression Substitute Replace variables Simplify Use PEMDAS
\(3x + 2\)
\(3(4) + 2\)
\(14\)
when \(x = 4\)

Order of Operations: PEMDAS

After substituting, you must simplify in the correct order. Remember PEMDAS:

P
Parentheses first: evaluate anything inside grouping symbols.
E
Exponents: calculate powers like \( x^2 \) or \( 3^3 \).
MD
Multiplication and Division: left to right.
AS
Addition and Subtraction: left to right.

Expression vs Equation

Students often confuse these two. Here's how they differ:

Expression Equation
Equals sign? No Yes
Example \( 4n + 7 \) \( 4n + 7 = 19 \)
What you do Evaluate (substitute and simplify) Solve (find the unknown)
Result A single number A value of the variable

How to Evaluate Step by Step

1
Write the expression. Copy it exactly as given.
2
Identify the variable values. Note what each variable equals.
3
Substitute with parentheses. Replace every variable with its value in parentheses.
4
Apply PEMDAS. Simplify using the order of operations.
5
State your answer. Write the final numerical result.

Example 1: Evaluate \( 5x - 3 \)

Evaluate \( 5x - 3 \) when \( x = 6 \).

Step 1:
Write expression: \(5x - 3\)
 Step 2:
Substitute \(x = 6\): \(5(6) - 3\)
 Step 3:
Multiply first: \(30 - 3\)
\( = 27\)

Example 2: Two Variables

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

Step 1:
Expression: \(2a + 3b\)
 Step 2:
Substitute: \(2(4) + 3(5)\)
 Step 3:
Multiply: \(8 + 15\)
\( = 23\)

Example 3: Exponents Involved

Evaluate \( x^2 + 4x - 1 \) when \( x = 3 \).

Step 1:
Substitute: \((3)^2 + 4(3) - 1\)
 Step 2:
Exponent first: \(9 + 4(3) - 1\)
 Step 3:
Multiply: \(9 + 12 - 1\)
 Step 4:
Add and subtract left to right: \(21 - 1\)
\( = 20\)

Example 4: A Real-World Problem

A taxi charges a base fare of \( \$3 \) plus \( \$2 \) per mile. The total cost is \( 3 + 2m \). How much for a \( 7 \)-mile ride?

1
Expression: \( 3 + 2m \)
2
Substitute \( m = 7 \): \( 3 + 2(7) \)
3
Multiply: \( 3 + 14 \)
4
Add: \( = 17 \). The ride costs \( \$17 \).

Example 5: Expressions with Fractions

Evaluate \( \frac{n + 8}{3} \) when \( n = 7 \).

1
Substitute: \( \frac{(7) + 8}{3} \)
2
Numerator first (grouping): \( \frac{15}{3} \)
3
Divide: \( = 5 \)

Common Mistakes to Avoid

✕ Forgetting Parentheses
Writing \( 34 \) instead of \( 3(4) = 12 \) when substituting \( x = 4 \) into \( 3x \).
✕ Wrong Order of Ops
Adding before multiplying: \( 2 + 3(5) \neq 25 \). Correct: \( 2 + 15 = 17 \).
✕ Substituting the Wrong Variable
Putting the value of \( a \) where \( b \) should go when there are two variables.

Summary Reference Table

Expression Given Values Substitution Result
\( 5x - 3 \) \( x = 6 \) \( 5(6) - 3 \) 27
\( 2a + 3b \) \( a = 4, b = 5 \) \( 2(4) + 3(5) \) 23
\( x^2 + 4x - 1 \) \( x = 3 \) \( (3)^2 + 4(3) - 1 \) 20
\( \frac{n+8}{3} \) \( n = 7 \) \( \frac{7+8}{3} \) 5

Question 1 of 5

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

Question 2 of 5

Evaluate \( n^2 - 5 \) when \( n = 4 \).

Question 3 of 5

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

Question 4 of 5

Evaluate \( \frac{2x + 6}{4} \) when \( x = 5 \).

Question 5 of 5

A store sells notebooks for \( \$3 \) each. The expression \( 3p \) gives the total cost for \( p \) notebooks. What is the cost of 9 notebooks?