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Order of Operations in Algebra: Evaluating Expressions Correctly

Capítulo 3

Estimated reading time: 3 minutes

The Order of Operations (with Variables)

When an expression includes several operations, you must follow a consistent order so everyone gets the same value. Variables do not change the order; you simply substitute a number for the variable first, then compute using the same rules.

Order to follow

Parentheses (and grouping symbols like brackets)
Exponents
Multiplication and Division from left to right
Addition and Subtraction from left to right

Important: Multiplication does not always come before division; you do whichever appears first when reading left to right. The same is true for addition and subtraction.

A Copyable Worked-Example Template

Use this template every time you evaluate an expression with a given variable value.

1) Rewrite the expression clearly (add parentheses if helpful). 2) Substitute the given value(s) for the variable(s). 3) Compute in order: parentheses → exponents → ×/÷ left-to-right → +/− left-to-right. 4) State the final value (with units if the problem has them).

Template Example 1: Evaluate 2x + 3 when x = 4

1) Rewrite: 2x + 3

2) Substitute: 2(4) + 3

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3) Compute in order: multiply first (it is multiplication, then addition): 2(4) = 8, so 8 + 3 = 11

4) Final value: 11

Template Example 2: Parentheses change everything

Evaluate both expressions when x = 5.

Expression	Substitute	Compute in order	Value
`3x + 2`	`3(5) + 2`	`15 + 2`	`17`
`3(x + 2)`	`3(5 + 2)`	parentheses first: `5 + 2 = 7`, then `3·7 = 21`	`21`

The expressions look similar, but the parentheses force you to add before multiplying.

Step-by-Step Evaluations with Multiple Operations

Example 3: Multiplication/division left-to-right

Evaluate 18 ÷ 3x when x = 2.

1) Rewrite clearly: 18 ÷ 3x means (18 ÷ 3)·x because division and multiplication are done left-to-right.

2) Substitute: (18 ÷ 3)·2

3) Compute in order: 18 ÷ 3 = 6, then 6·2 = 12

4) Final value: 12

Common mistake: Treating 18 ÷ 3x as 18 ÷ (3x). If you actually mean that, you must write parentheses: 18 ÷ (3x).

Example 4: Addition/subtraction left-to-right

Evaluate 20 - 3x + 4 when x = 2.

Substitute: 20 - 3(2) + 4

Compute: multiply: 3(2)=6, so 20 - 6 + 4. Now go left-to-right: 20 - 6 = 14, then 14 + 4 = 18.

Final value: 18

Common mistake: Doing -6 + 4 first because they are “next to each other.” Without parentheses, you still go left-to-right for + and −.

Example 5: Exponents and negatives (a frequent trap)

Evaluate each expression.

Expression	Meaning	Value
`-3^2`	The exponent applies to `3` only, then the negative sign stays outside: `-(3^2)`	`-(9) = -9`
`(-3)^2`	The exponent applies to the entire negative number inside parentheses	`9`

Key idea: Parentheses decide what is being squared. If the negative is inside parentheses, it is squared too.

Practice: Spot and Fix Common Mistakes

Mistake Type 1: Ignoring parentheses

Evaluate correctly when x = 3.

4(x - 1) + 2
4x - 1 + 2

Check yourself: These are not the same expression. In the first one, you must do (x - 1) before multiplying by 4.

Mistake Type 2: Distributing incorrectly during evaluation

Students sometimes try to “distribute” but change signs or forget a term. Evaluate when x = -2:

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

Reminder for accuracy: If you distribute, multiply the outside number by every term inside the parentheses, including signs. You can also avoid distributing by substituting first, then doing the parentheses arithmetic.

Mistake Type 3: Mishandling negative numbers

Evaluate when x = -3:

x^2
-x^2
(-x)^2

Tip: After substitution, add parentheses to make the structure visible. For example, if x = -3, then x^2 becomes (-3)^2.

Self-Check: Where Does Order Matter?

Answer each question and justify the step where order matters (for example: “I did parentheses first because…” or “I squared before multiplying because…”).

1) Evaluate 2(x + 3)^2 when x = 1. Which step must happen before multiplying by 2?
2) Evaluate 16 ÷ 4x when x = 2. Explain why left-to-right matters here, and rewrite the expression to show the intended grouping.
3) Evaluate 10 - 2x - 3 when x = 4. Identify the point where left-to-right affects the result.
4) Compare -2^3 and (-2)^3. Explain which symbol (parentheses or exponent) changes what is being powered.
5) A student evaluates 3(x - 2) + 5 at x = 6 and gets 3·6 - 2 + 5 = 21. Identify the first incorrect step and correct the work using the template.

Now answer the exercise about the content:

When evaluating the expression 18 ÷ 3x at x = 2, what is the correct way to interpret the operations before substituting and computing?

You are right! Congratulations, now go to the next page

You missed! Try again.

Division and multiplication have the same priority, so you compute them left-to-right. That means 18 ÷ 3x is interpreted as (18 ÷ 3)·x, not 18 ÷ (3x) unless parentheses are written.