Order of Operations with Variables

In arithmetic you learned PEMDAS — the order in which operations must be performed. In algebra, the same rules apply, but now expressions contain variables (letters that stand for unknown or changeable values). Being able to evaluate these expressions correctly — substituting numbers for variables and simplifying in the right order — is the gateway skill to every topic that follows.

Quick PEMDAS Review

The order of operations for any mathematical expression is:

1. P — Parentheses (and other grouping symbols: brackets, braces, fraction bars)
2. E — Exponents (and roots)
3. MD — Multiplication and Division, left to right
4. AS — Addition and Subtraction, left to right

Multiplication and division have equal priority — work left to right. The same is true for addition and subtraction.

Evaluating Expressions by Substitution

To evaluate an algebraic expression, replace every variable with its given value, then follow PEMDAS to simplify.

Example 1: Evaluate $3x + 7$ when $x = 4$

Step 1 — Substitute:

$3(4) + 7$

Step 2 — Multiply first (MD before AS):

$12 + 7$

Step 3 — Add:

$19$

Answer: $19$

Example 2: Evaluate $5a^2 - 2a + 1$ when $a = 3$

Step 1 — Substitute:

$5(3)^2 - 2(3) + 1$

Step 2 — Exponent first:

$5(9) - 2(3) + 1$

Step 3 — Multiply:

$45 - 6 + 1$

Step 4 — Add and subtract left to right:

$45 - 6 + 1 = 39 + 1 = 40$

Answer: $40$

Example 3: Evaluate $\dfrac{x + y}{x - y}$ when $x = 10$ and $y = 4$

A fraction bar acts as a grouping symbol — evaluate the numerator and denominator separately before dividing.

Numerator: $10 + 4 = 14$

Denominator: $10 - 4 = 6$

$\frac{14}{6} = \frac{7}{3}$

Answer: $\dfrac{7}{3}$

Nested Grouping Symbols

When parentheses appear inside brackets or braces, work from the innermost grouping outward.

Example 4: Simplify $2[3(x + 1) - 4]$ when $x = 5$

Step 1 — Innermost parentheses first. Substitute $x = 5$:

$2[3(5 + 1) - 4]$

$= 2[3(6) - 4]$

Step 2 — Multiply inside the brackets:

$= 2[18 - 4]$

Step 3 — Subtract inside the brackets:

$= 2[14]$

Step 4 — Multiply:

$= 28$

Answer: $28$

Example 5: Evaluate $-\{2[x^2 - (3x + 1)]\}$ when $x = 4$

Step 1 — Innermost parentheses:

$3(4) + 1 = 13$

Step 2 — Brackets next:

$x^2 - 13 = (4)^2 - 13 = 16 - 13 = 3$

Step 3 — Multiply inside braces:

$2(3) = 6$

Step 4 — Apply the negative sign (braces):

$-6$

Answer: $-6$

The Exponent Trap: $-x^2$ vs. $(-x)^2$

This distinction causes more errors than almost anything else in algebra.

• $-x^2$ means “take $x$, square it, then negate.” The exponent applies to $x$ only.
• $(-x)^2$ means “negate $x$ first, then square the result.” The exponent applies to $-x$.

Example 6: Evaluate both when $x = 3$

$-x^2 = -(3)^2 = -(9) = -9$

$(-x)^2 = (-3)^2 = 9$

These give different answers. The parentheses make all the difference.

Expressions with Multiple Variables

Many real-world formulas involve two or more variables. The process is the same: substitute all values, then simplify with PEMDAS.

Example 7: Evaluate $P = 2l + 2w$ when $l = 15$ and $w = 8$

$P = 2(15) + 2(8) = 30 + 16 = 46$

Answer: $P = 46$

This is the perimeter formula for a rectangle — you will see it in geometry and in many trades.

Real-World Application: Electrician — Ohm’s Law

Electricians use Ohm’s Law constantly:

$V = IR$

where $V$ is voltage (volts), $I$ is current (amps), and $R$ is resistance (ohms).

Scenario: A circuit has a current of $I = 3.5$ amps flowing through a resistance of $R = 12$ ohms. Find the voltage.

Step 1 — Substitute:

$V = (3.5)(12)$

Step 2 — Multiply:

$V = 42$

Answer: The voltage is $42$ volts.

Now suppose two resistors are in series ($R_1 = 12$ ohms, $R_2 = 8$ ohms). The total voltage is:

$V = I(R_1 + R_2) = 3.5(12 + 8) = 3.5(20) = 70 \text{ volts}$

Notice the parentheses — you must add the resistances before multiplying. Without them, $3.5 \times 12 + 8 = 42 + 8 = 50$, which is wrong.

Real-World Application: Nursing — Dosage by Weight

Nurses calculate medication dosages using body weight formulas. A common pattern is:

$D = \frac{W \times R}{C}$

where $D$ is the dose to administer, $W$ is the patient’s weight in kilograms, $R$ is the recommended dose rate (mg per kg), and $C$ is the concentration of the medication (mg per mL).

Scenario: A patient weighs $W = 70$ kg. The physician orders $R = 5$ mg/kg. The medication comes in a concentration of $C = 50$ mg/mL. Find the volume to administer.

Step 1 — Substitute:

$D = \frac{70 \times 5}{50}$

Step 2 — Multiply the numerator (grouping symbol — fraction bar):

$D = \frac{350}{50}$

Step 3 — Divide:

$D = 7 \text{ mL}$

Answer: Administer $7$ mL. The fraction bar acts as a grouping symbol — it tells you to evaluate the entire numerator ($70 \times 5 = 350$) before dividing by the denominator. Treating the fraction bar correctly is essential in clinical dosage calculations.

Common Mistakes to Avoid

1. Forgetting to square before negating. $-4^2 = -16$, not $16$. If you want a positive result, write $(-4)^2 = 16$.

2. Not treating the fraction bar as a grouping symbol. In $\frac{8 + 4}{3}$, you must add first: $\frac{12}{3} = 4$. Computing $8 + \frac{4}{3}$ is incorrect.

3. Multiplying before handling inner parentheses. In $2(3 + x)$, you cannot write $6 + x$. Evaluate or simplify inside the parentheses first — if $x$ is known, add it to 3; if $x$ is unknown, distribute: $6 + 2x$.

4. Substituting incorrectly for negative values. When $x = -2$ and the expression is $x^2$, write $(-2)^2 = 4$. Dropping the parentheses gives $-2^2 = -4$, which is wrong.

5. Working right to left instead of left to right for equal-priority operations. In $24 \div 4 \times 2$, divide first (left to right): $6 \times 2 = 12$. Working right to left gives $24 \div 8 = 3$, which is incorrect.

Practice Problems

Key Takeaways

• PEMDAS applies to algebraic expressions exactly the same way it applies to numeric ones
• To evaluate an expression, substitute the given values for every variable, then simplify step by step
• A fraction bar is a grouping symbol — evaluate the top and bottom separately before dividing
• Work innermost grouping symbols first when parentheses are nested inside brackets or braces
• Watch the critical difference between $-x^2$ (negative) and $(-x)^2$ (positive when $x$ is positive)
• Always wrap negative substitutions in parentheses to avoid sign errors
• Getting the order of operations right in real-world formulas (Ohm’s Law, dosage calculations) is not just a math skill — it is a safety issue

Return to Algebra for more topics in this section.