Algebraic Expressions & Simplifying

Why This Matters for Nursing: Dosing formulas often contain variables. "Dose = Weight × 2.5 + 10" is an algebraic expression. Understanding how to read and simplify these helps you use medical formulas correctly.

What You Need to Know

An algebraic expression combines numbers, variables, and operations:

• Variable: A letter representing an unknown value (x, y, n, w)
• Coefficient: The number multiplied by a variable (in 3x, the coefficient is 3)
• Constant: A number without a variable (in 3x + 5, the constant is 5)
• Term: Parts separated by + or - signs (in 3x + 5, there are 2 terms)

Like Terms

Like terms have the same variable raised to the same power:

• 3x and 5x are like terms ✓
• 3x and 3x² are NOT like terms ✗
• 4 and 7 are like terms (both constants) ✓

🧠 Memory Trick

"Combine the Cousins"

Like terms are like family members—you can only combine (add/subtract) terms that are "related" (same variable, same power).

3x + 5x = 8x ✓ (cousins!) 3x + 5y = 3x + 5y (strangers, can't combine)

Simplifying Expressions

Step-by-Step:

1. Identify like terms (same variable, same power)
2. Combine like terms by adding/subtracting coefficients
3. Write in standard form (highest power first, or alphabetical)

Example: Simplify 4x + 3 + 2x - 7

1. Group like terms: (4x + 2x) + (3 - 7)
2. Combine: 6x + (-4)
3. Answer: 6x - 4

The Distributive Property

a(b + c) = ab + ac

Multiply the outside term by EACH term inside the parentheses.

Example: 3(x + 4)

= 3·x + 3·4 = 3x + 12

✏️ Worked Examples

Example 1: Combining Like Terms

Problem: 5a + 3b - 2a + 7b

Step 1 — Identify the like terms. Like terms are terms with the SAME variable (same letter, same power). Here:

• "a" terms: 5a and -2a
• "b" terms: 3b and 7b

Step 2 — Group like terms together. (5a - 2a) + (3b + 7b).

Step 3 — Combine by adding/subtracting the coefficients (the numbers in front of the variables). Keep the variable the same.

• 5a - 2a = 3a (just do 5 - 2 = 3, keep the "a")
• 3b + 7b = 10b (just do 3 + 7 = 10, keep the "b")

Answer: 3a + 10b

💡 You can ONLY combine terms with the same variable. 3a and 3b are like strangers — they can't be combined.