Intro to Algebraic Expressions & Equations

TL;DR

Algebra uses letters, called variables, to represent unknown numbers in expressions and equations. An expression is a mathematical phrase without an equals sign, while an equation states that two expressions are equal. Changing the subject of a formula means rearranging an equation to isolate a different variable.

1. The Mental Model

Think of algebra like a puzzle where some numbers are hidden behind letters. Your goal is often to figure out what those hidden numbers are, or to rearrange the puzzle pieces to focus on a different hidden number.

2. The Core Material

In algebra, you'll work with variables, which are usually letters (like x, y, a, b) that stand for numbers you don't know yet.

An algebraic expression is a combination of variables, numbers, and operations (like +, -, *, /). It doesn't have an equals sign.
* Examples of expressions: 3x + 5, 2a - b, y/4

An algebraic equation is a statement that two expressions are equal. It always has an equals sign (=).
* Examples of equations: 3x + 5 = 11, A = L * W, y - 7 = 2

The subject of a formula (or equation) is the variable that's all by itself on one side of the equals sign. It's what the formula is "about."
* In A = L * W, A is the subject.
* In P = 2L + 2W, P is the subject.

When you change the subject of the formula, you're rearranging the equation so that a different variable is isolated on one side. You do this by performing the same operations on both sides of the equation to keep it balanced.

Here's how operations work in algebra:

graph TD
    A["Start with a variable/term"] --> B{What operation is applied to it?}

    B --> C["Addition (+ num)"]
    C --> C_INV["To undo, subtract the same number (- num)"]

    B --> D["Subtraction (- num)"]
    D --> D_INV["To undo, add the same number (+ num)"]

    B --> E["Multiplication (* num)"]
    E --> E_INV["To undo, divide by the same number (/ num)"]

    B --> F["Division (/ num)"]
    F --> F_INV["To undo, multiply by the same number (* num)"]

    B --> G["Powers (^ num)"]
    G --> G_INV["To undo, take the root (nth root)"]

    B --> H["Roots (nth root)"]
    H --> H_INV["To undo, raise to a power (^ num)"]

    C_INV --> I["Apply to BOTH sides of the equation"]
    D_INV --> I
    E_INV --> I
    F_INV --> I
    G_INV --> I
    H_INV --> I

The general idea when changing the subject is to "undo" operations in reverse order of operations (PEMDAS/BODMAS). So, you usually undo addition/subtraction first, then multiplication/division, and finally powers/roots.

Key Rules for Rearranging Equations:

Photo by Markus Winkler on Pexels

1. Do the same thing to both sides: Whatever you do to one side of the equals sign, you must do to the other side to keep the equation balanced.
2. Isolate the target variable: Your goal is to get the variable you want as the subject completely by itself on one side.
3. Use inverse operations: To "move" a term or factor from one side to the other, use its inverse operation.
   • + becomes -
   • - becomes +
   • * becomes /
   • / becomes *
   • ^2 (square) becomes sqrt (square root)

3. Worked Example

Let's change the subject of the formula V = I * R to make I the subject. This formula is common in physics, where V is voltage, I is current, and R is resistance.

Original Equation: V = I * R

Goal: Make I the subject (get I by itself).

Step 1: Identify the variable to isolate.
We want to isolate I.

Step 2: See what's "attached" to I and how.
I is being multiplied by R.

Step 3: Use the inverse operation to move R to the other side.
The inverse of multiplying by R is dividing by R.
So, divide both sides of the equation by R.

V / R = (I * R) / R

Step 4: Simplify.
On the right side, R divided by R cancels out, leaving just I.

V / R = I

Step 5: Rearrange (optional, but good practice).
It's common to write the subject on the left side.

I = V / R

Now, I is the subject of the formula.

4. Key Takeaways

• Variables are letters representing unknown numbers.
• An expression is a mathematical phrase without an equals sign.
• An equation is a statement that two expressions are equal, always containing an equals sign.
• The subject of a formula is the variable that is isolated on one side of the equals sign.
• Changing the subject involves using inverse operations to move terms and factors to get the desired variable by itself.
• Always perform the same operation on both sides of the equation to maintain balance.

Common Mistakes to Avoid:

Photo by KATRIN BOLOVTSOVA on Pexels

• Forgetting to do the same thing to both sides: This is the most common error and breaks the equation's balance.
• Mixing up inverse operations: Forgetting that addition undoes subtraction, multiplication undoes division, etc.
• Incorrect order of operations (when undoing): Start by undoing addition/subtraction, then multiplication/division, then powers/roots.
• Distributing incorrectly: If you have 2(x + 3), you can't just divide by 2 on one side without addressing the whole expression. You'd either distribute first or treat (x+3) as a single unit.

5. Now Try It

Change the subject of the formula P = 2L + 2W to make L the subject.

What to do: Start with the equation P = 2L + 2W. Your goal is to rearrange it so that L is completely by itself on one side of the equals sign.

What success looks like: Your final answer should be an equation where L is isolated and the other variables (P, W) are on the other side, looking something like L = ....