Algebraic Foundations

TL;DR

Algebra is like a puzzle where you use letters (variables) to represent unknown numbers. You'll learn to move these variables around to solve for their values. Mastering this helps you understand and solve many real-world problems.

1. The Mental Model

Think of algebra as a balancing scale. Whatever you do to one side, you must do to the other to keep it balanced. Your goal is to isolate the unknown (the variable) on one side of the scale.

2. The Core Material

What are Variables?

Variables are symbols, usually letters like 'x' or 'y', that stand in for numbers we don't know yet. They're like placeholders.

Expressions vs. Equations

• An expression is a mathematical phrase that can contain numbers, variables, and operations (like addition or subtraction). It doesn't have an equals sign.
  • Examples: x + 5, 3y - 7, 2a^2
• An equation is a statement that two expressions are equal. It always has an equals sign.
  • Examples: x + 5 = 10, 3y - 7 = 2, 2a^2 = 18

Solving Basic Equations

The main goal in algebra is often to "solve" an equation, which means finding the value of the variable that makes the equation true. We do this by using inverse operations.

• Addition and Subtraction are inverse operations: To undo adding 5, you subtract 5. To undo subtracting 3, you add 3.
• Multiplication and Division are inverse operations: To undo multiplying by 4, you divide by 4. To undo dividing by 2, you multiply by 2.

Rule of Thumb: Whatever you do to one side of the equation, you must do to the other side to keep it balanced.

Here's how to solve a common type:

Example: Solve for x in x + 7 = 12
1. Our goal is to get x by itself.
2. x has 7 added to it. The inverse operation of adding 7 is subtracting 7.
3. Subtract 7 from both sides of the equation:
x + 7 - 7 = 12 - 7
4. Simplify:
x = 5

Example: Solve for y in 4y = 20
1. Our goal is to get y by itself.
2. y is multiplied by 4. The inverse operation of multiplying by 4 is dividing by 4.
3. Divide both sides of the equation by 4:
4y / 4 = 20 / 4
4. Simplify:
y = 5

Combining Like Terms

"Like terms" are terms that have the same variables raised to the same powers. You can add or subtract like terms. You can't combine x with x^2, or x with y.

Example: Simplify 3x + 5 + 2x - 1
1. Identify like terms: 3x and 2x are like terms; 5 and -1 are like terms (constants).
2. Combine them:
(3x + 2x) + (5 - 1)
5x + 4

Distributive Property

The distributive property lets you multiply a single term by two or more terms inside parentheses.
a(b + c) = ab + ac

Example: Simplify 3(x + 4)
1. Multiply 3 by x: 3 * x = 3x
2. Multiply 3 by 4: 3 * 4 = 12
3. Combine them: 3x + 12

3. Worked Example

Let's solve the equation 2x + 5 = 17 for x.

1. Isolate the term with the variable: The term with x is 2x. To get it alone, we need to get rid of the + 5.
   • Subtract 5 from both sides:
     2x + 5 - 5 = 17 - 5
2x = 12
2. Isolate the variable: The variable x is multiplied by 2. To get x alone, we need to undo this multiplication.
   • Divide both sides by 2:
     2x / 2 = 12 / 2
x = 6

So, x = 6 is the solution. To check, substitute 6 back into the original equation: 2(6) + 5 = 12 + 5 = 17. It works!

4. Key Takeaways

• Use variables (letters) to represent unknown numeric values.
• Expressions are mathematical phrases, while equations have an equals sign.
• Solve equations by performing inverse operations to isolate the variable.
• Remember to always do the same operation to both sides of an equation.
• You can only add or subtract "like terms" (same variable, same power).
• The distributive property helps expand expressions with parentheses.

Common Mistakes to Avoid

• Forgetting to apply an operation to both sides of an equation.
• Trying to combine unlike terms (e.g., adding 3x and 5).
• Incorrectly applying the distributive property (e.g., 3(x+4) becoming 3x+4 instead of 3x+12).
• Making arithmetic errors when simplifying operations.

5. Now Try It

Solve the following equation for y: 3y - 8 = 16. Then, simplify the expression: 5(x - 2) + 3x.

What to do: First, solve the equation step-by-step for the value of y. Show each inverse operation. Then, apply the distributive property and combine like terms to simplify the expression.

What success looks like: You should find a single numerical value for y and a simplified expression for the second part.