Algebraic Foundations & Equations

TL;DR

Algebra uses letters (variables) to represent unknown numbers, allowing you to solve problems by finding those unknowns. Equations show that two expressions are equal, and you can manipulate them to isolate the variable. Think of solving equations as balancing a scale: whatever you do to one side, you must do to the other to keep it balanced.

1. The Mental Model

Imagine an algebraic equation like a balanced scale. Each side has some items, and the goal is to figure out the weight of a mystery item (your variable) while keeping the scale perfectly balanced. Whatever you add or remove from one side, you must do the exact same to the other.

2. The Core Material

When we talk about algebraic foundations, we're mostly talking about understanding variables, expressions, and how to solve equations.

What are Variables and Expressions?

A variable is simply a letter, like x, y, or a, that stands in for an unknown number. We use variables when we want to talk about "some number" without knowing exactly what it is yet, or when a number can change.

An expression is a combination of numbers, variables, and operation signs (+, -, , /). It doesn't have an equals sign.
* Examples of expressions:* x + 5, 2y - 7, 3a/4, x^2

What is an Equation?

An equation is a statement that two expressions are equal. It always has an equals sign (=). The goal is often to find the value(s) of the variable(s) that make the equation true.
* Examples of equations: x + 5 = 10, 2y - 7 = 3, 3a/4 = 6

Solving Basic Equations

The main idea for solving equations is to isolate the variable. This means getting the variable all by itself on one side of the equals sign. To do this, you'll perform inverse operations.

• Addition and Subtraction are inverse operations:

  • If you have x + 3 = 7, to get x alone, you subtract 3 from both sides:
    x + 3 - 3 = 7 - 3
x = 4
  • If you have y - 5 = 2, to get y alone, you add 5 to both sides:
    y - 5 + 5 = 2 + 5
y = 7

• Multiplication and Division are inverse operations:

  • If you have 2x = 8, to get x alone, you divide both sides by 2:
    2x / 2 = 8 / 2
x = 4
    (Remember, 2x means 2 * x)
  • If you have x / 3 = 5, to get x alone, you multiply both sides by 3:
    (x / 3) * 3 = 5 * 3
x = 15

Combining Like Terms

Before solving, you might need to simplify expressions by combining "like terms." Like terms have the same variable raised to the same power.
* Examples:
* 3x and 5x are like terms. You can combine them: 3x + 5x = 8x.
* 2y and 7y are like terms. 2y - 7y = -5y.
* 4 and 9 are like terms (they're just numbers, or "constants"). 4 + 9 = 13.
* 3x and 2y are NOT like terms.
* x^2 and x are NOT like terms, even though they both have x.

When solving equations, combine like terms on each side of the equation before you start moving terms across the equals sign.

Equations with Multiple Steps

Often, you'll need to use several inverse operations. A good general strategy is:
1. Simplify: Distribute anything, and combine like terms on each side of the equation.
2. Isolate the variable term: Use addition or subtraction to get all terms with the variable on one side, and all constant terms on the other.
3. Isolate the variable: Use multiplication or division to get the variable by itself.

3. Worked Example

Let's solve the equation: 5x + 7 - 2x = 22

1. Simplify (combine like terms on the left side):
   We have 5x and -2x. These are like terms. Also, 7 is a constant.
   5x - 2x + 7 = 22
3x + 7 = 22

2. Isolate the variable term (3x):
   We need to get rid of the + 7 on the left side. The inverse operation is to subtract 7. Remember to do it to both sides.
   3x + 7 - 7 = 22 - 7
3x = 15

3. Isolate the variable (x):
   We have 3x, which means 3 * x. The inverse operation is to divide by 3. Do it to both sides.
   3x / 3 = 15 / 3
x = 5

So, the solution is x = 5. You can always check your answer by plugging it back into the original equation: 5(5) + 7 - 2(5) = 25 + 7 - 10 = 32 - 10 = 22. It works!

4. Key Takeaways

• Variables are letters representing unknown numbers in math problems.
• An expression is a combination of numbers, variables, and operations, without an equals sign.
• An equation states that two expressions are equal and always contains an equals sign.
• Solving an equation means finding the value(s) of the variable(s) that make the statement true.
• To solve equations, use inverse operations to isolate the variable, always performing the same action on both sides to maintain balance.
• Combine "like terms" on each side of an equation before attempting to move terms across the equals sign.

Common mistakes you should avoid:
* Doing an operation to one side of the equation but forgetting to do it to the other.
* Forgetting to combine like terms before trying to isolate the variable.
* Mixing up inverse operations (e.g., adding when you should subtract, or multiplying when you should divide).
* Ignoring the signs (+ or -) in front of terms when moving them or combining them.

5. Now Try It

Solve the following equation for y:
4y - 6 + 2y = 18 - 3

What to do:
1. Combine any like terms on the left side of the equation.
2. Combine any like terms (constants) on the right side of the equation.
3. Use inverse operations to get the y term by itself on one side.
4. Use inverse operations to get y completely by itself.

What success looks like: You should arrive at a single numerical value for y, and if you substitute that value back into the original equation, both sides will simplify to the same number.