Foundations of Equations

TL;DR

Equations are like balanced scales, where both sides have the same value. Your main goal is to find the value of an unknown variable that makes the equation true. We use inverse operations to isolate this variable and solve the equation.

1. The Mental Model

Think of an equation as a perfectly balanced seesaw. Whatever you do to one side, you must do the exact same thing to the other side to keep it balanced. Our mission is to get the unknown "thing" (the variable) all by itself on one side.

2. The Core Material

An equation is a mathematical statement that shows two expressions are equal. It always has an equals sign (=).

For example: $x + 5 = 10$

Here, $x+5$ is one expression and $10$ is another. The = sign says they're the same value. Our job is to find what $x$ has to be for that statement to be true.

Variables and Constants

Photo by Rashed Paykary on Pexels

• A variable is usually a letter (like $x$, $y$, $a$) that represents an unknown number. Its value can vary.
• A constant is a number whose value never changes (like $5$, $10$, $-3$).

Expressions vs. Equations

Photo by Vanessa Garcia on Pexels

• An expression is a combination of numbers, variables, and operations (like $2x+3$ or $y-7$). It doesn't have an equals sign.
• An equation has an equals sign, showing that two expressions are equivalent (like $2x+3 = 11$).

Inverse Operations

Photo by David Kanigan on Pexels

To solve an equation, we use inverse operations. These are operations that "undo" each other.

• 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 $2$, you divide by $2$.
  • To undo dividing by $4$, you multiply by $4$.

The process of solving equations typically follows these steps:

graph TD
    A["Start with the equation"] --> B["Identify the variable to isolate"]
    B --> C{"Is there addition/subtraction near the variable?"}
    C -- Yes --> D["Apply inverse (subtraction/addition) to both sides"]
    C -- No --> E{"Is there multiplication/division near the variable?"}
    D --> E
    E -- Yes --> F["Apply inverse (division/multiplication) to both sides"]
    E -- No --> G["Variable is isolated!"]
    F --> G
    G --> H["Check your answer (optional but recommended)"]

Solving One-Step Equations

Photo by Vanessa Garcia on Pexels

Let's look at simple examples:

1. Addition/Subtraction:
$x + 7 = 12$
To get $x$ alone, we need to undo the "+ 7". The inverse is subtracting 7.
$x + 7 - 7 = 12 - 7$ (Do it to both sides!)
$x = 5$

2. Subtraction/Addition:
$y - 3 = 8$
To get $y$ alone, we need to undo the "- 3". The inverse is adding 3.
$y - 3 + 3 = 8 + 3$ (Do it to both sides!)
$y = 11$

3. Multiplication/Division:
$4z = 20$ (This means $4 \times z$)
To get $z$ alone, we need to undo the "multiply by 4". The inverse is dividing by 4.
$\frac{4z}{4} = \frac{20}{4}$ (Do it to both sides!)
$z = 5$

4. Division/Multiplication:
$\frac{a}{2} = 6$ (This means $a \div 2$)
To get $a$ alone, we need to undo the "divide by 2". The inverse is multiplying by 2.
$\frac{a}{2} \times 2 = 6 \times 2$ (Do it to both sides!)
$a = 12$

3. Worked Example

Let's solve the equation $3 + m = -9$.

1. Identify the variable: Our variable is $m$.
2. What's happening to the variable? It's being added to $3$. (It's the same as $m + 3 = -9$, the order of addition doesn't matter).
3. What's the inverse operation? To undo adding $3$, we need to subtract $3$.
4. Apply the inverse to both sides:
   $3 + m - 3 = -9 - 3$
5. Simplify:
   $m = -12$
6. Check your answer (optional but good practice): Substitute $m = -12$ back into the original equation.
   $3 + (-12) = -9$
   $-9 = -9$
   It works! So, $m = -12$ is the correct solution.

4. Key Takeaways

• An equation states that two expressions have the same value, marked by an equals sign =.
• A variable is an unknown value, usually represented by a letter.
• Your goal is to isolate the variable on one side of the equation.
• Use inverse operations (addition/subtraction, multiplication/division) to "undo" operations.
• Whatever you do to one side of the equation, you must do the exact same thing to the other side to keep it balanced.
• Always simplify both sides of the equation after performing an operation.

Common Mistakes to Avoid:
* Forgetting to perform the operation on both sides of the equation.
* Mixing up inverse operations (e.g., adding when you should subtract).
* Making sign errors, especially with negative numbers.
* Not simplifying completely after each step.

5. Now Try It

Solve the equation $x - 15 = 20$. Show each step clearly, indicating what operation you're performing on both sides. Once you have your answer for $x$, plug it back into the original equation to verify your solution.

Success looks like: You'll end up with $x$ by itself on one side, a number on the other, and when you check your answer, both sides of the original equation will be equal.