What are the rules for solving equations?

Solving equations involves a series of steps to isolate the variable and find its value. Here's a breakdown of the general rules:

General Steps for Solving Equations

The goal is to isolate the variable (e.g., x, y) on one side of the equation. Here's a step-by-step approach:

Simplify Each Side: According to the references, the first step is to simplify each side of the equation by removing parentheses and combining like terms. This involves:
- Removing parentheses using the distributive property. For example, 2(x + 3) becomes 2x + 6.
- Combining like terms on each side. For example, 3x + 2x - 1 becomes 5x - 1.
Isolate the Variable Term: The next step is to use addition or subtraction to isolate the variable term on one side of the equation. To do this, perform the opposite operation to move constants away from the variable term. For example, if you have x + 5 = 10, subtract 5 from both sides to get x = 5.
Solve for the Variable: Finally, use multiplication or division to solve for the variable. If the variable is multiplied by a number, divide both sides by that number. If the variable is divided by a number, multiply both sides by that number. For example, if you have 2x = 10, divide both sides by 2 to get x = 5.

Here's a more detailed look at the rules, along with examples:

Rule	Explanation	Example
1. Simplify Both Sides	Combine like terms and eliminate parentheses to make the equation easier to work with.	2(x + 1) + 3x = 5 -> 2x + 2 + 3x = 5 -> 5x + 2 = 5
2. Addition/Subtraction Property	You can add or subtract the same number from both sides of the equation without changing its solution.	x - 3 = 7 -> x - 3 + 3 = 7 + 3 -> x = 10
3. Multiplication/Division Property	You can multiply or divide both sides of the equation by the same non-zero number without changing its solution.	3x = 12 -> 3x / 3 = 12 / 3 -> x = 4
4. Distributive Property	Used to remove parentheses by multiplying the term outside the parentheses by each term inside.	2(x + 4) = 10 -> 2x + 8 = 10
5. Combining Like Terms	Combine terms that have the same variable and exponent.	4x + 2x - 3 = 9 -> 6x - 3 = 9

Keep it Balanced: Remember, an equation is like a balanced scale. Whatever you do to one side, you must do to the other to maintain the balance.
Work Backwards: Consider the order of operations (PEMDAS/BODMAS) when solving equations, but often you'll reverse the order (undoing addition/subtraction before multiplication/division) to isolate the variable.
Check Your Solution: After solving for the variable, substitute the value back into the original equation to verify that it is correct. This is a crucial step to avoid mistakes.

Example:

Solve for x: 3(x + 2) - 5 = 16