the concept of solving linear equations and solving for a letter

Solving linear equations involves finding the value of a variable (usually represented by a letter like \( x \), \( y \), or \( z \)) that makes the equation true. Linear equations are equations where the highest power of the variable is 1. The process of solving for a letter (variable) means isolating the variable on one side of the equation.

Here's a step-by-step explanation of how to solve a basic linear equation:

### General Form:
A linear equation usually looks like this:
\[
ax + b = c
\]
where:
- \( x \) is the variable (the letter we are solving for),
- \( a \) and \( b \) are constants (numbers), and
- \( c \) is the constant on the other side of the equation.

### Steps to Solve:
1. **Isolate the variable**:
   - Start by simplifying the equation if needed (e.g., combining like terms).
   - Move all terms with the variable on one side of the equation (usually to the left).
   - Move the constants (numbers without variables) to the other side.

2. **Use inverse operations**:
   - If there's an addition or subtraction involving the variable, undo it with the opposite operation (i.e., subtract if added or add if subtracted).
   - If the variable is multiplied by a constant, divide both sides by that constant to isolate the variable.

3. **Simplify**:
   - Once the variable is isolated, perform any remaining calculations to find its value.

### Example:

Solve the equation \( 2x + 4 = 12 \).

1. **Subtract 4 from both sides**:
   \[
   2x + 4 - 4 = 12 - 4 \quad \Rightarrow \quad 2x = 8
   \]

2. **Divide both sides by 2**:
   \[
   \frac{2x}{2} = \frac{8}{2} \quad \Rightarrow \quad x = 4
   \]

So, \( x = 4 \) is the solution.

### Key Points:
- **Inverse operations**: Addition ↔ Subtraction, Multiplication ↔ Division.
- **Simplify step by step**.
- **Check the solution** by plugging it back into the original equation to see if both sides are equal.

Would you like more details on any of the steps or a different example?

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Here are 5 related questions you might explore:

1. What if there are variables on both sides of the equation?
2. How do you solve linear equations with fractions?
3. What are linear equations with two variables?
4. How do you graph linear equations?
5. How can you check if a system of linear equations has no solution or infinitely many solutions?

**Tip**: Always simplify both sides of the equation as much as possible before trying to isolate the variable.

The concept of solving linear equations and solving for a letter.

Discover how to solve linear equations and isolate variables using inverse operations. This guide explains step-by-step methods, such as simplifying equations and solving for unknowns, suitable for students in Grades 6-8.

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