Solve 2x + 3 = 9

Blackboard with the equation 2x + 3 = 9 written in chalk

To solve the linear equation 2x + 3 = 9, isolate the variable x by performing inverse operations to both sides. This process involves subtracting 3 from both sides, then dividing by 2.

How Linear Equations Work

A linear equation is an equation that produces a straight line when graphed. These equations contain one or more variables with no exponents. The goal is to find the value of the variable that makes the equation true.

Main Types of Linear Equations

Linear equations can be categorized based on the number of variables and the form they take. The most common types include:

One-variable equations: These have only one variable, such as 2x + 3 = 9.
Two-variable equations: These involve two variables, typically represented as y = mx + b.
Multi-variable equations: These equations have more than two variables and require systems of equations to solve.

Step-by-Step Solution

Let's solve the equation 2x + 3 = 9 step-by-step:

Subtract 3 from both sides: This eliminates the constant term on the left side.
```
2x + 3 - 3 = 9 - 3
```
This simplifies to:
```
2x = 6
```
Divide both sides by 2: This isolates the variable x.
```
(2x)/2 = 6/2
```
This simplifies to:
```
x = 3
```

The solution to the equation 2x + 3 = 9 is x = 3.

Common Mistakes

When solving linear equations, it's easy to make mistakes. Here are a few common errors to avoid:

Forgetting to perform the same operation on both sides: Always ensure that whatever operation you perform on one side of the equation, you also perform on the other side.
Incorrectly combining like terms: Make sure to combine like terms correctly to avoid errors in simplification.
Misidentifying the variable: Ensure you are solving for the correct variable, especially in equations with multiple variables.

Verification

To verify the solution, substitute x = 3 back into the original equation:

2(3) + 3 = 9

This simplifies to:

6 + 3 = 9

Which is true. Therefore, x = 3 is the correct solution.

Further Examples

Let's consider another example to reinforce the concept. Solve the equation 2x - 5 = 7:

Add 5 to both sides:
```
2x - 5 + 5 = 7 + 5
```
This simplifies to:
```
2x = 12
```
Divide both sides by 2:
```
(2x)/2 = 12/2
```
This simplifies to:
```
x = 6
```

The solution to the equation 2x - 5 = 7 is x = 6.

For more complex equations, such as those involving fractions or decimals, the same principles apply. Always aim to isolate the variable by performing inverse operations systematically.

Understanding how to solve linear equations is a fundamental skill in algebra. It forms the basis for more advanced topics, including solving systems of equations and graphing linear functions. Practice regularly to build confidence and proficiency.