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## Solving Linear Systems by Addition

Linear systems, also known as systems of linear equations, are a set of simultaneous equations that can be solved to find the values of the variables they contain. One method for solving linear systems is by addition, where we add the two equations together to eliminate one variable and simplify the system.

### Adding Equations to Solve Systems

To add equations in a linear system, simply align the equations vertically and add the corresponding coefficients of each variable. The result is a new equation that represents the sum of the original equations.

For example, let’s consider the following system of equations:

```
2x + 3y = 7
x - y = 1
```

Adding these equations, we get:

```
(2x + 3y) + (x - y) = 7 + 1
```

Simplifying the left-hand side:

```
3x + 2y = 8
```

This new equation represents the result of adding the original equations.

### Advantages of Adding Equations

Adding equations can be a useful technique for solving linear systems, particularly when:

**One variable has opposite coefficients:**If one variable has opposite coefficients in the two equations, adding the equations will eliminate that variable.**Coefficients need to be multiplied:**If one or both equations require multiplying coefficients to make them compatible, adding the equations can simplify the process.**Elimination is required:**Adding equations can help eliminate a variable, making the system easier to solve.

### Subtracting Equations

In some cases, it may be advantageous to subtract equations instead of adding them. This can be useful when:

**One variable has the same coefficient with opposite signs:**Subtracting the equations will eliminate that variable.**Coefficients need to be divided:**If one or both equations require dividing coefficients to make them compatible, subtracting the equations can simplify the process.

### Conclusion

Adding equations is a valuable technique for solving linear systems when it can simplify the process and eliminate variables. By carefully aligning the equations and adding the corresponding coefficients, we can obtain a new equation that represents the sum of the original equations. This can make the system easier to solve, leading to the correct values for the variables.

### Frequently Asked Questions

**What are the benefits of using addition to solve linear systems?**

- Eliminates variables with opposite coefficients
- Simplifies the process of multiplying coefficients
- Makes the system easier to solve

**When should I use subtraction instead of addition to solve a linear system?**

- When one variable has the same coefficient with opposite signs
- When division of coefficients is required

**How do I determine which equations to add or subtract?**

- Identify equations with opposite coefficients for addition
- Identify equations with the same coefficients with opposite signs for subtraction

**What is the goal of adding or subtracting equations in a linear system?**

- To eliminate variables and simplify the system

**Is adding or subtracting equations the only method for solving linear systems?**

- No, there are other methods such as substitution, elimination, and matrix methods

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