Question

System of linear equations

Answer 1

A system of linear equations consists of two or more linear equations involving the same set of variables. The main goal is to find the values of these variables that satisfy all equations in the system simultaneously.

### Example of a System of Linear Equations

Consider the following system of two equations with two variables \(x\) and \(y\):

1. \(2x + 3y = 6\) (Equation 1)
2. \(4x - y = 5\) (Equation 2)

### Methods to Solve a System of Linear Equations

1. **Graphical Method**: This involves graphing each equation on the same set of axes and finding the point(s) where the lines intersect.

2. **Substitution Method**:
- Solve one of the equations for one variable in terms of the other.
- Substitute that expression into the other equation.

For the example above, we can solve Equation 2 for \(y\):
\[
y = 4x - 5
\]
Now substitute \(y\) into Equation 1:
\[
2x + 3(4x - 5) = 6
\]

3. **Elimination Method**:
- Add or subtract the equations to eliminate one variable, making it easier to solve for the remaining variable.

We can multiply Equation 2 by 3 to eliminate \(y\):
\[
12x - 3y = 15
\]
Now we have:
- \(2x + 3y = 6\)
- \(12x - 3y = 15\)

Adding the two equations together will eliminate \(y\).

4. **Matrix Method (Row Reduction or Inverse)**:
- Represent the system as an augmented matrix and use Gaussian elimination or find the inverse of the coefficient matrix (if it exists).

The augmented matrix for our example is:
\[
\begin{pmatrix}
2 & 3 & | & 6 \\
4 & -1 & | & 5
\end{pmatrix}
\]

### Solutions

A system can have three types of solutions:
1. **Unique Solution**: One point of intersection (the lines are not parallel).
2. **No Solution**: Parallel lines that never intersect (the system is inconsistent).
3. **Infinitely Many Solutions**: The equations represent the same line (the system is dependent).

### Conclusion

To solve a system of linear equations, choose a method that you find most convenient. Ensure to check your solution by substituting back into the original equations. If you have a specific system you would like me to help with, feel free to provide the equations!