Question

Consistent or Inconsistent Systems of Linear Equations
example

Answer 1

In linear algebra, systems of linear equations can be classified as consistent or inconsistent based on whether they have solutions.

### Consistent Systems
A system of linear equations is considered *consistent* if there is at least one solution. This can happen in two ways:

1. **Unique Solution**: When the equations represent lines (or planes in higher dimensions) that intersect at a single point.
2. **Infinitely Many Solutions**: When the equations represent the same line (or plane), meaning they coincide.

#### Example of Consistent System
1. **Unique Solution**:
\[
\begin{align*}
2x + y &= 4 \quad (1)\\
x - y &= 1 \quad (2)
\end{align*}
\]
This system has a unique solution at the point \((x, y) = (2, 0)\).

2. **Infinitely Many Solutions**:
\[
\begin{align*}
x + 2y &= 4 \quad (1)\\
2x + 4y &= 8 \quad (2)
\end{align*}
\]
This system has infinitely many solutions because the second equation is just a multiple of the first.

### Inconsistent Systems
A system of linear equations is considered *inconsistent* if there are no solutions. This typically occurs when the equations represent parallel lines (or planes) that do not intersect.

#### Example of Inconsistent System
\[
\begin{align*}
2x + y &= 3 \quad (1)\\
2x + y &= 5 \quad (2)
\end{align*}
\]
In this case, both lines have the same slope but different y-intercepts, meaning they are parallel and will never meet. Therefore, there is no solution to this system.

### Summary
- **Consistent System**: Has at least one solution (either unique or infinitely many).
- **Inconsistent System**: Has no solutions (the equations represent parallel lines).