Determine if the following statement is true or false:

    A linear system with three equations and two variables must be inconsistent.

The given statement is **false**. Here's why:

A linear system with three equations and two variables does not necessarily have to be inconsistent. 

### Key Points:

- **Two variables:** Means we are dealing with equations of lines in a two-dimensional plane.
- **Three equations:** Means we have three lines in this plane.

There are several possibilities for such a system:

1. **Consistent (One solution):** The three lines could intersect at a single point. This means that there is one unique solution for the system, where all three equations are satisfied by the same pair of values for the two variables.
   
2. **Consistent (Infinitely many solutions):** The three lines could be coincident, meaning all three are actually the same line. In this case, there are infinitely many solutions since all points on the line are solutions to all three equations.

3. **Inconsistent (No solution):** The lines could be parallel or otherwise arranged such that no common intersection exists for all three lines.

Thus, a system with three equations and two variables can be consistent (either with one or infinitely many solutions), or inconsistent. It is not necessarily inconsistent.

### Do you want further details or have any questions?

Here are 5 related questions you can consider:
1. What does it mean for a system of linear equations to be inconsistent?
2. How can you determine if a system of linear equations has infinitely many solutions?
3. What is the geometric interpretation of a system of two linear equations in two variables?
4. How do you classify linear systems by the number of solutions?
5. Can a system of equations with more unknowns than equations be inconsistent?

**Tip:** Always visualize or draw the lines corresponding to equations in two variables for better understanding of their relationships!

Determine if the following statement is true or false: A linear system with three equations and two variables must be inconsistent.

Explore whether a system with three equations and two variables must be inconsistent. This problem covers key linear algebra concepts, including system consistency, line intersections, and geometric interpretations. Suitable for high school students in Grades 10-12.

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