Consider the following system of equations in three real variables x1, x2 and x3 2×1 – x2 + 3×3 = 1 3×1 – 2×2 + 5×3 = 2 -x1 – 4×2 + x3 = 3 This system of equations has A. no solution B. a unique solution C. more than one but a finite number of solutions D. an infinite number of solutions

The correct answer is $\boxed{\text{A. no solution}}$.

To solve a system of equations, we can use Gaussian elimination. In this case, we can eliminate $x_1$ by adding the first equation to the second equation and the third equation. This gives us the following system of equations:

$$5x_2 + 8x_3 = 3$$
$$-5x_2 + 5x_3 = 6$$

We can then eliminate $x_2$ by adding the two equations together. This gives us the following system of equations:

$$13x_3 = 9$$

This equation has no solutions, since the left-hand side is not divisible by the right-hand side. Therefore, the original system of equations has no solutions.

Here is a brief explanation of each option:

• Option A: No solution. This is the case when the system of equations is inconsistent, meaning that there are no values of $x_1$, $x_2$, and $x_3$ that satisfy all of the equations.
• Option B: A unique solution. This is the case when the system of equations is consistent and has a unique solution.
• Option C: More than one but a finite number of solutions. This is the case when the system of equations is consistent and has more than one solution.
• Option D: An infinite number of solutions. This is the case when the system of equations is consistent and has an infinite number of solutions.