Let \(a, b, c\) be positive real numbers, the following systems of equations in \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\). \(\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)-\left(z^{2} / c^{2}\right)=1\) \(-\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)-\left(z^{2} / c^{2}\right)=1\) \(-\left(\mathrm{x}^{2} / \mathrm{a}^{2}\right)-\left(\mathrm{y}^{2} / \mathrm{b}^{2}\right)+\left(\mathrm{z}^{2} / \mathrm{c}^{2}\right)=1\), has (a) unique solution (b) no solution (c) finitely many solutions (d) infinitely many solutions.

Real numbers are the building blocks of algebra and encompass both rational numbers (like \frac{1}{2}, 0.75, or -2) and irrational numbers (such as \( \sqrt{2} \), \pi, or e). They are found on the number line extending in both directions infinitely, including positive numbers, negative numbers, and zero. In the system of equations presented in the exercise, we are working with positive real numbers \( a, b, \)and \( c \). Positive real numbers have crucial properties that affect the outcome of equations, especially when discussing systems that include squares of these numbers. Since squares of real numbers are always non-negative (\(x^2 \geq 0\) for any real number \(x\)), this fact comes into play when seeking solutions for the given system of equations.

For instance, the final step of our solution process leads to asserting that \( z^2/c^2 = -1 \), which contradicts with the property that squares of real numbers are non-negative. Thus, such an equation has no solution in the realm of real numbers, directly influencing our reasoning in determining that the system has no consistent solution.

Quadratic equations are second-degree polynomial equations in one variable, generally expressed in the form \( ax^2 + bx + c = 0 \), where \( a, b, \)and \( c \) are coefficients and \( a \eq 0 \). These equations are pivotal in algebra and have diverse applications in various fields, including physics, engineering, and finance. Quadratic equations usually have two solutions that can be found using methods such as factoring, completing the square, or the quadratic formula.

Within the context of the exercise, the system involves quadratic terms such as \( x^2/a^2, y^2/b^2, \)and \( z^2/c^2 \). However, the contradiction we encounter suggests something fundamental about quadratic equations: if a quadratic equation yields a negative value for the square of a variable (such as \( x^2 = -1 \) in this case about \( z^2/c^2 = -1 \) ), it has no solution within the real number system. The recognition of this property allows us to correctly conclude that the system of equations provided does not have a solution.

A consistent solution in the realm of systems of equations refers to a set of values for the variables that satisfies all the equations simultaneously. When all the equations work together, without any contradictions, and yield a specific solution (or set of solutions), the system is considered to be consistent. There are different types of consistent systems; some have a unique solution, others have infinitely many solutions, depending on the number of equations relative to the number of unknowns.

Systems with no solution are labeled inconsistent, meaning there is no possible set of values for the variables that would make all the equations true at the same time. In the step-by-step solution of the given exercise, we identify a contradiction implying that our supposed solution does not fit within the confines of the real numbers, making the system inconsistent. By understanding the characteristics of real numbers and quadratic equations, we can see why the system cannot have a consistent solution, leading to the conclusion of no solution.