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When dealing with systems of equations, the goal is to find the values of variables that satisfy all equations simultaneously. These systems can include two or more equations involving the same set of variables. In our exercise, we have two equations with two variables, which forms a simple system.

One common method to solve such systems is the substitution method, which involves expressing one variable in terms of the other and then substituting this expression into the other equation. This effectively reduces the system to a single equation with one variable. After finding the value of one variable, you can easily find the value of the other by substituting it back into any of the original equations.

The substitution method is particularly useful when one of the equations can be easily rearranged to isolate one variable, which is exactly what was done in the step-by-step solution.

The quadratic formula is a powerful tool used to find the roots of quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). The quadratic formula is expressed as \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\). This formula allows us to find solutions by simply substituting the coefficients \(a\), \(b\), and \(c\) from the equation.

In the provided exercise, we apply the quadratic formula to solve for \(y\) after substituting \(x\) with \(1 - y\) and simplifying the equation. However, we encounter a negative value under the square root, which is known as the discriminant. In the case where the discriminant is negative, it tells us that the roots of the quadratic equation are complex numbers, indicating there are no real solutions to the equation.

When we encounter a negative discriminant using the quadratic formula, this leads us to the realm of complex numbers. A complex number is composed of a real part and an imaginary part and is usually represented in the form \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit with the property that \(i^2 = -1\).

In our exercise example, the discriminant is \(9 - 24\), which is negative. Using the quadratic formula, the roots of the quadratic equation become complex numbers. Given that the original problem is set within the realm of real numbers, complex solutions indicate that there is no solution to the system in the set of real numbers.

Understanding complex numbers is critical when working with quadratic equations, as they expand our ability to find solutions beyond the real number system. Complex numbers are used extensively in fields such as engineering, physics, and applied mathematics.