91Ó°ÊÓ

When you encounter two or more equations containing the same variables, this is known as a system of equations. The goal is to find a set of values for the variables that satisfy all equations simultaneously. In this exercise, you have two equations:

• \( x - y = -1 \)
• \( y = x^2 + 1 \)

Each equation gives us valuable information about the relationship between the variables \(x\) and \(y\). By combining and working through these equations, you aim to discover the specific \(x\) and \(y\) that fit both relationships observed in the given equations.

A quadratic equation is any equation that can be rearranged in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In our problem, we encountered a quadratic equation after substituting \( y \) in the first equation.
Through substitution, the original system became \( x^2 - x + 2 = 0 \). This is a classic quadratic form, representing a parabola when graphed. Quadratic equations are solved using techniques like factoring, completing the square, or the quadratic formula, which we employed here. The quadratic formula allows us to find the roots of the equation, giving us the potential \( x \)-values needed to solve our system.

Complex solutions arise when solving quadratic equations, particularly when the term under the square root in the quadratic formula, known as the discriminant, is negative. In the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), if \( b^2 - 4ac < 0 \), then the solutions include imaginary numbers.
In this exercise, we found that \( b^2 - 4ac = -7 \), leading to complex solutions. These solutions were expressed in the form \( x = \frac{1}{2} \pm \frac{\sqrt{7}i}{2} \), where \( i \) denotes the imaginary unit, satisfying \( i^2 = -1 \). Working with complex numbers often involves understanding the real and imaginary parts of a number, crucial when interpreting solutions to typical algebra problems involving imaginary components.

Algebraic substitution is a method used to simplify and solve equations or systems of equations. It involves replacing one variable with an equivalent expression involving the other variable(s). This technique can make complex problems more manageable by reducing them to simpler forms.
In our exercise, we substituted \( y = x^2 + 1 \) into the first equation \( x - y = -1 \). By substituting \( y \) from the second equation, the original system transitions into a single equation involving just \( x \). This simplification is what allowed us the strategic advantage to solve for \( x \) easily through the quadratic formula, eventually leading to the determination of both variables.

To "solve for variables" simply means finding the values of the variables that satisfy the equations in a system. This process often involves different strategies and methods depending on the type of equations you're working with.
In this scenario, after transforming the system using substitution, we solved a quadratic equation to find the \( x \)-values. With these \( x \)-values: \( \frac{1}{2} \pm \frac{\sqrt{7}i}{2} \), the corresponding \( y \)-values were calculated using \( y = x^2 + 1 \). The outcome is a pair of solutions for \( x \) and \( y \), ensuring that for each \( x \), the \( y \) derived also satisfies the original equations. This verification step confirms the integrity and completeness of the solution process.