Problem 9 - Algebra A Combined Function

Question

Solve each nonlinear system of equations. $$ \left\\{\begin{array}{l} x^{2}+2 y^{2}=2 \\ x-y=2 \end{array}\right. $$

Step-by-Step Solution

Answer

No real solutions for the system of equations.

Step 1: Analyze the Second Equation

The second equation given is \(x - y = 2\). We can solve this for one of the variables. Let's solve for \(x\) in terms of \(y\). Add \(y\) to both sides to get:\[x = y + 2\].

Step 2: Substitute in the First Equation

Now that we have \(x\) expressed in terms of \(y\), substitute \(x = y + 2\) into the first equation:\[(y+2)^2 + 2y^2 = 2\].

Step 3: Simplify the Substituted Equation

Now, expand and simplify the equation:1. Expand \((y+2)^2\): \((y+2)^2 = y^2 + 4y + 4\).2. Substitute back: \(y^2 + 4y + 4 + 2y^2 = 2\).3. Combine like terms: \(3y^2 + 4y + 4 = 2\).

Step 4: Form a Quadratic Equation

Rearrange the terms to set the quadratic equation to zero:\[3y^2 + 4y + 4 - 2 = 0\]which simplifies to:\[3y^2 + 4y + 2 = 0\].

Step 5: Solve the Quadratic Equation

Solve the quadratic equation \(3y^2 + 4y + 2 = 0\) using the quadratic formula:\[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\],where \(a = 3\), \(b = 4\), \(c = 2\).Calculate the discriminant:\[ b^2 - 4ac = 4^2 - 4 \times 3 \times 2 = 16 - 24 = -8\].Since the discriminant is negative, there are no real solutions for \(y\).

Step 6: Conclusion from the Solution

Since there are no real solutions for \(y\) due to a negative discriminant, the system of nonlinear equations has no real solutions in terms of \(x\) and \(y\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). It allows you to find the values of \(x\) (also known as the roots or solutions) by simply plugging in the coefficients \(a\), \(b\), and \(c\) into the formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula works for any quadratic equation, regardless of whether the roots are real numbers or complex numbers.
It is particularly useful when the equation cannot be factored conveniently. The "\(\pm\)" symbol indicates that there are usually two solutions: one with a plus and one with a minus.
This method is a lifesaver when dealing with complex systems such as the one presented in a nonlinear system of equations.

Discriminant

The discriminant is a crucial part of the quadratic formula, located under the square root: \(b^2 - 4ac\). It provides essential information about the nature of the roots of the quadratic equation.

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If it is negative, the roots are complex (non-real).

In our problem, the discriminant turned out to be \(-8\), which is negative.
This means that the quadratic equation we derived from our system of equations does not have real solutions.
This result heavily affects our approach to finding real number solutions for the system of nonlinear equations.

System of Equations Solving

Solving systems of equations involves finding the values of variables that satisfy all equations simultaneously.
A system can have:

• A unique solution
• Infinitely many solutions
• No solution at all

For non-linear systems, like the given example with quadratic terms, the complexity increases.
The first step is to manipulate the equations to express variables in terms of each other, or to eliminate variables.
In this exercise, we used substitution, a common technique for solving systems of equations, where one variable is expressed in terms of another using one of the equations. This is replaced back into another equation to simplify and solve further.

Variable Substitution

Variable substitution is a fundamental method used in solving systems of equations.
This technique involves taking one of the equations and solving it for one variable in terms of the others, i.e., isolating a variable on one side.
In the original exercise, the second equation \(x - y = 2\) was solved for \(x\) as \(x = y + 2\).

• The next step was to replace \(x\) in the first equation with \(y + 2\), reducing the number of variables and simplifying the complex equations.
• This allows one to focus on solving a single equation with a single variable outcome.

Substitution is especially useful because it reduces the complexity and makes it easier to handle the equations.
This technique is vital for solving both linear and nonlinear systems effectively.