Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} 2 x^{2}-y^{2} &=4 \\ |x| &=|y| \end{aligned}$$

Step-by-Step Solution

Answer

The solutions are (2, 2), (-2, -2), (2, -2), and (-2, 2).

Step 1: Analyze the absolute value equation

The equation \(|x| = |y|\) implies that either \(x = y\) or \(x = -y\). These are two separate cases that we will have to analyze in detail.

Step 2: Solve for Case 1 (x = y)

Substitute \(x = y\) into the first equation: \[2x^2 - y^2 = 4\] becomes \(2x^2 - x^2 = 4\), which simplifies to \(x^2 = 4\). The solutions to this are \(x = 2\) and \(x = -2\). Thus, for Case 1, the solutions are \((x, y) = (2, 2)\) and \((-2, -2)\).

Step 3: Solve for Case 2 (x = -y)

For this case, substitute \(x = -y\) into the first equation: \[2x^2 - y^2 = 4\] becomes \(2x^2 - (-x)^2 = 4\). Simplifying gives \(x^2 = 4\), resulting in the solutions \(x = 2\) and \(x = -2\). Thus, for \(x = -y\), we have the solutions \((x, y) = (2, -2)\) and \((-2, 2)\).

Step 4: Combine the solutions

Gather all potential solutions from both cases. From Case 1, we have \((2, 2)\) and \((-2, -2)\). From Case 2, we have \((2, -2)\) and \((-2, 2)\). These are all valid solutions.

Key Concepts

Absolute Value Equations

Absolute value equations often create multiple scenarios for solving because they deal with the distance from zero on the number line rather than the directed value. The absolute value equation \(|x| = |y|\) tells us that the distance of \(x\) from zero is the same as that of \(y\). This results in two different cases: either \(x\) and \(y\) are the same or they are exact opposites. So \(x = y\) or \(x = -y\). It's important to consider both to cover all possible solutions.
When solving such equations, always remember to split into these basic cases.
Think of absolute value as a distance that can go in either direction, providing multiple cases to solve for.

Real Solutions

Finding real solutions is about identifying the actual values that satisfy the system of equations. These solutions lie on the real number line and aren't imaginary.
In our exercise, we solved for real solutions by analyzing each case provided by the absolute value equation.

• In Case 1, where \(x = y\), the solutions are \((2, 2)\) and \((-2, -2)\).
• In Case 2, where \(x = -y\), the solutions are \((2, -2)\) and \((-2, 2)\).

All these solutions are on the real number line and satisfy both given equations in the system.

Analytical Methods

Analytical methods involve solving equations using algebraic manipulations rather than numerical methods or approximation techniques. These methods are precise and give exact solutions.
In the given problem, we used an analytical approach by substituting potential values, simplifying expressions, and solving quadratic equations. Each expression \(x = y\) and \(x = -y\) was analyzed separately, showing how both must be checked analytically to find results.

We relied on simplification of equations to achieve solutions that are exact, without any trial and error or numeric approximation.

Substitution Method

The substitution method is frequently used for solving systems of equations, especially when they are nonlinear. It involves replacing one variable with another equivalent expression and is ideal when congruent expressions are evident.
In our exercise, we substituted \(x = y\) and \(x = -y\) into the nonlinear equation \(2x^2 - y^2 = 4\).

• For \(x = y\), the equation simplified to \(x^2 = 4\).
• For \(x = -y\), it also simplified to the same quadratic \(x^2 = 4\).

Each substitution was key in reducing the system to more manageable forms, allowing us to solve for \(x\) and consequently \(y\). Through this method, the system was simplified from its nonlinear form to simpler quadratic equations, which are easier to solve.