Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} x^{2}+y^{2} &=10 \\ 2 x^{2}-y^{2} &=17 \end{aligned}$$

Step-by-Step Solution

Answer

The real solutions are \((3, 1), (-3, 1), (3, -1), (-3, -1)\).

Step 1: Understand the System of Equations

We have two equations: 1. \( x^2 + y^2 = 10 \) 2. \( 2x^2 - y^2 = 17 \) Our goal is to find \( x \) and \( y \) values that satisfy both equations simultaneously.

Step 2: Subtract the Second Equation from the First

Subtract the second equation from the first to eliminate \( y^2 \):\[x^2 + y^2 - (2x^2 - y^2) = 10 - 17\]This simplifies to:\[-x^2 + 2y^2 = -7\]Rearranging gives:\[x^2 - 2y^2 = 7\]

Step 3: Solve for One Variable

We now solve for one of the variables. Let's solve for \( x^2 \) using the equation from Step 2:\[x^2 = 2y^2 + 7\]

Step 4: Substitute Back into the First Equation

Substitute \( x^2 = 2y^2 + 7 \) from Step 3 into the first equation:\[(2y^2 + 7) + y^2 = 10\]Simplifying gives:\[3y^2 + 7 = 10\]\[3y^2 = 3\]\[y^2 = 1\]Thus, \( y = 1 \) or \( y = -1 \).

Step 5: Solve for x Using Each y Value

Using \( y = 1 \), plug into the rearranged equation for \( x^2 \):\[x^2 = 2(1)^2 + 7 = 9\]Thus, \( x = 3 \) or \( x = -3 \).Using \( y = -1 \), plug into the rearranged equation for \( x^2 \):\[x^2 = 2(-1)^2 + 7 = 9\]Thus, \( x = 3 \) or \( x = -3 \).

Step 6: Analyze Solutions

Both values of \( y \) give the same values of \( x \). Therefore, the solutions are:1. \((x, y) = (3, 1)\)2. \((x, y) = (-3, 1)\)3. \((x, y) = (3, -1)\)4. \((x, y) = (-3, -1)\)

Key Concepts

Analytical Solution

An analytical solution involves solving equations using algebraic manipulations rather than numerical approximations or graphical interpretations. When dealing with nonlinear systems of equations, the goal is to find exact expressions for the unknown variables. In the given problem, our task is to solve the system:

• \( x^2 + y^2 = 10 \)
• \( 2x^2 - y^2 = 17 \)

To find solutions analytically, we use algebraic techniques that include substitution and elimination to rewrite the equations in a form where we can more readily find the unknowns. This is often preferable in educational settings where understanding the precise relationships between variables is crucial.

Elimination Method

The elimination method is a strategy used to eliminate one variable by adding or subtracting equations. This method can simplify the system of equations and make it easier to solve for the remaining variable. In our exercise:

• We start with the equations: \( x^2 + y^2 = 10 \) and \( 2x^2 - y^2 = 17 \).
• By subtracting the second equation from the first, we effectively remove \( y^2 \) from the equations:

\[x^2 + y^2 - (2x^2 - y^2) = 10 - 17\]which simplifies to:\[-x^2 + 2y^2 = -7\]After rearranging, this transforms into:\[x^2 - 2y^2 = 7\]The elimination of \( y^2 \) in this step allows us to express \( x^2 \) directly in terms of \( y^2 \), making it easier to substitute into other equations.

Substitution Method

The substitution method involves solving one equation for a variable and then substituting that expression into another equation. This method simplifies the nonlinear system by reducing the number of variables in one of the equations.From the exercise, after the elimination method, we had:\[x^2 = 2y^2 + 7\]We used this expression in place of \( x^2 \) in the original equation \( x^2 + y^2 = 10 \). This allowed us to focus only on one variable:\[(2y^2 + 7) + y^2 = 10\]Simplifying gives:

• \(3y^2 + 7 = 10\)
• \(3y^2 = 3\)
• \(y^2 = 1\)

With \( y^2 = 1 \), we find \( y = 1 \) or \( y = -1 \). These values are then plugged back to find corresponding \( x \) values, leading us to the final solutions. This approach highlights the interplay of substitution and algebraic manipulation in solving nonlinear systems.