Problem 68 - A Graphical Approach to College Algebra

Question

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &2 x^{2}+3 y^{2}=5\\\ &3 x^{2}-4 y^{2}=-1 \end{aligned}$$

Step-by-Step Solution

Answer

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

Step 1: Analyze the System

We have two equations: \((1) \ 2x^2 + 3y^2 = 5\) and \((2) \ 3x^2 - 4y^2 = -1\). Both equations are quadratic in form and involve the squares of \(x\) and \(y\). We aim to find real solutions for \(x\) and \(y\).

Step 2: Eliminate One Variable

To eliminate a variable, let's align the coefficients of \(x^2\) by multiplying equation (1) by 3 and equation (2) by 2. Thus, the new equations become:\[ \begin{align*} 3(2x^2 + 3y^2) &= 3(5) \ 2(3x^2 - 4y^2) &= 2(-1) \end{align*} \] This results in:\[ \begin{align*} 6x^2 + 9y^2 &= 15 \ 6x^2 - 8y^2 &= -2 \end{align*} \]

Step 3: Subtract the Equations

Subtract the second equation from the first to eliminate \(x^2\):\[ (6x^2 + 9y^2) - (6x^2 - 8y^2) = 15 - (-2) \] Simplify the equation:\[ 17y^2 = 17 \] Solve for \(y^2\), we find \(y^2 = 1\).

Step 4: Solve for y

Since \(y^2 = 1\), we take the square root to find \(y = \pm 1\).

Step 5: Solve for x Using y=1

Substitute \(y = 1\) into equation (1): \[ 2x^2 + 3(1)^2 = 5 \] which simplifies to \[ 2x^2 + 3 = 5 \] \[ 2x^2 = 2 \] Solve for \(x^2\), \(x^2 = 1\). Take the square root to find \(x = \pm 1\).

Step 6: Solve for x Using y=-1

Substitute \(y = -1\) into equation (1): \[ 2x^2 + 3(-1)^2 = 5 \] which simplifies to \[ 2x^2 + 3 = 5 \] \[ 2x^2 = 2 \] Solve for \(x^2\), \(x^2 = 1\). Take the square root to find \(x = \pm 1\).

Step 7: Verify Solutions

Now we have possible pairs: \((x, y) = (1, 1), (-1, 1), (1, -1), (-1, -1)\). Substitute each pair into both original equations to verify they satisfy the system. All pairs satisfy both equations, confirming they are real solutions.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) represents the variable. In nonlinear systems, like the one we are considering, these equations form the foundation of our calculations. Here, instead of a single equation, we have two quadratic equations that relate variables \( x \) and \( y \) through their squares. By solving these, we are seeking values for \( x \) and \( y \) that simultaneously satisfy both equations.
Quadratic equations have three potential solutions depending on the discriminant value \( \Delta = b^2 - 4ac \):

• If the discriminant is positive, the equation has two real solutions.
• If the discriminant is zero, there is exactly one real solution (a repeated root).
• If the discriminant is negative, no real solutions exist, only complex ones.

In our specific case, the equations are already simplified to highlight the squares of \( x \) and \( y \), allowing us to manage two related expressions that we manipulate to find real solutions.

Real Solutions

Real solutions in nonlinear systems of equations are the pairs of numbers \( (x, y) \) that solve every equation in the system with real values. This means that when these numbers are substituted back, they make each equation true. Here's how we find them and ensure they are real:First, when solving, aim for numbers that adhere all equations in the system.This approach confirms general consistency across the equations. You can do this in systems like ours by:

• Aligning or equalizing terms, such as matching coefficients to eliminate one variable.
• Simplifying to identify clear, solvable expressions of the variable squares.

Next, after finding possible solutions like \( x = \pm 1 \) and \( y = \pm 1 \), substitute these values back into the original equations. Each valid pair must satisfy both equations.
Verifying solutions by checking their validity in initial equations, ensures that calculations haven't overlooked potential inconsistencies.

Analytical Methods

Analytical methods are systematic procedures used to identify solutions to mathematical equations, often involving eliminating one or more variables to simplify complex systems. In the scenario provided, we applied a particular analytical method where we:

• Multiplied each equation by strategic constants to align coefficients and cancel a variable.
• Subtracted the modified equations from each other, isolating one variable.
• Solved the resulting simpler equation for the remaining variable.

The strategic multiplication and subtraction were chosen because they provided a clear path to isolating \( y^2 \), which was critical in minimizing complexity and guiding us toward finding \( y \) first.
Once \( y \) was known, this known value was substituted back to identify \( x \), streamlining the process by reducing a two-variable system to a single-variable problem in manageable steps.
Analytical approaches emphasize careful handling of algebraic manipulation, ensuring that logical deductions lead to solvable expressions and real solutions.