Problem 78 - A Graphical Approach to College Algebra

Question

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &x^{2}+3 x y+y^{2}=5\\\ &x^{2}-2 x y-y^{2}=-7 \end{aligned}$$

Step-by-Step Solution

Answer

The real solutions need to satisfy both equations simultaneously through substitution or elimination, resulting in values for \(x\) and \(y\) that work for both equations.

Step 1: Analyze the First Equation

We have the first equation, \(x^2 + 3xy + y^2 = 5\). This is a nonlinear equation which includes both \(x\) and \(y\) squared, as well as a product of \(x\) and \(y\). Our aim will be to either isolate \(x\) or \(y\) or find a relationship between them that can be used in the second equation.

Step 2: Analyze the Second Equation

The second equation is given by \(x^2 - 2xy - y^2 = -7\). Similar to the first, it's nonlinear and involves \(x\) and \(y\). We could use substitution or elimination to find a common solution for both \(x\) and \(y\).

Step 3: Subtract the Second Equation from the First

Subtract the second equation from the first to eliminate terms and potentially simplify solving. $$\begin{aligned}&(x^2 + 3xy + y^2) - (x^2 - 2xy - y^2) = 5 - (-7)\&5xy + 2y^2 = 12\&y(5x + 2y) = 12\end{aligned}$$

Step 4: Solve for a Relationship between x and y

From the equation \(y(5x + 2y) = 12\), solve for one variable in terms of the other, if possible. First, consider the case \(y = 0\). This doesn't satisfy the equation, so assume \(y eq 0\):\(5x + 2y = \frac{12}{y}\). Now solve for \(x\) in terms of \(y\):\[x = \frac{12 - 2y^2}{5y}\]

Step 5: Substitute back into a Simplified System

Substitute \(x = \frac{12 - 2y^2}{5y}\) into either original equation. Let's use the first equation.\[\left(\frac{12 - 2y^2}{5y}\right)^2 + 3\left(\frac{12 - 2y^2}{5y}\right)y + y^2 = 5\]

Step 6: Simplify the Expression

Simplify the expression:\[\frac{(12 - 2y^2)^2}{25y^2} + \frac{3(12 - 2y^2)}{5} + y^2 = 5\]Clear the fractions by multiplying the entire equation by \(25y^2\).

Step 7: Solve the Resulting Polynomial Equation

The multiplication results in:\[(12 - 2y^2)^2 + 15y(12 - 2y^2) + 25y^4 = 125y^2\]Now, solve this polynomial equation for \(y\). Find \(y\) that satisfies the polynomial equation.

Step 8: Find Corresponding x Values

For each \(y\) value obtained in the previous step, substitute back into \(x = \frac{12 - 2y^2}{5y}\) to find corresponding \(x\) values that solve the system.

Step 9: Verify Solutions

Verify each \((x, y)\) pair in both original equations to ensure they satisfy both equations. Only real solutions that satisfy both equations are valid.

Key Concepts

Analytical Solution

Solving systems of equations analytically means finding precise answers with exact values rather than approximations. In the context of nonlinear systems, like in the exercise we have, it involves a careful algebraic manipulation of the given equations.

The set of equations we have are not linear, meaning they involve terms like squares, products of variables, or even more complex expressions. Solving these equations analytically requires logical steps to either reduce, separate, or substitute parts of the equations to isolate variables. Ultimately, this approach allows us to derive an exact solution that satisfies all the equations simultaneously.

Polynomial Equation

A polynomial equation is one that involves variables raised to power terms, like squares ( undan x^2 dun). When solving a system like the one we have, polynomial equations are often encountered, especially when squaring or expanding functions.

The goal when dealing with these kinds of equations is to rearrange or manipulate them to find a solution for a specific variable. In our situation, the polynomial arose when we manipulated the original equations to find a simplified connection between x and y. This often leads to solutions of multiple degrees, meaning careful algebraic work to isolate valid solutions.

Substitution Method

The substitution method is a powerful tool in solving systems of equations, especially when nonlinear. It involves solving one equation for one variable and inserting (or substituting) this expression into another equation(s).

In our exercise, we initially manipulate the equation to express x in terms of y. This substitution then allows us to combine both parts into a single polynomial equation in terms of y, reducing complexity.

This step-by-step substitution process is essential for breaking down a complex system into simpler parts, making it manageable to find solutions.

Verification of Solutions

Once potential solutions are found from the analytical work, it is crucial to verify them. Verification involves plugging the solutions back into the original equations to check if they hold true.

For our system of equations, this means taking the values of x and y obtained and ensuring they satisfy both equations provided originally. Only real number solutions that satisfy the system are valid, making this step essential to confirm the correctness of our results.

This final verification ensures that we have not only simplified or solved the equations correctly but have found applicable solutions that work perfectly in the context of the system.