Problem 4 - Algebra A Combined Function

Question

Solve each nonlinear system of equations. $$ \left\\{\begin{aligned} 4 x^{2}+y^{2} &=10 \\ y &=x \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solutions are \((\sqrt{2}, \sqrt{2})\) and \((-\sqrt{2}, -\sqrt{2})\).

Step 1: Analyze the Equations

The given nonlinear system of equations is:\[\begin{align*} 4x^2 + y^2 &= 10 \ y &= x\end{align*}\]Here, the second equation \( y = x \) tells us that the solution for \( y \) is directly dependent on \( x \). We will substitute \( y \) with \( x \) in the first equation.

Step 2: Substitute and Simplify

Substitute \( y = x \) into the first equation:\[4x^2 + (x)^2 = 10\]This simplifies to:\[4x^2 + x^2 = 10\]Combine like terms:\[5x^2 = 10\]

Step 3: Solve for x

Divide both sides of the equation \( 5x^2 = 10 \) by 5:\[x^2 = 2\]Take the square root of both sides to find \( x \):\[x = \pm \sqrt{2}\]

Step 4: Solve for y Using x

Since \( y = x \), substitute the values of \( x \) to find \( y \):\[y = x = \pm \sqrt{2}\]This gives two solutions for \( (x, y) \):\[ (\sqrt{2}, \sqrt{2}) \] and \[ (-\sqrt{2}, -\sqrt{2}) \]

Step 5: Verify the Solutions

Substitute the solutions back into the original equations to verify correctness. For \( (\sqrt{2}, \sqrt{2}) \):\[4(\sqrt{2})^2 + (\sqrt{2})^2 = 8 + 2 = 10 \]For \( (-\sqrt{2}, -\sqrt{2}) \):\[4(-\sqrt{2})^2 + (-\sqrt{2})^2 = 8 + 2 = 10\]Both solutions satisfy the first equation and since \( y = x \), they satisfy both equations of the system.

Key Concepts

Understanding the Substitution Method

In the realm of nonlinear systems of equations, leveraging the substitution method can be a powerful tool for finding solutions. This method is particularly useful when at least one equation in the system allows us to express one variable in terms of another, simplifying the system to make it easier to solve.

In our given exercise, the system consists of the equations: \( 4x^2 + y^2 = 10 \) and \( y = x \).

• Step one involves identifying that the second equation, \( y = x \), already provides a simple substitution. By expressing \( y \) in terms of \( x \), this equation allows us to replace \( y \) in the first equation.
• Next, substitute the expression for \( y \) into the first equation. This reduces the system's complexity from two equations with two variables to a single equation: \( 4x^2 + x^2 = 10 \).

This method enables us to concentrate our efforts on solving just one equation, making the problem more manageable.

Demystifying the Square Root

The square root is a fundamental concept in mathematics, often encountered when solving equations like the one in our nonlinear system. In the example \( 5x^2 = 10 \), we isolate \( x^2 \) to find \( x \) by dividing both sides by 5, resulting in \( x^2 = 2 \).

Taking the square root is our next step:

• Applying the square root to both sides of \( x^2 = 2 \) yields \( x = \pm \sqrt{2} \).
• The symbol \( \pm \) is crucial here as it indicates two solutions: \( \sqrt{2} \) and \( -\sqrt{2} \), representing both the positive and negative roots.

Understanding that squaring a negative number results in a positive outcome helps to clarify why both positive and negative roots exist for the equation \( x^2 = 2 \).

This step is key in solving equations that have been reduced to a quadratic form through earlier methods like substitution.

Solution Verification - Checking Your Work

Once solutions are derived from a set of equations, it's important to verify that they satisfy all original conditions of the system. Solution verification not only confirms correctness but also reinforces understanding of the problem-solving process.

In our example:

• Substitute \( (\sqrt{2}, \sqrt{2}) \) back into the first equation: the left side becomes \( 4(\sqrt{2})^2 + (\sqrt{2})^2 = 8 + 2 = 10 \), which equals the right side.
• Similarly, check \( (-\sqrt{2}, -\sqrt{2}) \): the calculation yields \( 8 + 2 = 10 \), verifying it also satisfies the equation.

Additionally, since both solutions maintain \( y = x \), they meet the conditions of the second equation trivially. This verification step assures us that the solutions found are indeed the legitimate answers to the nonlinear equation system.