Problem 8 - Algebra A Combined Function

Question

Solve each nonlinear system of equations. $$ \left\\{\begin{array}{l} 4 x^{2}+3 y^{2}=35 \\ 5 x^{2}+2 y^{2}=42 \end{array}\right. $$

Step-by-Step Solution

Answer

Solutions: \((x, y) = (2\sqrt{2}, 1), (2\sqrt{2}, -1), (-2\sqrt{2}, 1), (-2\sqrt{2}, -1)\).

Step 1: Identify the Equations

The given system of equations is: \( 4x^2 + 3y^2 = 35 \) and \( 5x^2 + 2y^2 = 42 \). We need to solve for \( x \) and \( y \).

Step 2: Eliminate One Variable

Start by multiplying the first equation by 2: \( 2(4x^2 + 3y^2) = 70 \), which gives us \( 8x^2 + 6y^2 = 70 \). Then multiply the second equation by 3: \( 3(5x^2 + 2y^2) = 126 \), which results in \( 15x^2 + 6y^2 = 126 \).

Step 3: Subtract the Modified Equations

Subtract \( 8x^2 + 6y^2 = 70 \) from \( 15x^2 + 6y^2 = 126 \) to eliminate \( y^2 \):\[(15x^2 + 6y^2) - (8x^2 + 6y^2) = 126 - 70\]which simplifies to \( 7x^2 = 56 \).

Step 4: Solve for First Variable

Solve \( 7x^2 = 56 \) to find \( x^2 \):\[x^2 = \frac{56}{7} = 8\]Thus, \( x = \pm \sqrt{8} = \pm 2\sqrt{2} \).

Step 5: Substitute Back to Solve for Second Variable

Substitute \( x^2 = 8 \) into the first original equation: \( 4(8) + 3y^2 = 35 \), giving us:\[32 + 3y^2 = 35\]which simplifies to \( 3y^2 = 3 \), leading to \( y^2 = 1 \) and \( y = \pm 1 \).

Step 6: Verify Solutions

The possible solutions are \((x, y) = (2\sqrt{2}, 1), (2\sqrt{2}, -1), (-2\sqrt{2}, 1), (-2\sqrt{2}, -1)\). Plug these back into the original equations to verify both conditions are met. Both satisfy the equations, confirming the solutions.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest power of the variable is two. They take the form \( ax^2 + bx + c = 0 \). In our exercise, we have a system of equations where both involve quadratic terms, such as \( 4x^2 \) and \( 3y^2 \). These types of equations can have up to two solutions for each variable, which can result in multiple combined solutions when part of a system.
Understanding the nature of quadratic equations is essential in solving systems of nonlinear equations. Every solution step must carefully address the square terms to reveal valid potential solutions. Here, solving for the variables involves recognizing the need to eliminate one variable to solve for the other, which is more complex than solving linear equations.

Elimination Method

The elimination method is a technique used to solve systems of equations. It involves combining equations in such a way that one of the variables is eliminated, allowing the remaining variable to be solved directly.
For our nonlinear system, elimination is performed by adjusting the equations to align coefficients of one of the variables before subtracting the equations. This was done by multiplying the first equation by 2 and the second by 3, resulting in a new pair of equations:

• \( 8x^2 + 6y^2 = 70 \)
• \( 15x^2 + 6y^2 = 126 \)

By subtracting these, we effectively removed the \( y^2 \) term, simplifying our system to solve for \( x^2 \). This calculated value of \( x^2 \) is then used to find \( x \) without the interference of \( y \).
This step is crucial because it converts the system into a simpler form that is more straightforward to solve.

Square Roots

Square roots are a fundamental concept that often comes into play when solving quadratic equations, especially after isolating \( x^2 \) or \( y^2 \). Once you solve for the square of a variable, the square root function is used to find the possible values of the variable itself.
In this exercise, after finding \( x^2 = 8 \), we determine \( x = \pm \sqrt{8} = \pm 2\sqrt{2} \). This notation shows both the positive and negative roots, crucial as quadratic equations can have two solutions for each square term. Finding the square root requires us to consider both potential solutions and ensure we do not miss any valid answers.
Taking square roots can sometimes yield complex numbers, but in this particular problem, real-number solutions were adequate. Always verify your answers in the original equation to ensure they make mathematical sense in the given context.

Verification of Solutions

Verifying solutions is a highly important step in any problem involving systems of equations. It confirms that the computed solutions truly satisfy all given equations.
In this context, the possible solutions \((x, y)\) included \((2\sqrt{2}, 1), (2\sqrt{2}, -1), (-2\sqrt{2}, 1), (-2\sqrt{2}, -1)\). Verification involves plugging each pair back into the original equations \( 4x^2 + 3y^2 = 35 \) and \( 5x^2 + 2y^2 = 42 \) to see if both sides equal for each case.
By verifying, we ensure no extraneous solutions are mistakenly accepted. This step can also reveal whether solutions are consistent across both equations, helping identify calculation errors or assumptions made during elimination or simplification.