Problem 35 - College Algebra

Question

For the following exercises, use any method to solve the nonlinear system. $$\begin{aligned} 3 x^{2}-y^{2} &=12 \\ x^{2}+y^{2} &=16 \end{aligned}$$

Step-by-Step Solution

Answer

The solutions are \((\sqrt{7}, 3), (-\sqrt{7}, 3), (\sqrt{7}, -3), (-\sqrt{7}, -3)\).

Step 1: Understanding the System of Equations

We have a system of two nonlinear equations:\[ 3x^2 - y^2 = 12 \]\[ x^2 + y^2 = 16 \] Our goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously.

Step 2: Expression Rearrangement

From the second equation, \(x^2 + y^2 = 16\), we can express \(x^2\) in terms of \(y^2\):\[ x^2 = 16 - y^2 \]

Step 3: Substitution Method

Substitute \(x^2 = 16 - y^2\) into the first equation:\[ 3(16 - y^2) - y^2 = 12 \]This simplifies to:\[ 48 - 3y^2 - y^2 = 12 \]

Step 4: Simplify and Solve for y

Simplify the equation:\[ 48 - 4y^2 = 12 \]Subtract 48 from both sides:\[ -4y^2 = -36 \]Divide both sides by -4:\[ y^2 = 9 \]Solve for \(y\):\[ y = \pm 3 \]

Step 5: Solve for x

Use \(y = 3\) and \(y = -3\) to find corresponding \(x\) values.For \(y = 3\):\[ x^2 = 16 - 3^2 = 16 - 9 = 7 \]\[ x = \pm \sqrt{7} \]For \(y = -3\), the same computation applies:\[ x = \pm \sqrt{7} \]

Step 6: Identify the Solution Set

The solutions are the sets of \((x, y)\) pairs:\[ (\sqrt{7}, 3), (-\sqrt{7}, 3), (\sqrt{7}, -3), (-\sqrt{7}, -3) \]

Key Concepts

Substitution Method

The substitution method is a popular technique for solving systems of equations, especially when dealing with one linear and one nonlinear equation. Here, we first isolate one variable in one of the equations and then substitute that expression into the other equation. This method simplifies the system into a single equation with one variable, making it easier to solve.
In our problem, we started with the equations:

• \(3x^2 - y^2 = 12\)
• \(x^2 + y^2 = 16\)

From the second equation, we rearranged to find \(x^2\) in terms of \(y^2\):
\(x^2 = 16 - y^2\)
We substituted this expression into the first equation. This process helps us eliminate one variable and streamline solving for the other. By focusing on one variable at a time, substitution reduces complexity and reveals solutions more directly.
Using substitution not only aids algebraic manipulation but also illuminates the interdependency of the variables, shedding light on how changes in one affect the other.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form \(ax^2 + bx + c = 0\). In our exercise, the presence of quadratic terms \(3x^2\) and \(y^2\) indicates that we are dealing with quadratic relationships.
When we encounter a quadratic equation during the substitution process as seen in the solution,

• \(48 - 4y^2 = 12\)

We simplify to find \(y^2 = 9\). Solving this yields two possible solutions for \(y\):

• \(y = 3\)
• \(y = -3\)

Quadratic equations have this characteristic because they describe parabolic graphs with non-linear curvature. The solutions are symmetrical, represented by plus and minus values due to the square root operation, which is key to fully exploring the set of possible solutions.
The quadratic aspect of these equations presents more varied solutions than linear equations, making them crucial in analyzing systems of nonlinear equations.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables that are solved together because they share the same set of variables. The goal is to find a common solution that satisfies all equations in the system.
In our case, we dealt with:

• \(3x^2 - y^2 = 12\)
• \(x^2 + y^2 = 16\)

The challenge comes from finding values for \(x\) and \(y\) such that both these relationships hold true simultaneously.
The process involves transforming the system into workable forms—such as through substitution—and leveraging algebraic techniques (like solving quadratics) to uncover all potential solutions. Each solution pair, like \((\sqrt{7}, 3)\), must make both original equations true.
In solving these, maintaining a balance between equations and checking possible combinations is key. This often requires back-substitution to ensure all solutions satisfy the initial system. Simultaneous systems showcase the interconnected nature of equations where variables link different expressions into a coherent set.