Problem 81 - A Graphical Approach to Precalculus with Limits

Question

Use substitution to solve the nonlinear system of equations in three variables. Note that solutions are ondered triples. $$\begin{aligned} &\frac{x^{2}}{16}+\frac{y^{2}}{4} \leq 1\\\ &x^{2}-y^{2} \geq 1 \end{aligned}$$

Step-by-Step Solution

Answer

The ordered pairs are (2, √3), (2, -√3), (-2, √3), and (-2, -√3). Both inequalities are satisfied.

Step 1: Analyze the inequality system

The first inequality \( \frac{x^2}{16} + \frac{y^2}{4} \leq 1 \) describes an ellipse centered at the origin. The second inequality \( x^2 - y^2 \geq 1 \) describes a hyperbola. Both are conic sections in the xy-plane.

Step 2: Express y in terms of x from the second inequality

The second inequality can be rearranged to express \( y^2 \leq x^2 - 1 \). Solve for \( y \) to obtain two expressions: \( y \leq \sqrt{x^2 - 1} \) or \( y \geq -\sqrt{x^2 - 1} \).

Step 3: Substitute into the first inequality

Substitute \( y = \sqrt{x^2 - 1} \) and \( y = -\sqrt{x^2 - 1} \) into the first inequality \( \frac{x^2}{16} + \frac{y^2}{4} \leq 1 \) to check for valid values of \( x \).

Step 4: Solve \( y = \sqrt{x^2 - 1} \) substitution

Substituting, we have \( \frac{x^2}{16} + \frac{x^2 - 1}{4} \leq 1 \). Multiply through by 16 to clear fractions: \( x^2 + 4x^2 - 4 \leq 16 \), simplifying to \( 5x^2 \leq 20 \), which gives \( x^2 \leq 4 \) or \( -2 \leq x \leq 2 \).

Step 5: Verify for \( y = -\sqrt{x^2 - 1} \) substitution

Substituting, \( \frac{x^2}{16} + \frac{x^2 - 1}{4} \leq 1 \) gives the same inequality \( x^2 \leq 4 \) or \( -2 \leq x \leq 2 \). Therefore, the solutions for both substitutions are equivalent.

Step 6: Determine possible ordered pairs (x, y)

Since \( -2 \leq x \leq 2 \) and \( y = \pm\sqrt{x^2 - 1} \), feasibly \( x = \pm 2 \); at these points \( y = \pm \sqrt{3} \). Thus, ordered pairs are \((2, \sqrt{3})\), \((2, -\sqrt{3})\), \((-2, \sqrt{3})\), and \((-2, -\sqrt{3})\).

Step 7: Check solutions back against both inequalities

Verify the ordered pairs in both original inequalities. \( x^2 - y^2 = 1 \) checks out for all pairs. Also, \( \frac{4}{16} + \frac{3}{4} = 1 \) checks out, confirming validity.

Key Concepts

Substitution Method

The substitution method is a powerful tool used to solve nonlinear systems of equations. It involves substituting one variable with an expression derived from another equation in the system. This method is especially useful when dealing with equations that are complex or interdependent.

In the context of the exercise, we first rearrange one of the inequalities, such as solving for \( y \) in terms of \( x \) in the inequality \( x^2 - y^2 \geq 1 \).

• First, rearrange to find \( y^2 \leq x^2 - 1 \), allowing substitution into the other equation.
• Substitute \( y = \pm \sqrt{x^2 - 1} \) back into the other inequality \( \frac{x^2}{16} + \frac{y^2}{4} \leq 1 \) to find possible \( x \) values.

This substitution helps reduce the complexity of the system by allowing you to work with one equation at a time. Utilizing the substitution method effectively requires understanding variable relationships and simplifying equations accordingly.

Conic Sections

Conic sections, as seen in this exercise, are curves obtained from slicing a cone with a plane. They include ellipses, parabolas, hyperbolas, and circles. Each has distinctive properties and equations.

In our problem:

• The equation \( \frac{x^2}{16} + \frac{y^2}{4} \leq 1 \) forms an ellipse. This represents a set of points such that the sum of distances from two focal points remains constant.
• The equation \( x^2 - y^2 \geq 1 \) forms a hyperbola. For a hyperbola, the difference of distances from two foci is constant.

Understanding these shapes helps in visualizing the solution set on a graph. The intersection of these conic sections often provides us with the solution set to our system, as seen when solving the problem using the substitution method. Real-life applications of conic sections include orbits of planets (ellipses) and the design of reflective surfaces (parabolas).

Inequalities in Mathematics

Inequalities play a crucial role in defining ranges and boundaries within mathematics. They are expressions that define an order relation between values, indicating whether one value is larger, smaller, or not equal to another.

In nonlinear systems, inequalities often outline regions of solutions rather than fixed points, as seen in the exercise:

• The ellipse described by the inequality \( \frac{x^2}{16} + \frac{y^2}{4} \leq 1 \) sets boundaries within the coordinate plane.
• The hyperbola from \( x^2 - y^2 \geq 1 \) defines another region, where solutions must overlap with those of the ellipse.

Solving inequalities involves finding where these regions intersect, particularly when visualized on a graph. It’s critical to continually evaluate solutions against all given constraints to ascertain their validity. Being adept with inequalities enables one to solve and understand complex systems like those in physics or engineering problems, where they are used to define safe operational limits or optimal conditions.