Problem 29 - Intermediate Algebra

Question

What happens if you try to graph the system $$ \left(\begin{array}{rr} x^{2}+4 y^{2}= & 16 \\ 2 x^{2}+5 y^{2}= & -12 \end{array}\right) $$

Step-by-Step Solution

Answer

The second equation has no real solutions; thus, the system has no graph or intersecting points in the real plane.

Step 1: Understand the Equations

The system consists of two equations: \(x^2 + 4y^2 = 16\) and \(2x^2 + 5y^2 = -12\). These equations are meant to represent conic sections, typically ellipses or hyperbolas, in the plane.

Step 2: Analyze the First Equation

The first equation \(x^2 + 4y^2 = 16\) can be rewritten as \(\frac{x^2}{16} + \frac{y^2}{4} = 1\). This represents an ellipse centered at the origin, with a semi-major axis of length 4 along the x-axis and a semi-minor axis of length 2 along the y-axis.

Step 3: Analyze the Second Equation

The second equation \(2x^2 + 5y^2 = -12\) hints that it should represent a conic section. However, since the sum of squares is set equal to a negative number, it cannot be a real ellipse or any other real conic section. In reality, no real solutions exist for this equation.

Step 4: Conclusion on Graphing

Since there are no real solutions for \(2x^2 + 5y^2 = -12\), it means this equation does not represent any graph in the real plane. Thus, the system of equations has no common real solutions and hence, they do not intersect in the real plane.

Key Concepts

Conic Sections

In mathematics, conic sections refer to the curves obtained by intersecting a plane with a cone. These curves are very diverse and appear in various physical phenomena and applications. The main types of conic sections are:

• Ellipse - Formed when the cutting plane is less inclined than the cone’s side, but not parallel to its base. It resembles an elongated circle.
• Circle - A special case of an ellipse where the plane cuts perpendicular to the axis of the cone.
• Parabola - Achieved when the plane is parallel to the cone’s edge.
• Hyperbola - Arises when the plane cuts through both nappes of the cone.

Understanding these shapes is vital because they model various real-world structures and phenomena. For example, satellite dishes and reflective surfaces are typically parabolic.
Conic sections are described using quadratic equations. Variations in the coefficients of these equations determine the type of conic section formed.

Ellipses

Ellipses are fascinating members of the conic sections family. They occur frequently in both nature and synthetic applications. An ellipse is defined by the equation:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.

A crucial property of ellipses is that the sum of the distances from any point on the ellipse to the two foci is constant. When these distances are equal, we have a circle.

In the given exercise, the first equation \(x^2 + 4y^2 = 16\) can be rewritten in the standard form as \(\frac{x^2}{16} + \frac{y^2}{4} = 1\), indicating an ellipse centered at the origin with axes aligned along the x and y axes.

• The semi-major axis (length 4) lies along the x-axis, indicating a wider stretch horizontally.
• The semi-minor axis (length 2) aligns with the y-axis.

Studying ellipses helps in understanding planetary orbits and technical designs involving lenses and mirrors.

Systems of Equations

Solving systems of equations is a fundamental task in algebra, where we find values of variables that satisfy all given equations simultaneously. There are several methods employed to solve systems of equations:

• Graphical Method - Involves graphing each equation on the same set of axes to find the intersection points, if any. The intersections are the solutions.
• Substitution Method - Solving one equation for a variable and substituting it into the other equations.
• Elimination (or Addition) Method - Involves adding or subtracting equations to eliminate variables successively.

In our exercise, the system consists of two quadratic equations. However, the second equation \(2x^2 + 5y^2 = -12\) equates to a negative, which means it cannot describe a real conic section. This results in no solution when solving for intersections between the two equations. Understanding this scenario helps highlight that not all systems of equations will have real solutions, especially when they involve quadratic expressions aiming to express conic sections.