Problem 56 - College Algebra

Question

A system for tracking ships indicates that a ship lies on a path described by \(2 y^{2}-x^{2}-1 .\) The process is repeated and the ship is found to lie on a path described by \(2 x^{2}-y^{2}-1\). If it is known that the ship is located in the first quadrant of the coordinate system, determine its exact location.

Step-by-Step Solution

Answer

The exact location of the ship in the first quadrant of the coordinate system is at point (1, 1).

Step 1: Preparation

First, rewrite the equations to express y in terms of x for each equation. From the first equation \(2 y^{2}-x^{2}=1\), express y^2 as \((x^2 + 1) / 2\). Using the second equation \( 2 x^{2}-y^{2} = 1\), express y^2 as \(2x^2 - 1\).

Step 2: Equate the two y^2 expressions

Since both expressions from Step 1 equal y^2, we can create a single equation with x by putting these expressions equal to each other which results in \((x^2 + 1) / 2 = 2x^2 - 1\).

Step 3: Solve for x

Upon simplifying the equation from Step 2, we multiply everything by 2 to eliminate fractions resulting in \(x^2 + 1 = 4x^2 - 2\). This simplifies further to \(3x^2 = 3\) and finally \(x = \sqrt{1} = 1\).

Step 4: Solve for y

Substitute \(x = 1\) into \(y^2 = (x^2 + 1) / 2\) to get \(y^2 = (1 + 1) / 2 = 1\). Therefore, the y-coordinates are \(y = \sqrt{1}\) and \(y = -\sqrt{1}\). Because the ship is located in the first quadrant, the y-coordinate should be positive. Thus, \(y = 1\).

Step 5: Check the solution

Check whether \(x=1, y=1\) is a solution to the second equation in the system. Substituting into \(2 x^{2}-y^{2}=1\), we confirm that the left-hand side equals the right-hand side. This means the solution is correct.

Key Concepts

Coordinate System

A coordinate system is a mathematical construct that allows us to precisely identify the location of points in a space, most commonly the plane or three-dimensional space. The most familiar of these is the Cartesian coordinate system, which uses two perpendicular axes—the x-axis (horizontal) and the y-axis (vertical)—to define locations.
In the plane, any point can be represented by an ordered pair of numbers, written as \(x, y\). This means that the first number describes the position along the x-axis, and the second number describes the position along the y-axis.
For example, the point \(3, 4\) is located three units to the right of the origin (where the axes intersect) and four units up. The origin itself is labeled as (0, 0).
To solve algebraic systems like the one in this problem, the coordinate system helps visualize where the solution points lie. When specified to be in the first quadrant, both x and y are positive, limiting the solution's potential location.

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). They manifest in many mathematical problems, including the exercise in question, where we equate expressions to solve for unknowns.
In general, there are several strategies for solving quadratic equations:

• Factoring: Expressing the quadratic expression as a product of binomials. This method is efficient if the roots are rational numbers.
• Using the Quadratic Formula: The formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) gives solutions for any quadratic equation, and is especially useful when the quadratic doesn't factor neatly.
• Completing the Square: This method involves rearranging the equation to form a perfect square trinomial, which can then be solved by taking the square root of both sides.

In the context of the exercise solution, the quadratic nature appears when expressing \(y^2\) in terms of \(x\), setting us up to solve for \(x\) by simplifying and equating the expressions to handle the unknowns.

First Quadrant

In the Cartesian coordinate system, the plane is divided into four regions known as quadrants. These are numbered counterclockwise starting from the upper right quadrant known as the first quadrant.
The first quadrant is the area where both the x and y coordinates are positive. This means any point within this quadrant has both positive x and y values, such as (2, 3).
Understanding the concept of quadrants is crucial when solving system equations as they give clues about the nature of the solutions. For instance, if the problem states the solution lies in the first quadrant, it automatically tells us the possible values of x and y are greater than zero.
In the exercise at hand, knowing that the ship's location is in the first quadrant significantly guides us to choose the positive value when solving for y, thereby ensuring our solution is both mathematically sound and contextually appropriate.