Problem 25 - College Algebra

Question

For the following exercises, use any method to solve the nonlinear system. $$x^{2}-y^{2}=9$$ $$x=3$$

Step-by-Step Solution

Answer

The solution is \( x = 3 \) and \( y = 0 \).

Step 1: Understand the Given System of Equations

We have a system of nonlinear equations:1. \( x^2 - y^2 = 9 \)2. \( x = 3 \)The first equation is a hyperbola and the second equation is a vertical line where \( x = 3 \). Our goal is to find the value of \( y \) when \( x = 3 \).

Step 2: Substitute \( x = 3 \) into the First Equation

Take the value of \( x \) from the second equation and substitute it into the first equation:\[ (3)^2 - y^2 = 9 \]

Step 3: Simplify the Equation

Calculate \( 3^2 \), which is 9. Substitute this into the equation:\[ 9 - y^2 = 9 \]

Step 4: Solve for \( y^2 \)

Subtract 9 from both sides to isolate \( y^2 \):\[ 9 - y^2 = 9 \]\[ y^2 = 0 \]

Step 5: Find \( y \)

Take the square root of both sides to solve for \( y \):\[ y = 0 \]

Step 6: Verify the Solution

Substitute \( x = 3 \) and \( y = 0 \) back into the original equations to verify.1. Substitute into \( x^2 - y^2 = 9 \): \[ 3^2 - 0^2 = 9 \] which simplifies to \( 9 = 9 \), so it is correct.2. Ensure \( x = 3 \) holds, which it does.

Key Concepts

Hyperbola

A hyperbola is a set of all points \(x, y\) in a plane such that the absolute difference of the distances from two fixed points (foci) is always constant. Unlike circles or ellipses, hyperbolas have two branches that open either side-by-side (horizontally) or one above the other (vertically).
In mathematical terms, a hyperbola can be generally represented as:

• For a horizontal hyperbola: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \)
• For a vertical hyperbola: \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \)

The first equation in our system \(x^2 - y^2 = 9\) indicates a hyperbola, where the center is at the origin (0, 0) and the axes aligned with the x and y axes. This particular hyperbola has an equation of the form \(x^2 - y^2 = c^2\), where \(c = 3\). The branches of this hyperbola extend along the x-axis.

Vertical Line

The concept of a vertical line in mathematics is quite straightforward. It is a line where every point on the line has the same x-coordinate. This means it runs parallel to the y-axis.
In our exercise, the given equation \(x = 3\) describes a vertical line. All the points on this line will have an x-coordinate of 3, regardless of the y-coordinate. Therefore, any point on this line can be written as \( (3, y) \).
Vertical lines are important when solving systems of equations because they can easily intersect other equations in the system, such as parabolas or hyperbolas. This intersection helps in finding the solution to the system, which in this exercise, is done through substitution.

Substitution Method

The substitution method is a powerful algebraic tool used for solving systems of equations. It involves solving one of the equations for a single variable and then substituting that expression into the other equation(s).
For the exercise provided, we used the substitution method easily because one of the equations \(x = 3\) directly provides the value of x. In essence:

• We took the equation \(x^2 - y^2 = 9\) which is our hyperbola.
• Substituted \(x = 3\) into the hyperbola equation, resulting in \(9 - y^2 = 9\).
• Simplified the equation to solve for y by isolating \(y^2\).
• Finally, solved for y, finding that \(y = 0\).

This method is particularly useful when the equations are easily manipulated, such as having one equation in terms of a single variable. It often simplifies the process of solving non-linear systems and verifies if real solutions exist when graphically interpreted.