Problem 67 - Precalculus with Limits

Question

In Exercises 57-70, find any points of intersection of the graphs algebraically and then verify using a graphing utility. \(x^2+2y^2-4x+6y-5=0\) \(-x+y-4=0\)

Step-by-Step Solution

Answer

The intersection points are found by solving the system of equations. After finding the x values from the quadratic equation, the y values can be calculated from the linear equation. Validate these points by graphically plotting both equations and checking if the intersection points match with the graphically obtained ones.

Step 1: Rearrange the linear equation to express y in terms of x

Rearrange the linear equation \( -x+y-4=0 \) to express y in terms of x. This can be done like this: \( y = x + 4 \).

Step 2: Substitute y in the quadratic equation

Substitute \( y = x + 4 \) in the quadratic equation \( x^2+2y^2-4x+6y-5=0 \). Performing this substitution leads to a new equation that only includes x.

Step 3: Simplify the equation

Simplify this equation by using algebroaic tools, such as expanding and combining similar terms. This should result in a quadratic equation solely in terms of x.

Step 4: Solve the quadratic equation for x

Solve the quadratic equation obtained in the previous step for x. The solutions can be real or imaginary numbers.

Step 5: Determine the corresponding y coordinates

Substitute the x-values obtained in the previous step in the equation \( y = x + 4 \) to get the corresponding y coordinates.

Step 6: Verify the solutions graphically

Using a graphing utility, plot both equations on the same graph. The intersection points found algebraically should be visible on the graph.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra and represent curves known as parabolas. They have the standard form:

• \( ax^2 + bx + c = 0 \)

In this definition, "a," "b," and "c" are constants, and "x" represents the variable.
A quadratic equation can have two solutions, one solution, or no real solutions at all.
To solve them, you can employ several methods such as factoring, completing the square, or using the quadratic formula. The solutions typically describe the x-values where the parabola meets the x-axis.
For instance, in the given problem, the quadratic part \( x^2 + 2y^2 - 4x + 6y - 5 = 0 \) describes the parabola's general behavior.

Graphing Utility

A graphing utility is a tool that serves to visualize equations in a graphical format effectively.
These tools can range from handheld calculators to sophisticated software programs.
They help in understanding complicated relations visually by plotting them on a coordinate plane.

• Graphing utilities are crucial for verifying algebraic solutions.
• They provide a visual representation to confirm the intersection points found through calculations.

In this way, they act as digital assistants standing by to prove that what you've solved theoretically can also be observed on a graph.

Algebraic Manipulation

Algebraic manipulation involves various techniques used to simplify, rearrange, or solve equations and expressions.
In the context of the exercise, this manipulation is used to simplify a system of equations in order to find solutions.

• Rearranging equations to isolate variables.
• Substituting isolated values into other equations.
• Simplifying and solving resulting expressions.

For example, you start by rearranging the linear equation into \( y = x + 4 \), then substitute it into the quadratic equation to simplify it to only involve one variable.
Finally, these simplifications lead you to solve for the unknown values systematically.

Intersection Points

Intersection points occur where two or more graphs meet on a coordinate plane.
They represent the solutions to systems of equations and are significant in algebra and calculus alike.
For any two functions graphed together, their intersection points are the x and y coordinates that satisfy both equations simultaneously.

• They provide practical solutions to complex mathematical problems by showing common solutions in real-world applications.
• Common in studies involving rates of change, optimization, and geometry.

In the given problem, intersection points are the solutions that satisfy both the quadratic and linear equations, showing exactly where the curves meet on the graph.