Problem 58 - Precalculus with Limits

Question

In Exercises 57-70, find any points of intersection of the graphs algebraically and then verify using a graphing utility. \(-x^2-y^2-8x+20y-7=0\) \(x^2+9y^2+8x+4y+7=0\)

Step-by-Step Solution

Answer

Therefore, the points of intersection of the graphs are (-1,0), (7,0), (-5,-3), and (13,-3).

Step 1: Simplify Equations

Firstly, rewrite the equations to create a more convenient form:1. Equation 1: \(-x^2 - 8x - y^2 + 20y = 7\)2. Equation 2: \(x^2 + 8x + 9y^2 + 4y = -7\)

Step 2: Solve the System of Equations

Now, add these two equations together:\(( -x^2 - 8x - y^2 + 20y) + (x^2 + 8x + 9y^2 + 4y) = 7 - 7 \)This simplification leads to:\(8y^2 + 24y = 0\)

Step 3: Solve for y

Since 8y^2 + 24y = 0, we can factor out 8y, yielding:\(8y(y + 3) = 0\)Setting y=0 yields \(y=0\).Setting y+3=0 yields \(y=-3\)

Step 4: Substitution and Solution for x

Substitute the values of \(y\) back into one of the original equations, for example, equation (1), and solve for \(x\). For \(y=0\), Equation (1) simplifies to -\(x^2 - 8x = 7\), yielding \(x = -1\) or \(x = 7\). For \(y=-3\), Equation (1) simplifies to -\(x^2 - 8x and -81=-7\), yielding \(x = -5\) or \(x =13\)

Key Concepts

Intersection Points

When two graphs meet at a certain point, that point is called an intersection. Finding intersection points is essential when solving systems of equations. This is because the solutions to these systems correspond to the locations where the graphs intersect.

For an algebraic form like in the given problem, intersection points reveal which

• values of \( x \) and \( y \) satisfy both equations simultaneously.
• By solving both equations together, you identify the exact \( (x, y) \) pairs.

In our specific exercise, we used algebra to find these critical points, suggesting locations where the curves described by our equations meet. Understanding intersection points gives clarity to how different equations relate to each other on a graph.

Graphing Utility

Using a graphing utility is an effective way to visually verify the solutions of a system of equations. Once you solve the equations algebraically, a graphing utility, like a graphing calculator or software, helps confirm these results visually.

• Firstly, input the simplified equations into the graphing utility.
• The utility then draws the curves corresponding to each equation.
• Observe where these graphs intersect; these are your equation solutions.

Graphing utilities are great for verifying algebraic work because they give you a visual look at that mathematical intersection in a straightforward manner. They are especially beneficial in situations where it's hard to solve equations manually or check for accuracy.

Algebraic Simplification

Simplifying equations means rewriting them in an easier form without changing their meaning, making them more convenient to solve. This process often involves:

• Re-grouping like terms to clean up the equation.
• Eliminating unnecessary terms by performing algebraic operations.
• Factoring to simplify the expressions into solvable bits.

In the exercise's context, we adjusted both equations by grouping like terms to form new expressions. This step directly influenced solving the system successfully, indicating which particular values hold across both original equations. Simplification is an invaluable first step in solving systems of equations as it makes more complex algebra much more manageable.

Equation Solving

Equation solving in systems of equations involves finding values for variables that satisfy all equations within the system. This requires:

• Determining a method to solve – substitution, elimination, or graphical methods.
• Substituting found variable values back into one of the original equations to confirm solutions.
• Verifying solutions, often using different methods to ensure consistency.

In our problem, we solved the simplified equations by factorizing and substituting. This yielded specific points for \( y \), which we then used to compute corresponding \( x \) values by plugging back into either equation. The solutions obtained signify the intersection between graphs, where both conditions are met. Solving equations is the core step where all preparatory work comes together to unravel the problem.