Problem 58 - Precalculus

Question

Find any points of intersection of the graphs of the equations algebraically and then verify using a graphing utility. $$\begin{aligned} -16 x^{2}-y^{2}+24 y-80 &=0 \\ 16 x^{2}+25 y^{2}-400 &=0 \end{aligned}$$

Step-by-Step Solution

Answer

The steps involve solving for one variable in one equation, substituting into the other equation, and then solving for the second variable. All potential points found need to be verified by substituting back into the original equations and should be plotted on a graph to verify intersection visually.

Step 1 - Rearrange the First Equation for y

Rearrange the first equation to get y as the subject. The equation is \( -16 x^{2} - y^{2} + 24y - 80 = 0 \). It can be rewritten as \( y^{2}-24 y +80 + 16x^{2} = 0 \) then factored into \( (y - 20) (y - 4) -16x^{2} = 0\). So when \(16x^{2}\) is less than 80, there will be two solutions for y.

Step 2 - Substitute y into the Second Equation

Next step is substituting obtain values for y from equation 1 into equation 2. The second equation is \( 16 x^{2} + 25 y^{2} - 400 = 0\). Substituting \( (y - 4) \) and \( (y - 20) \) into the equation will give two values of x when each is equal to zero.

Step 3 - Verify the Intersecting Points

After getting potential values for both x and y, substitute these into both the original equations. If they satisfy both the equations, then these are points of intersection. Both pairs of (x, y) coordinate points are needed to be verified.

Step 4 - Use a Graphing Utility to Verify

Once potential points of intersections are found, plot the two equations on a graph and confirm whether the points found algebraically are the points where the graphs intersect. If they are, the solution is correct.

Key Concepts

Algebraic Equations

Algebraic equations involve expressions set equal to each other, often containing variables like \(x\) and \(y\). Solving these equations means finding the values of the variables that make the equation true. In our exercise, we have two algebraic equations:

• \(-16x^2 - y^2 + 24y - 80 = 0\)
• \(16x^2 + 25y^2 - 400 = 0\)

To solve them, we rearrange and manipulate these equations to find points of intersection between the curves they represent. This usually involves expressing one variable in terms of the other or finding common solutions.

Systems of Equations

A system of equations is a set of equations with multiple variables that you solve together. Each equation in the system contains two or more unknowns. In our exercise, the two equations form a system:

• The first equation simplifies to \((y-20)(y-4) = 16x^2\)
• The second equation stays the same: \(16x^2 + 25y^2 - 400 = 0\)

The goal is to find values for \(x\) and \(y\) that solve both equations simultaneously. Often in these problems, you solve one equation for one variable and substitute into the other, allowing you to find the solutions for the entire system.

Graphing Utility

A graphing utility is a tool, like a graphing calculator or software, that helps visualize equations by plotting their graphs. It assists in checking answers found algebraically. To use a graphing utility:

• Enter each equation separately.
• Observe where the graphs intersect.
• Check predicted intersection points against these graph intersections.

This tool provides a visual confirmation, making it easier to verify solutions as well as see errors if the calculated points are incorrect.

Verification of Solutions

Verifying solutions ensures the accuracy of your answers. Here’s how you verify the intersection points for this exercise:

• First, substitute the found values of \(x\) and \(y\) back into the original equations.
• Check if both equations are satisfied with these values.
• Use the graphing utility to confirm that these points lie at the intersection of the graphs.

Verification is crucial because it confirms that the solutions work in all parts of the system, providing confidence in the accuracy and completeness of your solutions.