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Chapter 10: Problem 105

Solve using a graphing calculator. Find all real solutions. $$\begin{aligned} &14.5 x^{2}-13.5 y^{2}-64.5=0\\\ &5.5 x-6.3 y-12.3=0 \end{aligned}$$

Short Answer

Expert verified
Use a graphing calculator to identify the intersection points of the equations: 14.5x^2 - 13.5y^2 - 64.5 = 0 and 5.5x - 6.3y - 12.3 = 0. Record the intersection points as the solutions.

Step by step solution

01

- Enter Equations into the Graphing Calculator

Enter the equations into the graphing calculator. For the first equation, enter: 14.5x^2 - 13.5y^2 - 64.5 = 0For the second equation, enter: 5.5x - 6.3y - 12.3 = 0
02

- Graph the Equations

Graph both equations on the graphing calculator. Make sure that the viewing window is set such that you can see the intersection points of the curves.
03

- Identify the Points of Intersection

Use the graphing calculator's intersection feature (often found in the 'calc' or 'trace' menu) to find the points where the two graphs intersect. These points represent the real solutions to the system of equations.
04

- Record the Real Solutions

Write down the coordinates of the intersection points that the graphing calculator provides. These coordinates are the real solutions for the variables x and y.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equations
Quadratic equations are equations of the form ax^2 + bx + c = 0a, b, and c here are constants, with 'a' not being zero. In our given exercise, we notice that the first equation, 14.5 x^{2}-13.5 y^{2}-64.5=0 is a quadratic equation in both x and y. It involves squared terms and is more complex than basic linear equations. Quadratic equations can have up to two real solutions, meaning that when graphed, they can intersect with linear or other quadratic equations at up to two points.
Linear Equations
Linear equations are much simpler and follow the formax + by = cwhere a, b, and c are constants. In our exercise, the second equation, 5.5x - 6.3y - 12.3 = 0,is a linear equation. It represents a straight line when graphed. Linear equations can have only one intercept (crossing point) on the graph per variable, making them relatively easier to solve. However, when combined with quadratic equations, we are often looking for points where these different types of equations intersect.
Graphing Calculator Usage
Using a graphing calculator can greatly simplify solving complex systems of equations. Here are the basic steps to follow:
  • Enter the equations exactly as given. Be mindful of all terms and constants.
  • Proceed to graph them. Ensure your viewing window includes a sufficiently wide range to see potential intersection points. Adjust the window settings if necessary.
  • Use the graphing calculator's 'calc' or 'trace' functions to find intersections. This feature automatically computes the x and y values where the graphs overlap.
Mastering these steps can turn difficult equation systems into more manageable problems. Make sure you鈥檙e familiar with your calculator's specific functions and settings.
Intersection of Graphs
The intersection points of graphs represent the solutions to the system of equations. These points show where the graphs of the equations meet. In graphing calculator terms, this means the x and y values are true for both equations simultaneously. For our exercise:
  • When you graph 14.5x^{2} - 13.5y^{2} - 64.5 = 0and 5.5x - 6.3y - 12.3 = 0,you're looking for points where these graphs cross.
  • Using the intersect function on your graphing calculator, you will identify these crossing points, signifying the real solutions.
  • Finally, recording the coordinates given by the calculator will provide the specific values for x and y that solve both equations.
Understanding this technique is crucial for tackling more complex algebraic problems.

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Most popular questions from this chapter

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