Problem 62 - Precalculus

Question

Find any points of intersection of the graphs of the equations algebraically and then verify using a graphing utility. $$\begin{aligned} x y+x-2 y+3 &=0 \\ x^{2}+4 y^{2}-9 &=0 \end{aligned}$$

Step-by-Step Solution

Answer

The points of intersection are obtained by solving the equations and would be confirmed respectively by substituting the coordinates into the equations. The actual points would require actual computations in the named steps.

Step 1: Rearrange the First Equation

Rearrange the first equation to solve for x. This gives: \( x = (2y-3)/(y+1) \).

Step 2: Substitute x in the Second Equation

Substitute \( x = (2y-3)/(y+1) \) into the second equation \( x^{2}+4y^{2}-9 = 0 \). This gives a quadratic equation in terms of y: \( ((2y-3)/(y+1))^2 +4y^{2}-9 = 0 \).

Step 3: Solve the Quadratic Equation

Solve the quadratic equation \( ((2y-3)/(y+1))^2 +4y^{2}-9 = 0 \) to find the y-coordinates of the points of intersection.

Step 4: Find Corresponding x-coordinates

Substitute the y-coordinates obtained in Step 3 into the rearranged first equation \( x = (2y-3)/(y+1) \) to find the corresponding x-coordinates of the points of intersection.

Step 5: Verify Using a Graphing Utility

Lastly, plot the two original equations \( xy+x-2y+3 = 0 \) and \( x^{2}+4y^{2}-9 = 0 \) on the same graph using a graphing utility to visually check if the intersection points found algebraically coincide with points of intersection on the graph.

Key Concepts

Understanding Algebraic Solutions

Finding the points of intersection between two equations is essential in algebra, especially when dealing with system of equations. An algebraic solution involves manipulation and simplification of equations to find common solution sets. In our exercise, the first step requires rearranging one of the equations to express one variable in terms of the other. Here, we solve the first equation for x, leading to the formulation x = (2y-3)/(y+1).

Next, we must substitute this expression into the second equation to eliminate x, allowing us to find a relation solely in terms of y. This helps us isolate one of the variables, making it easier to solve the system and find the intersection points. It's crucial to simplify the resulting equation properly to get a quadratic equation that can be solved by factoring, completing the square, or using the quadratic formula. This process underscores the importance of manipulation skills in algebra to provide clarity and reduce complex systems to solvable equations.

Using Graphing Utility for Verification

After obtaining the algebraic solution, verification is an essential step to ensure accuracy. A graphing utility is a powerful tool students can use to verify their algebraic solutions. Once you input the original equations into the graphing utility, it will display the graphs, and you can visually identify the points of intersection.

In the exercise, we plot both equations: xy+x-2y+3 = 0 and x²+4y²-9 = 0. The points on the graph where the curves intersect correspond to the solution of the system. This not only confirms the algebraic solution but also provides a visual understanding of how two equations can relate to each other in a coordinate space. Using technology enhances comprehension and is an excellent way to double-check work, ensuring that the solutions make sense in both mathematical and graphical contexts.

Solving Quadratic Equations

The quadratic equation is a fundamental concept in algebra that appears in various forms across mathematics. It’s an equation of the form \(ax^2+bx+c=0\). Our exercise involves a rearranged quadratic equation \( ((2y-3)/(y+1))^2 +4y^2-9 = 0 \) that emerged after substitution. Solving quadratic equations is a two-step process that typically involves simplifying the equation to its standard form and then finding the roots through methods such as factoring, using the quadratic formula, or graphing.

The solutions to the quadratic equation represent the y-coordinates of the points of intersection of the original equations. Once y is found, it's substituted back into one of the original equations to find x, completing the set of intersection points. Mastery of solving quadratic equations is crucial for interpreting and predicting various physical phenomena, making it an indispensable tool in both pure and applied sciences.