Find the solution set for each system by graphing both of the ...

Question

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.

$\begin{cases}\frac{x^2}{25} + \frac{y^2}{9} = 1 \y = 3\end{cases}$

Accepted Answer

Hello, today we're going to be solving for the system of equations, then verify the solution. So we are given a system of equations. We're going to select the first equation as one and set the second equation to two. So quickly taking a look at the second equation we are given X is equal to seven. This is the graph of a line, specifically a vertical line located at positive seven along the X axis. So this is going to be the shape of the graph for the second equation. Then we want to go ahead and determine the graph of the first equation. The graph of the first equation is going to be X squared over plus Y squared over 64 is equal to one. This is the equation of an ellipse. But we need to figure out what the major axis of the ellipse is. Well notice that the value underneath the Y term is greater than the value underneath the X term. That means that the major axis of this ellipse is going to be the Y axis. And this is going to give us the shape of the equation which is X squared over B squared plus Y squared over A S A squared is equal to one. And that also means that the rough shape of the graph of the ellipse is going to have the major axis or the major vertices along the Y axis. And the minor vertices along the X axis, we represent the values of the major vertices with A and we represent the values of the minor vertices with B. So we just need to find the values of A and B. In this case, we're going to set A squared equal to the bigger denominator, which in this case is 64. And we're going to set B squared equal to the smaller denominator, which is 49. If we take the square root of A squared, we get A is equal to plus or minus eight. And if we take the square root of the B equation, we get B is equal to plus or minus seven. So that means that we're going to have an ellipse whose major vertices are located at zero, comma eight zero, comma negative eight and minor vertices at 07, comma zero and negative seven comma zero. And the shape of the lips is roughly gonna look something like this. Now keep in mind that we've determined the shape of the second equation previously, that is going to be a vertical line that passes through the value of X is equal to seven. So we know that the equation of the second or the shape of the second equation is going to be a vertical line passing through X is equal to seven. And that means that this line is going to intercept with the 0. comma zero. So that means that the solution to the system of equations should be seven comma zero. Now, what we wanna do is we want to verify the solution by plugging it into the original equations. Now keep in mind the second equation is given to us as X is equal to seven. And we want to test to see that seven comma zero is the solution to the system of equations. While the X value here is seven. And if we plug it into this equation, we get seven is equal to seven, which is a true statement. So this value works for the second equation. Now let's verify the same point for the first equation. The first equation is given to us as X squared over 49 plus Y squared over 64 is equal to one. If we plug in the values of X and Y, we get seven squared which is 49/ plus zero is equal to one 49. Over 49 is going to simplify, to give us one and one plus zero is just one. So we're left with the statement, one is equal to one, which is a true statement. And since the one solution verifies both of the equations. That means that the solution to the system of equations is going to be seven comma zero. And we also have the rough graph of the system of equations drawn. So that means that the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.
$\begin{cases}\frac{x^2}{25} + \frac{y^2}{9} = 1 \\y = 3\end{cases}$

Key Concepts

Graphing Conic Sections (Ellipses)

Graphing Linear Equations

Finding Points of Intersection

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