Verify that the points of intersection specified on the graph of ...

Question

Verify that the points of intersection specified on the graph of each nonlinear system are solutions of the system by substituting directly into both equations. y = 3x^2 x^2 + y^2 = 10

Graph showing the intersection points of y=3x^2 and x^2+y^2=10 at (-3,2) and (3,2).

Accepted Answer

Hey, everyone in this problem for the indicated intersection points on the graph, we're asked to check if they are solutions to the nonlinear system of equations given using the substitution method. Now we're given two equations in our system. We have equation one. We're gonna label that as equation one which is nine Y is equal to two X squared that represents the equation of a parabola. So we've labeled this in red and to correspond with the red parabola, we're given in our graph in the second equation which we're gonna label S two is X squared plus Y squared is equal to 13. That's the equation of a circle. And we've labeled that in green to correspond with the green circle we're given in the graph. Now, in this problem, we have two answer choices. Option A is at the given points are solutions and option B is that they are not. Now, the points were given on this graph, we have negative three comma two, we have positive three comma two. We're told to use the substitution method. We need to be a little bit careful here. We want to check if there are solutions using this method, we are not asked to solve the system using this method. OK. So if we were to solve the system using the substitution method, that would be very different than just checking if these are solutions. All right. So to check if these are solutions, what we wanna do is substitute each point into each equation. So we're gonna start with the first point and that point is negative three comma two. And we're gonna substitute that point and we're gonna substitute it first into equation one and then we're gonna substitute it into equation two. And we're gonna see what happens in each case. So for equation one, OK, we have nine multiplied by Y which in this case is too, is equal to two multiplied by X squared. So negative three squared, let me just rewrite the equations on the right here. So we can see them as we work through this as well. We have nine Y is equal to two X squared in red and we have X squared plus Y squared is equal to 13 in green. OK? So back to our substitution method, OK, when we're checking a point, the substitution method just means substitute that point into the equation and see if it satisfies it. So we're gonna work on the left hand side and the right hand side separately on the left, we have nine multiplied by two which gives us 18. On the right, we have two multiplied by negative three squared. So that gives us two multiplied by nine. And so we get that 18 is equal to 18. And this is true. We give this a checkmark, this is a true statement. And so the point negative three comma two satisfies the first equation. Now we do the same with the second. OK, we have X squared which is negative three squared plus Y squared, which is two squared is equal to 13. Simplifying the left hand side, negative three squared gives us nine plus two squared. So that's gonna be plus four. On the right hand side, we have this is equal to 13. And then adding on the left hand side, nine plus four gives us 13. So we get that 13 is equal to 13. And so this is true. OK? Another checkmark here, this is true. And so what this tells us is that the point negative three comma two is a solution to our system of equations. I'm just gonna write two system to short form it here. Maybe because it satisfies both of those equations. Now, we're gonna go through the exact same process with the second one. The second point we were given was positive three comma two. We're gonna substitute that. And first, we're gonna substitute it into equation. One equation, one is nine multiplied by X square or sorry, multiplied by Y. So nine multiplied by two is equal to two multiplied by X squared. So two multiplied by three squared simplify on the left, we have 18, on the right, we have two multiplied by nine. So we get 18 is equal to 18. Again, maybe the only thing that changed in this point was the sign of the X value. But in our equation X is getting squared. So it doesn't matter if it's negative or positive, we're gonna get the same result. And so this is a true statement. Again, the point satisfies the first equation. And now we need to double check with that second equation. So we're gonna substitute this point into equation two as well. We have X square. So three square plus Y squared plus two squared is equal to 13. And again, our X value is being squared. So it's gonna be the exact same as before we have nine plus four is equal to 13. This gives us that 13 is equal to 13, which is a true statement. And so what that means is that three comma two is also a solution to the system. Mhm So going back up to our answer choices. OK, we evaluated the two points using the substitution method and found that both of them satisfy the equations, which means that they are solutions to the nonlinear system that corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Verify that the points of intersection specified on the graph of each nonlinear system are solutions of the system by substituting directly into both equations. y = 3x^2 x^2 + y^2 = 10

Key Concepts

Substitution Method

Nonlinear Equations

Points of Intersection

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