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. 2x^2 = 3y + 23 y = 2x - 5

Graph showing intersection points (0,-1) and (3,6) of a quadratic and linear equation.

Accepted Answer

Everyone in this problem for the indicated intersection points on the graph or asked to check if they are solutions to the nonlinear system of equations given using the substitution method. Now, the nonlinear system we have is made up of two equations. The first is seven X squared equals nine Y plus nine. And the second is three, Y is equal to seven X minus three. We're given our graph and then we're given two answer choices. Option A is that the given points are solutions. And option B is that the given points are not solutions. So if we look at our graph, we have a quadratic looking function drawn in green. OK. That's gonna co correspond with that first equation because it is a quadratic equation. We have an X squared and then a Y term OK. And then we have in red, the equation of a straight line. OK. We have this straight line drawn and red in the equation three Y equals seven X minus three. This question is asking us to use the substitution method. Yeah. Remember they're asking us to check if they're solutions not to solve the system. OK. So the substitution method when we're checking solutions versus when we're actually solving a system is a little bit different. And all they mean in this case is that we're gonna take each of these points, the zero comma negative 13, comma six. We're gonna substitute them into the equations we have and check whether or not they satisfy those equations. Let's give ourselves some more room to work. And I'm just gonna rewrite the equations we were given. So we have seven X squared is equal to nine Y plus nine. And we're gonna call this equation one. And then we have three, Y is equal to seven X minus three. And we're gonna call this equation two. So we're gonna start with that 1st 0.0 comma negative one. And we're gonna start by substituting that point into equation one. So when we do that, we have seven multiplied by X which is zero squared is equal to nine, multiplied by Y which is negative one plus nine. On the left hand side, we have seven multiplied by zero, which gives a zero. On the right hand side, we get negative nine plus nine, which is also going to give a zero. So you have zero is equal to zero. This is a true statement. Give that a green check mark and this is a true statement. And so what that means is that zero, negative one satisfies the first equation. Now, what about the second equation? So we're gonna take zero negative one still. And we're gonna substitute it into equation two. OK? So we substitute it into equation one. Now we're substituting into equation two, we have three multiplied by Y which is negative one is equal to seven, multiplied by zero minus three. On the left hand side, we get negative three. On the right hand side, we have zero minus three, which also gives us negative three. So we have negative three is equal to negative three. And again, that's a true statement. OK? Both sides are equal, this is true. And so what this means is that, that first point zero comma negative one is a solution to this system because it satisfies both equations. Now we're gonna do the exact same thing with the second point. So we're gonna substitute in this second point which is three comma six. OK? Into the first equation seven multiplied by X which is three squared is equal to nine, multiplied by Y just six was not on the left hand side, we have seven multiplied by nine. And on the right hand side, we get 54 plus nine. And if we simplify each of these sides, we get that 63 is equal to 63. And again, that's a true statement. The left and the right sides are equal. So this point satisfies equation one. And now we can move to equation two, we have three multiplied by Y which is six is equal to seven multiplied by X which is three minus three. Simplifying on the left hand side, we have 18 and on the right, we have 21 minus three. This gives us 18 is equal to 18. And again, that's a true statement, green check mark here. And so this tells us that our point three comma six is also a solution. So we found that both of these points are solutions to the system of equations. If we go back up to our answer choices, this 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. 2x^2 = 3y + 23 y = 2x - 5

Key Concepts

Systems of Equations

Substitution Method

Graphical Interpretation

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