In Exercises 19–28, solve each system by the addition method. ...

Question

In Exercises 19–28, solve each system by the addition method. $\begin{cases}x^2 + y^2 = 25 \(x - 8)^2 + y^2 = 41\end{cases}$

Accepted Answer

Hey everyone in this problem we are asked to solve the system of equations given below and were told to use the addition method. Okay, the system of equations were given here is x squared plus y squared equals 16 and x squared plus y minus three squared equals 31. Okay, so let's start off just by labeling these equations. So it's easier for us to refer to them. So we get x squared plus y squared equals 16 is equation one, An equation two is x squared plus y minus three. All squared is equal to 31. All right. Now we're told to use the addition method and in the addition method, what we want to do is we want to choose one of the variables to eliminate. Okay. Either the X or the Y. And we want the coefficient in equation one and equation to to be the same magnitude but the opposite sign. Okay, So that when we add them together to cancel, they add up to zero. Okay? Alright, so here we have x squared and x squared. And so all we need to do is multiply one of these equations by negative one and then we will have the Form that we want for the addition method. Okay, so let's take equation one. Okay and multiply it by -1. And then we're gonna add equation too. Okay, so equation one times negative one is going to give us minus x squared minus y squared equals negative 16. Equation two is x squared plus y minus three squared equals 31. All right. And you want to be careful when you're doing addition method with systems that are non linear case. In our case our equations are nonlinear because we have these squared terms and if you have terms like this, y minus three all squared you have y squared terms. And you have just y terms. So it can be quite messy to eliminate those. Okay, so in this case we've chosen X squared and then we don't have to deal with any of those Y terms and y squared terms. Alright, so adding these together minus x squared plus x squared gives us zero exactly what we wanted. We have minus y squared plus y minus three squared is equal to negative 16 plus 31 is 15. Okay, expanding what we have we have negative y squared OK, y minus three. All squared where we're gonna get a y squared term? We're going to get minus six. Y plus nine equals minus y squared plus y squared. Those are going to cancel We have -6. Y Is equal to moving the nine to the right hand side is equal to six. Okay. And that gives us AY Value of -1. Okay? Alright, so we found a Y value of minus one. Now we need to find a corresponding X value. Okay, y value alone doesn't solve the system. We need some X values as well. So we only have one Y value. So we're going to substitute it into our equations to figure out what the corresponding X value is. And we can set it into either equation. Okay so this needs to satisfy both equations. So we can set it into either equation in this case we're going to choose one. Um but you can choose to if you wanted into equation one and so we're gonna have X squared plus negative one squared is equal to 16, So x squared plus one is equal to 16. X squared is equal to 15. And taking the square root X is equal to plus or minus the square of 15. Okay? And don't forget that when you take the square root you get a positive and a negative route. Ok because if you square a positive or you square negative you're going to get the same thing out of it. Alright so now we have X values corresponding with the y value found. Okay we have two X values and the single y values that's gonna give us two points. Okay so our solution set is going to be the points where we have the negative square root of 15. Okay and that corresponded with the y. Value of negative one and we have our positive square root of 15 which corresponded with the y value negative one as well. Okay so our solution set is the two points negative square root of 15, negative one and the square root of 15 -1 which corresponds with answer a thanks everyone for watching See you in the next video.

In Exercises 19–28, solve each system by the addition method. $\begin{cases}x^2 + y^2 = 25 \\(x - 8)^2 + y^2 = 41\end{cases}$

Key Concepts

Systems of Equations

Addition Method (Elimination Method)

Equations of Circles and Their Expansion

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