For Exercises 11–22, use Cramer's Rule to solve each system.

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Question

For Exercises 11–22, use Cramer's Rule to solve each system. 4x - 5y = 17 2x + 3y = 3

System of equations for Cramer's Rule: 4x - 5y = 17 and 2x + 3y = 3.

Accepted Answer

Hey everyone. Welcome to this video in this problem, we are asked to solve the following system using Cramer's rule. Okay? And the system we're given is five, X minus two. Y. Is equal to 26 7 X plus six. Y. Is equal to 10. The first thing we want to do when we're using Cramer's rule is go ahead and write this system of equations in matrix vector form. Okay, So we want to write this system as a X equals B. A. Is a matrix, It's gonna be two by two, and X and B are vectors. Alright, so what we're gonna do is a is gonna be the coefficient matrix. Okay, so we're gonna take the coefficients from the left hand side of these equations and write them A. Okay, so in the first row we're gonna have the coefficients corresponding with the first equation. In the first column we're gonna have a corresponding to X. So we have five and in the second column corresponding to why? So negative two. Okay, And then in the second row the same thing, but for the second equation, so we get five and six, okay, X is going to be the vector with the variables in it. Okay, so X is going to have X and y. And the vector B is going to have the right hand side of these equations. 26. 10. Okay. And you can think of this if we multiply this out on the top, we're gonna have five, X minus two, Y equals 26. And then seven X plus six, Y equals 10-K. So that's exactly what that system of equations is just rewritten in a different form. Alright, so now let's get started with Cramer's rule. Cramer's rule tells us that the value of X. That solves this system will be equal to the determinant of a. one divided by the determinant of A. And the one comes from, we're talking about X. Which is in the first position of the X. Vector. Okay, so index one in the X vector means that we have the one here when we're talking a one that matrix is going to be the matrix A. Where the first column is replaced by B. Okay, when Cramer's rule, that's what that subscript means. So we're gonna have the determinant, we're gonna replace the first column of A. With B. So we're going to get 26 10 in the first column and negative two and six in the second column. Okay, we're going to divide by the determinant A 57 in the first column negative 26 in the second column in the numerator we're gonna have 26 times six. Okay, these are determinants of two by two matrices, recall that you multiply cross on the angles and then subtract. So 26 times six minus 10 times negative two. Okay, watch your signs here, it can be easy to mix them up five times six minus seven times negative two in the denominator. This is going to give us 176 divided by 44. Which is going to equal four. Okay, so we found our X value of four. Now let's go ahead and find a. Y value. So we have a full solution. So our Y value in this case we have the determinant of A. And what subscript are we going to use? Well we have why here why is the second position of the vector? X. So we're gonna have a. Two. Okay. All right. And we're gonna divide by the determinant of a. The determinant of A to well this is going to be the matrix where the second column is replaced by B. Okay, so the first column is going to be 57. The regular column one of a. Okay. In column two is going to be replaced by B. 26 10. We're dividing by the determinant of a. Which we found for the x value is 44. OK, it's the same determinant of a um and so we can use that value. Save ourselves doing the calculation twice. Alright, in the numerator we get five times 10 - times 26 divided by 44. This is going to give us negative 132 divided by 44 which is equal to negative three. Okay, so we have our X value and we have a Y value and so we can write our solution which is going to be X equals four And Y is equal to -3. So using Cramer's rule, we found the solution is A. X equals four. Y is equal to negative three. That's it for this video. I hope that helped see you in the next one.

For Exercises 11–22, use Cramer's Rule to solve each system. 4x - 5y = 17 2x + 3y = 3

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