Use Cramer's rule to solve each system of equations. If D = 0, th...

Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method
to determine the solution set. See Examples 5–7. 
5x + 4y = 10
 3x - 7y = 6

Accepted Answer

Hello, today we're going to solve the system of equations by using Kramer's rule. Now, the first step into doing this is to convert our system of equations into a system of matrices. The first matrix we're going to create is going to be a two by two matrix that's gonna hold the coefficient of our X and Y terms. And the second matrix that we're going to make is going to be the solution matrix. So the two by two matrix is going to have the coefficient of all of our X and Y terms the coefficients of X and Y in the first row are going to be 10 and three. And the solution to the first row is going to be 21. The coefficients of X and Y in the second row are going to be four and negative nine. And the solution to the second row is going to be negative 63. Now, in order to determine whether we can use Kramer's rule or not, we need to find the determinant of our current two by two matrix. So again, the two by two is 10, 34 and negative nine. And in order to take the determinant of a two by two matrix, we're going to multiply the upper left and lower right element and subtract the product of the lower left and upper right element. The determinant is going to be defined as 10 times negative nine minus four times three, 10 times negative nine is going to give us negative 90 negative four times three is going to give us negative 12. Finally negative 90 minus 12 is going to give us the value of negative 102. And one thing to note is that since this determinant is not equal to zero, that verifies that we can use Kramer's rule. So we're going to make a note that the general determinant is equal to negative 102. So how do we use Kramer's rule to solve for X and Y? Well, XX is defined as the determinant with respect to X divided by the general determinant and Y is equal to the determinant with respect to Y divided by the general determinant. So we're going to need to solve for the determinant with respect to X and the determinant with respect to Y. In order to get the determinant with respect to X, we're going to take our original two by two which is 10, 34 and negative nine. And what we're going to do is we're going to replace the X coefficient, which is the first column with our solution matrix. That is going to give us a new two by two matrix which is going to be 34. I'm sorry, not four, negative 63 and negative nine. And now what we wanna do is you want to take the determinant of this two by two matrix that's going to be 21 times negative nine minus negative 63 times three. Now 21 times negative nine is going to give us the value of negative 189 and negative 63 times three is going to give us negative 1 89. And if we multiply that by negative one, that's going to give us positive 189 a negative 89 plus 1 89 is going to give us zero. That means that the determinant with respect to X is going to be zero. Now we're gonna repeat this process for the determinant with respect to why? Now, just like before we took our original two by two, which is 10, 34 and negative nine. And in order to get the determinant with respect to X, we replace the X coefficient which was the first column. But in this case, in order to get the determinant with respect to Y, we're going to replace the Y coefficients which is the second column with our solution matrix that is going to give us a new det new two by two matrix which is going to be 10, 21 4 and negative 63. Now we want to take the determinant of this two by two matrix. So the determinant is going to be 10 times negative 63 minus four times 10 times negative 63 is going to give us negative and four times 21 is going to give us the value of 84 times negative one is negative 84. Finally negative 630 minus is going to give us the value of negative 714. So the determinant with respect to Y is equal to negative 714. Now that we have all the necessary values to solve for X and Y, we're gonna go ahead and plug in these values. So again, X is defined as the determinant with respect to X divided by the general determinant that is going to give us zero over negative 102 and zero divided by any value is going to give us zero. So X is going equal to zero, Y is defined as the determinant with respect to Y divided by the general determinant. So Y is equal to negative 714 over negative 102 714 is divisible by 102 as it divides into seven and a negative number over a negative number is going to give us a positive number. So that means that Y is going to equal to positive seven. So we just solved that X is equal to zero and Y is equal to seven. And if we put that together the solution to the system of equations is going to be zero comma seven. And with that being said, the answer to this problem is going to be C So I hope this video helps you in understanding how to use Kramer's rule. And I'll go ahead and see you all in the next video.

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. 5x + 4y = 10 3x - 7y = 6

Key Concepts

Cramer's Rule

Determinants

Alternative Methods for Solving Systems

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