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

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.

x + y = 4

2x - y = 2

Accepted Answer

Hello, today we're going to be solving the system of equations by using Kramer's rule. So the first thing we're gonna want to do is we're going to want to convert the system of equations into a matrix form. In order to do that, we're going to list out the coefficients of all of our X and Y terms in the main matrix. And we're going to list out the solutions to the system of equations in a solution matrix. So the coefficients of X and Y in the first row are going to be one and one and the solution to the first row is going to be 12. The coefficients of X and Y in the second row are going to be and negative one. And the solution to the second row is going to be 84. So the first thing we wanna do in Kramer's rule is we want to take the derivative of the general matrix. The general matrix is in this case is going to be the two by two matrix that we made to the left that's going to be 11 11 and negative one. Now, in order to take a determinant of the two by two matrix. We're gonna multiply the upper left and lower right element and subtract that by the product of the lower left and upper right element. So the determinant is going to be one times negative one minus 11 times one, one times negative one is going to give us negative one and 11 times one is going to give us 11 times negative one is going to give us negative 11 and negative one minus 11 is going to give us negative 12. Now, one thing to note is that the general determinant is not equal to zero. So we can go ahead and use Kramer's rule. So just to make a note, we're going to be having the general determinant as negative 12. Now, in order to solve for X and YX is going to equal to the determinant with respect to X divided by the general determinant and the determinant with respect to why divided by the general determinant. So how do we find the determinant with respect to X? Well, in this case here we have the matrix 11 11 and negative one. What we're going to do is we're going to replace the X column, which is the first column with the solution matrix. This is going to give us the determinant with respect to X. So if we replace the first column with the solution matrix, the new two by two matrix you get is going to be 12 84 and negative one. And now we want to take the determinant of this two by two matrix. So the determinant with respect to X is going to be 12 times negative one minus 84 times one, 12 times negative one is going to give us negative 12 and negative times one is going to give us negative 84. Finally negative 12 minus 84 is going to give us a value of negative 96. So the determinant with respect to X is going to equal to negative 96. Now we want to find the determinant with respect to Y, how do we do that? Well, before, in order to get the determinant with respect to X, we replace the X column with the solution matrix. In order to get the determinant with respect to Y, we're going to replacing the Y column with the solution matrix. So if we do that the matrix with respect to Y is going to be one 12, 11 and 84. And now we want to take the determinant of this two by two matrix. The determinant of this two by two matrix is going to be one times minus 11 times 12, one times 84 is going to give us and 1 to 11 times 12 is going to give us the value of 100 and 32. So we have 84 minus 1, 32 and 84 minus 100 and is going to give us the value of negative 48. So the determinant with respect to Y is going to be negative 48. Now, in order to solve for X, we're going to divide the determinant with respect to X by the overall determinant. Now, earlier we said the determinant with respect to X is negative and the determinant with re the general determinant, I'm sorry is going to be negative 12. So we just have to reduce this fraction while 96 is divisible by 12, 96/12 is going to reduce to give us eight and a negative number over a negative number gives us a positive number. So X is going to equal to eight in this case. Now we're gonna repeat this process for Y we're gonna divide the determinant with respect to Y by the general determinant. That's going to be negative 48 over negative 12. A negative number over a negative number gives us a positive number and 48/12 reduces to give us four. So the solution to Y is going to be four. And if we were to combine the solutions together the solution to X and Y for our systems of equations is going to be eight comma four. And that means that the answer to this problem is going to be D. 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.
x + y = 4
2x - y = 2

Key Concepts

Cramer's Rule

Determinant of a Matrix

Alternative Methods for Solving Systems

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