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.

4x + 3y = -7

2x + 3y = -11

Accepted Answer

Hello, today we're going to be solving the system of equations by using Kramer's rule. So the first thing we want to do in order to use Kramer's rule is to convert our system of equations into matrices. The first matrix is going to be a two by two matrix and this is going to contain the coefficients of the X and Y terms. And we're going to create a solution matrix which contains the solutions to the system of equations. So what are the coefficients of X and Y in the first group or in the first row? Well, those coefficients are going to be nine and two and the solution to the first row is going to be negative 55. Next, we're going to write the coefficient of X and Y in the second row and that is going to be five and three and the solution to the second row is going to be negative 23. Now, we need to check to see if it's possible to use Kramer's rule in order to do that we're going to solve for the general determinant of the two by two matrix we created again the two by two, matrix is 925 and three. And in order to get the determinant of this two by two, we're gonna multiply the upper left and lower right element and subtract the product of the lower left and upper right element. The general determinant is going to be nine times three minus five times two, nine times three is going to give us 27 and five times two is going to give us times negative one is going to give us negative 10 and 27 minus 10 is going to give us 17. Now, since the determinant is not equal to zero, that clarifies or verifies that we can use Kramer's rule. So we're going to make a note that the general determinant is 17. Now, how do we use Kramer's rule to solve for X and Y? Well, using Kramer's rule, X is defined as the determinant with respect to X divided by the general determinant and Y is defined as the determinant with respect to Y divided by the general determinant. So we're going to need to get 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 matrix which is 952 and three. And what we're going to do is we're going to replace the X column or the first column with the solution matrix and doing so is going to give us a new two by two which is going to be negative 55 negative 23 2 and three. Now, if we take the determinant of this new two by two, that is going to give us the determinant with respect to X. So the determinant of the new two by two is going to be negative times three minus negative times two, negative 55 times three is going to give us the value of negative 165 and negative 23 times two is going to give us negative 46 times negative one is going to give us positive 46 and negative 165 plus 46 is going to give us the value of negative 119. So that means that the determinant with respect to X is going to equal to negative 119. Now we're gonna repeat this process by solving for the determinant with respect to why. Now previously, in order to get the determinant with respect to X, we replaced the first column which is the X column with the solution matrix. In order to get the determinant with respect to Y, we're going to replace the Y column with the solution matrix. And doing so is going to give us a new two by two matrix which is going to be 95, negative 55 and negative 23. Now we're going to take the determinant of this two by two matrix. The determinant is going to be nine times negative 23 minus five times negative 55 nine times negative 23 is going to give us negative and negative five times fit negative 55 is going to give us positive 275. Finally negative 207 plus 275 is going to give us the value of positive 68. So the determinant with respect to Y is going to equal to 68. Now we can use these values to solve for X and Y. Now keep in mind, X is defined as deter determinant with respect to X divided by the general determinant. So that means X is going to equal to negative 119 divided by 17. Now 100 and 19 is divisible by 17 as it's going to reduce to seven, but negative 119 divided by 17 is going to give us negative seven. So X is going to equal to negative seven. Now why is, is defined as the determinant with respect to Y divided by the general determinant that is going to be 68/ and 68 is divisible by 17 as this is going to reduce to four. So Y is going to equal to four. And if we put our solutions together the solutions for X and Y are going to be negative seven and four. And this is going to be the solution to the system of equations by using Kramer's rule. And with that being said, the answer to this problem is going to be a. 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.
4x + 3y = -7
2x + 3y = -11

Key Concepts

Cramer's Rule

Determinant of a Matrix

Alternative Methods for Solving Systems

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