In Problems 15- 12 , solve each system of equations using...

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}x+2 y-z=-3 \\ 2 x-4 y+z=-7 \\ -2 x+2 y-3 z=4\end{array}\right.$

Key Concepts

Cramer's Rule

Cramer's Rule is a technique for solving a system of linear equations by replacing columns of the coefficient matrix with the constant terms and computing the resulting determinants. It is applicable only when the determinant of the coefficient matrix is nonzero, ensuring that the system has a unique solution.

Determinants

The determinant is a scalar value computed from a square matrix, reflecting certain properties of the matrix such as invertibility and the scaling factor associated with the linear transformation it defines. Determinants are central in methods like Cramer's Rule, as a nonzero determinant indicates a unique solution for a system of equations.

Systems of Linear Equations

A system of linear equations consists of multiple linear equations that are solved simultaneously to find the common values for the variables. These systems are foundational in linear algebra and are used to model and solve problems across various disciplines, from engineering to economics.

Transcript

0:00 Hello.

00:01 So here we have this system of equations.

00:04 We have x minus y plus z equal to negative 4, 2x minus 3y plus 4 z equals negative 15, and 5x plus y minus 2 z equal to 12.

00:15 So to use kramer's rule, you want to find the determinant of the coefficient matrix.

00:20 So d is equal to the determinant.

00:23 We just take the coefficients on all of our variables.

00:26 So we have the determinant of this matrix here.

00:28 Now we have a determinant of a 3x3 matrix.

00:30 So we can go ahead and expand along the first column.

00:34 So we just have one times the determinant of its minor.

00:38 We just cross out the row and column where one is in and look at the corresponding 2x2 matrix.

00:44 And then we have minus, well, the next entry times the determinant of its minor, but the next entry is negative.

00:55 So we have minus a negative 1, which is a plus 1 times its determinant, and then plus one times its minor times its determinant of its minor here.

01:04 Okay, so what we have here, this is going to be equal to, well, one times we get negative three times negative two, which is a negative or positive six minus one times four.

01:17 So that's just six minus four.

01:22 And then we get a plus one times while two times negative two is a negative four.

01:30 And then minus five times four.

01:32 So that's minus 20.

01:34 And then plus one times two times one is two, minus five times a negative three.

01:40 So that's minus a negative 15, which becomes plus 15.

01:44 Okay, so this becomes what? it becomes just equal to one times six minus four.

01:49 That's just one times two, which is two.

01:52 And then we get a plus, well, one times minus four, minus 20 is minus 24.

01:57 So we have two minus 24.

01:59 And then we get a plus one times 17.

02:03 So we get a plus 17.

02:05 And this works out to be equal to negative 5.

02:08 So d here is negative 5, which is not equal to 0.

02:13 So since the determinant here is not equal to 0, yes, we can use kramer's rule.

02:18 Kramer's rule just says that, well, x is equal to d subx over d.

02:27 And then every variable, y is equal to d sub y over d, and in this case, z is equal to d sub z over d...

Please give Ace some feedback