Solve each system of equations using Cramer's Rule if is...

Solve each system of equations using Cramer's Rule if is applicable. If Cramer's Rule is not applicable, write, "Not applicable"
$\left\{\begin{array}{rr}x-y+z= & -4 \\ 2 x-3 y+4 z= & -15 \\ 5 x+y-2 z= & 12\end{array}\right.$

Key Concepts

Determinant

A determinant is a specific scalar value that can be calculated from the elements of a square matrix. It provides crucial information about the matrix, including whether the matrix is invertible. In the context of applying Cramer's Rule, a non-zero determinant of the coefficient matrix indicates that the system of linear equations has a unique solution.

Cramer's Rule

Cramer's Rule is a method used in linear algebra for solving systems of linear equations with as many equations as unknowns, provided that the determinant of the coefficient matrix is non-zero. It involves replacing one column at a time with the constants from the right-hand side of the equations and finding the solution by taking the ratio of the determinant of this modified matrix to the determinant of the original coefficient matrix.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The solution to such a system is a set of values for the variables that satisfies all the equations simultaneously. These types of systems can often be solved by algebraic methods, including substitution, elimination, or matrix techniques like Cramer's Rule.

Transcript

00:01 Okay, let's start solving this problem.

00:03 So first we turn it into a matrix.

00:06 So i have 1 negative 1, 1, minus 4.

00:11 I have 2, minus 3, 4, and minus 15.

00:18 I have 5, 1, minus 2, and 12.

00:34 Okay, so now let's use the kramer's rule.

00:39 So x equals to the bottom is, 1 minus 1, 1, 2, minus 3, 4.

00:59 5, 1, minus 2, 1, 2.

01:04 And the top is minus 4, minus 1, 1, minus 15, minus 3, 4.

01:19 12, 1, minus 2.

01:23 So, equals to minus 4 times minus 1, 1, minus 3, 4, 1, minus 2, minus, or plus, minus, or plus, minus, 15, 4, 12, and minus 2.

01:56 And the minus, minus 15, minus 3, 12, 1.

02:16 And the bottom is minus 3, 4, 1, minus 2 times 1, and minus, plus, plus 2, 5, plus 2, 5 ,000, plus 2, minus 2 and the last one is minus uh sorry plus plus two minus three and the five and the one okay so let's solve this let's solve the bottom part first okay so the top uh let's just solve the bottom first so six minus 4 plus minus 4 minus 20 plus 2 minus minus or plus 15.

03:14 Okay, the top is minus 4 times 6 minus 4 is just 2 and a plus 30 minus 48 minus 16 minus 21.

03:40 So let's further solve this thing.

03:53 So i have minus 8, minus 16, minus 21.

04:00 Okay, so minus 8, minus 16 is minus 4, uh, 24.

04:07 And the minus 21 is minus 45.

04:13 Okay.

04:20 And the bottom is minus 5.

04:31 So the total is 9.

04:33 Okay.

04:37 Now let's solve for y...