In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If

step 1 Hello. So here we have this given system of three equations. So what we're going to do is find t

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

Key Concepts

Systems of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all the equations simultaneously. This concept is fundamental in algebra and is applied in various fields, such as engineering, economics, and computer science, often using methods such as substitution, elimination, and matrix techniques.

Cramer's Rule

Cramer's Rule is a theorem in linear algebra that provides an explicit formula for the solution of a system of linear equations with as many equations as unknowns, using determinants. Specifically, each variable is solved by replacing the corresponding column of the coefficient matrix with the constant terms and dividing the determinant of this new matrix by the determinant of the original coefficient matrix. This method is applicable only when the determinant of the coefficient matrix is non-zero, ensuring a unique solution.

Determinants

A determinant is a scalar value computed from the elements of a square matrix and is a key concept in linear algebra. Determinants are used to determine whether a matrix is invertible or singular. In the context of solving systems of linear equations using Cramer's Rule, the determinant of the coefficient matrix is crucial because a non-zero determinant indicates that the system has a unique solution, while a zero determinant suggests that the system may be either inconsistent or have infinitely many solutions.

Applicability Conditions of Cramer's Rule

Cramer's Rule can only be applied to square systems (i.e., the number of equations equals the number of unknowns) and requires that the determinant of the coefficient matrix is non-zero. If the determinant is zero, the system is either dependent or inconsistent, meaning Cramer's Rule cannot be used, and other methods such as row reduction or parameterization must be considered.

Transcript

0:00 Hello.

00:02 So here we have this given system of three equations.

00:05 So what we're going to do is find the determinant d of the coefficients on our variables.

00:13 So we look at the coefficients on our variables, and we take the determinant d of that corresponding matrix.

00:21 So we have the determinant.

00:23 So d here is equal to the determinant of, well, the coefficients on our first equation are 1 -2.

00:30 And 3, and then we have 3, 1, and negative 2, and then we have 2, negative 4, and 6.

00:41 Okay, so computing a 3 -by -3 determinant, we can go ahead and expand along the first row.

00:48 So if we do that, we have, well, 1 times the determinant of, well, it's minors.

00:56 We cross out the row and column here.

00:57 So we look at 1 -9 -2 -4 -6.

01:01 So the determinant there is 1 times 6, which is 6 minus negative 4 times negative 2.

01:08 That's going to be 6 minus 8, which becomes negative 2.

01:14 So we have negative 2.

01:15 Then we have minus, well, minus the next value here, the next value is negative 2.

01:21 If we have minus a negative 2 or plus 2 times the determinant of its minor.

01:26 So crossing out the row and column here, looking at 3 negative 2, 2.

01:31 6...

Please give Ace some feedback