Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-s

Use matrices to solve the system of equations, if possible....

Key Concepts

System 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 forms the foundation for many problems in algebra and numerical methods, and is essential in modeling real-world situations where multiple conditions must be met at once.

Matrix Representation

Matrix representation involves expressing a system of linear equations in a compact form using arrays. In this representation, the coefficients of the variables form a coefficient matrix, and an additional column (often appended as the augmented column) contains the constants from the equations. This approach helps in applying algebraic methods and computational algorithms such as Gaussian elimination.

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations by transforming the augmented matrix into an upper triangular form. This process involves using elementary row operations to simplify the equation system progressively. The method is widely used because it provides a straightforward procedure for simplifying complex systems into a form that can be solved easily.

Row Operations

Row operations are the fundamental manipulations performed on the rows of a matrix during Gaussian elimination. They include row swapping, scaling a row by a nonzero scalar, and adding a multiple of one row to another. These operations are crucial because they simplify the matrix without changing the solution set of the system, allowing for systematic reduction towards an upper triangular form.

Back-Substitution

Back-substitution is the process used after transforming the system into an upper triangular matrix. It involves solving for the variables starting from the last equation and substituting the known values into the previous equations. This sequential approach enables the determination of all variable values efficiently once the matrix is in the appropriate triangular form.

Transcript

00:01 We have given the equation here x plus x plus 2y minus 3g equals to minus 28 and second equation 4y minus this is plus 2 z equals to 0 and minus x plus y minus z equals to minus 5.

00:22 And now the augmented matrix will be 1, 2, minus 3 and then this will be 2 minus 3 and then this will with a minus 28, 4, this is the 0, 4, 2 and also 0, minus 1, minus 1 and this is minus 5.

00:44 So this is the augmented matrix.

00:47 And now we want this term as unity and then this term as 0 and this term as unity.

01:00 So now first we are doing in operation number row that is operation number 1 that is row 2 replaced with row 2 divide with 4.

01:13 1 2 minus 3 this will be the minus 28.

01:17 So this will be the 0 1 and this will be the 1 half and then this is 0 and then here minus 1, minus 1 and then minus 5.

01:29 And now we are doing the next operation that is r3 this term we want this term as 0.

01:39 So r3 replaced with r3 plus r1.

01:43 So doing this we get so do this operation.

01:48 So nothing changes in row number 1.

01:51 It is as previous 1.

01:54 Nothing changes in row number 2.

01:57 And now for the row number 3 this will be the 0 and this will be the 0.

02:00 And this will be again here 3 and this will be the minus 4 equals to this is the minus 32 and now we have to do next operation so for this we have to make here row number 3 that is here we have to make this as unit so for this r3 replaced with also we want here 1 then we want here 0 so for this r3 is replaced with also we want here 0 so for this r 3 is replaced with with r3 minus 3 times of r2...

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