Solve each system of equations using matrices (row operations). If the system has no solution, s

Solve each system of equations using matrices (row...

Key Concepts

System of Linear Equations

A collection of two 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. These systems can have a unique solution, infinitely many solutions, or no solution at all.

Augmented Matrix

A matrix that includes both the coefficients of the variables and the constants from the right-hand side of the equations. This compact representation allows one to apply matrix operations systematically to solve the system.

Row Operations

Fundamental manipulations on the rows of a matrix, including swapping rows, scaling a row by a nonzero constant, and adding a multiple of one row to another row. These operations are used to simplify the matrix to a form where the solutions of the system become apparent.

Gaussian Elimination

A method that uses row operations to transform an augmented matrix into an upper triangular form or row-echelon form. This procedure helps in systematically solving for the variables by back-substitution.

Inconsistent Systems

A situation in which the system of equations has no solution because the corresponding augmented matrix contains a row that leads to a contradictory statement (such as 0 = nonzero). Recognizing inconsistency is critical when solving systems using row operations.

Transcript

00:01 The idea or the concept behind this problem is to solve a system of linear equations using matrices.

00:11 As we solve this problem using matrices, we'll go through a variety of different row operations.

00:18 Now, the goal of what we want to try to accomplish is to take the matrix associated with the system and write the matrix using some different row operations, write the matrix.

00:31 In what is called reduced row echelon form.

00:35 If you can get the matrix into this form, then you should be able to determine the solutions to your system.

00:44 This form is going to tell you that x is equal to some rule number a because you have 1x, 0y, and 0 z is equal to a.

00:56 And in a similar way you'll be able to say y is equal to some wheel number b and then z is equal to some wheel number c meaning the order triple that would solve the system you're working with is abc now also as i've worked through this problem i'm going to use some notations lowercase r1 r2 and r3 are going to refer to the row that i am currently or the row from the matrix that i'm currently working with.

01:35 Okay, so it's the row of the current matrix and uppercase r1, r2, r3 is going to indicate the row operation that i will work with.

01:55 Now, when you have a system that you're trying to solve using matrices and row operations, there are sometimes different ways to work through it.

02:07 And you've just got to work with what comes to your mind, but keep you.

02:12 In mind the goal of what you want you want to get this reduced row at your arm all let's get started on this problem and i am going to set up the matrix associated with this system so i'm going to set up two negative two negative two vertical bar from the second equation i will have the second row two three one and two and then a similar thing with the third equation to get the third row.

02:49 Three, two, there's no z term in the third equation, so i'll put a zero, and then it equals zero.

02:57 Okay, so there is our matrix that's associated with the given system.

03:02 Again, as i said, our goal is to go through some series of row operations to try to get this reduced row echelon form.

03:12 Well, on this particular problem, i noticed my first row.

03:18 Every term in it is divisible by two.

03:21 Every term in it has a factor of two.

03:24 So i'm going to make my first row operation to change row one by taking the current row one times a half...

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