Solve the following simultaneous equations for x1, x2 and x3, using matrix methods: x1+2 x2

Solve the following simultaneous equations for x1, x2 and...

Key Concepts

Matrix Inversion

The inverse matrix method involves finding the inverse of the coefficient matrix A, provided that A is square and invertible, and then computing the solution vector x as x = A?¹b. This approach offers a direct formula for the solution and is fundamental to understanding the relationship between linear transformations and their inverses in linear algebra.

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations by applying a sequence of row operations to transform the augmented matrix into an upper triangular, or row echelon, form. Once in this form, back substitution can be used to easily determine the values of the unknowns. This method is one of the most standard approaches in linear algebra for solving linear systems.

Augmented Matrix

An augmented matrix is formed by appending the constant vector b to the coefficient matrix A, combining both the coefficients and the constants into a single matrix. This representation is fundamental when using row operations to solve the system, as it allows for all necessary information to be manipulated simultaneously during the elimination process.

Matrix Representation

This concept involves expressing a system of linear equations in the form of a matrix equation, Ax = b. Here, the coefficient matrix A, the variable vector x, and the constant vector b encapsulate all the information from the system. It provides a compact and structured way to handle multiple equations simultaneously, which is especially useful for applying algorithmic solution techniques.

Transcript

00:01 Okay, we are going to solve here three equations, x1 plus 2x2 plus 3x3 equals 1, and the other two equations at the same time.

00:11 We are going to use the same technique as before, we are going to isolate x1 from the first equation.

00:19 To do so, we are going to write the first equation as x1 equals 1 minus 2x2 minus 3x3.

00:29 Then we substitute this equation into the second and the third equation.

00:38 So these two equations then would become, the first one will be 3 1 minus 2x2 minus 3x3 plus 4x2 plus 5x3 equals 2.

00:57 Now simplify it, 3 minus 6x2 minus 9x3 plus 4x2 plus 5x3 equals 2.

01:08 Now add the similar terms, and i can move the 3 to the right side, it will be 2 minus 3.

01:17 So minus 6x2 plus 4x2 would be minus 2x2, minus 9x3 plus 5x3 would be minus 4x3, and that will be equals 2 minus 3, and that is minus 1.

01:36 All right, so now here we have a new equation which is going to be minus 2x2 minus 4x3 equals minus 1.

01:45 All right, so this is here the second equation.

01:48 Also we are going to substitute this equation into the third equation, and let's see how it will become.

01:59 So it's going to be 1 minus 2x2 minus 3x3 plus 2, sorry, plus 3x2 plus 4x3 equals 3.

02:14 Now we can sum up the, we can simplify it.

02:20 To simplify it, okay, we will go with, we move one to the right side, minus 2x2 plus 3x2 will be x2, so here will be x2, minus 3x3 plus 4x3 will be plus x3 equals 3 minus 1 will be 2.

02:41 All right, so this is the second equation.

02:44 Now we have two equations here.

02:48 Keep in mind what we are doing right now is substitution.

02:52 So for example, i can isolate x2 from this equation.

02:56 I can say, okay, x2 will be equals to 2 minus x3.

03:01 So now i substitute this equation back into this equation.

03:06 So it will be minus 2, 2 minus x3 minus 4x3 equals minus 1.

03:16 So this will be minus 4 plus 2x3 minus 4x3 equals minus 1.

03:23 Simplify it, will be minus 4.

03:26 I mean, i can move the minus 4 to the right side...