System of Linear Equation power point presentation

system of linear equation

1. Systems of Linear Equations

2. A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. Systems of Linear Equations

3. The general form of a linear equation is:

4. Types of Systems and Solutions: Independent System *Exactly one solution pair (x, y). *Lines intersect at a single point. Dependent System *Infinitely many solutions. *Lines coincide, representing the same line. *Every coordinate pair on the line satisfies both equations. Inconsistent System *No solution exists. *Parallel lines with no common point of intersection.

5. AX = B Where: A is the coefficient matrix (a matrix containing the coefficients of the variables), X is the variable matrix (a column matrix containing the variables), B is the constant matrix (a column matrix containing the constants from the equations). A system of linear equations can be represented in matrix form as:

6. MATrIces A matrix is an array or table consisting of rows and columns. It can be represented as m x n, where m is the number of rows and n is the number of columns. A matrix with only one column is a column vector, and with only one row is a row vector. Column vector Row vector

7. We can associate to a linear system three matrices: (1) the coefficient matrix, (2) the output column vector, and (3) the augmented matrix. For example, for the linear system 5x1 − 3x2 + 8x3 = −1 x1 + 4x2 − 6x3 = 0 2x2 + 4x3 = 3

8. Solving linear systemS Three basic operations, called elementary operations, can be performed on linear systems: Interchange two equations. Multiply an equation by a nonzero constant. Add a multiple of one equation to another.

9. to solve the associated linear system. Solution. Notice that the augmented matrix has a triangular structure. The third row corresponds to the equation x3 = 1. The second row corresponds to the equation x2 − x3 = 0 and therefore x2 = x3 = 1. The first row corresponds to the equation x1 − 2x3 = −4 and therefore x1 = −4 + 2x3 = −4 + 2 = −2. Therefore, the solution is (−2, 1, 1). EXAMPLES

10. EXAMPLE

11. activity

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