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This section covers the fundamental methods for solving systems of linear equations, focusing on diagonal form. It introduces the three elementary row operations—rearranging equations, multiplying by a nonzero number, and adding multiples of equations—that are crucial for transforming equations. The Gaussian elimination method is detailed, demonstrating how to achieve diagonal form through iterative application of these operations. Additionally, the augmented matrix representation is explained for easier manipulation of coefficients and constants, simplifying the solution process.
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Section 2.1 Solving System of Linear Equations
2.1 Solving Systems of Linear Equations I • Diagonal Form of a System of Equations • Elementary Row Operations • Elementary Row Operation 1 • Elementary Row Operation 2 • Elementary Row Operation 3 • Gaussian Elimination Method • Matrix Form of an Equation • Using Spreadsheet to Solve System
Diagonal Form of a System of Equations • A system of equations is in diagonal form if each variable only appears in one equation and only one variable appears in an equation. • For example:
Elementary Row Operations • Elementary row operations are operations on the equations (rows) of a system that alters the system but does not change the solutions. • Elementary row operations are often used to transform a system of equations into a diagonal system whose solution is simple to determine.
Elementary Row Operation 1 • Elementary Row Operation 1 Rearrange the equations in any order.
Example Elementary Row Operations 1 • Rearrange the equations of the system • so that all the equations containing x are on top.
Elementary Row Operation 2 • Elementary Row Operation 2 Multiply an equation by a nonzero number.
Example Elementary Row Operation 2 • Multiply the first row of the system • so that the coefficient of x is 1.
Elementary Row Operation 3 • Elementary Row Operation 3 Change an equation by adding to it a multiple of another equation.
Example Elementary Row Operation 3 • Add a multiple of one row to another to change • so that only the first equation has an x term.
Gaussian Elimination Method • Gaussian Elimination Method transforms a system of linear equations into diagonal form by repeated applications of the three elementary row operations. • Rearrange the equations in any order. • Multiply an equation by a nonzero number. • Change an equation by adding to it a multiple of another equation.
Example Gaussian Elimination Method • Continue Gaussian Elimination to transform into diagonal form
Example Gaussian Elimination ( 3) The solution is(x,y,z) = (4/5,-9/5,9/5).
Matrix Form of an Equation • It is often easier to do row operations if the coefficients and constants are set up in a table (matrix). • Each row represents an equation. • Each column represents a variable’s coefficients except the last which represents the constants. • Such a table is called the augmented matrix of the system of equations.
Example Matrix Form of an Equation • Write the augmented matrix for the system Note: The vertical line separates numbers that are on opposite sides of the equal sign.
Summary Section 2.1 - Part 1 • The three elementary row operations for a system of linear equations (or a matrix) are as follows: • Rearrange the equations (rows) in any order; • Multiply an equation (row) by a nonzero number; • Change an equation (row) by adding to it a multiple of another equation (row).
