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Section 2.1

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

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Section 2.1

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  1. Section 2.1 Solving System of Linear Equations

  2. 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

  3. 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:

  4. 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.

  5. Elementary Row Operation 1 • Elementary Row Operation 1 Rearrange the equations in any order.

  6. Example Elementary Row Operations 1 • Rearrange the equations of the system • so that all the equations containing x are on top.

  7. Elementary Row Operation 2 • Elementary Row Operation 2 Multiply an equation by a nonzero number.

  8. Example Elementary Row Operation 2 • Multiply the first row of the system • so that the coefficient of x is 1.

  9. Elementary Row Operation 3 • Elementary Row Operation 3 Change an equation by adding to it a multiple of another equation.

  10. 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.

  11. 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.

  12. Example Gaussian Elimination Method • Continue Gaussian Elimination to transform into diagonal form

  13. Example Gaussian Elimination (2)

  14. Example Gaussian Elimination ( 3) The solution is(x,y,z) = (4/5,-9/5,9/5).

  15. 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.

  16. 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.

  17. 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).

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