1 / 19

SECTION 6.1

SECTION 6.1. SYSTEMS OF LINEAR EQUATIONS: SUBSTITUTION AND ELIMINATION. MOVIE THEATER TICKET SALES. SEE EXAMPLE 1 SEE EXAMPLE 2. EQUIVALENT SYSTEMS OF EQUATIONS. Linear System More than one linear equation considered at a time.

cliff
Download Presentation

SECTION 6.1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. SECTION 6.1 • SYSTEMS OF LINEAR EQUATIONS: • SUBSTITUTION AND ELIMINATION

  2. MOVIE THEATER TICKET SALES SEE EXAMPLE 1 SEE EXAMPLE 2

  3. EQUIVALENT SYSTEMS OF EQUATIONS Linear System More than one linear equation considered at a time. Solution - ordered pair (or triple) that satisfies both (or all) equations simultaneously.

  4. CONSISTENT VS. INCONSISTENT When a system of equations has at least one solution, it is said to be consistent; otherwise, it is called inconsistent.

  5. THREE POSSIBILITIES FOR A LINEAR SYSTEM x - y = 1 x - y = 3 x - y = 1 2x - y = 4 x - y = 1 2x - 2y = 2 No Solution One Solution Infinitely Many Solutions

  6. SOLVING A SYSTEM BY SUBSTITUTION 2x + y = 5 - 4x + 6y = 12

  7. RULES FOR OBTAINING AN EQUIVALENT SYSTEM 1. Interchange any two equations. 2. Multiply (or divide) each side of an equation by a nonzero constant. 3. Replace any equation in the system by the sum (or difference) of that equation and a nonzero multiple of any other equation in the system.

  8. SOLVING A SYSTEM BY ELIMINATION • 2x + 3y = 1 • x + y = - 3 • Multiply equation 2 by 2 • Replace equation 2 with the sum of equations 1 and 2.

  9. MOVIE THEATER TICKET SALES DO EXAMPLE 5

  10. AN INCONSISTENT SYSTEM 2x + y = 5 4x + 2y = 8

  11. AN DEPENDENT SYSTEM 2x + y = 4 - 6x - 3y = - 12

  12. 3 EQUATIONS, 3 UNKNOWNS 2 x + 4y - 2 z = - 10 - 3x + 4y - 2 z = 5 5x + 6y + 3 z = 3 x + 2 y - z = - 5 - 3x + 4y - 2 z = 5 5x + 6y + 3 z = 3 Dividing 1st equation by 2 to make leading coefficient equal to 1.

  13. EXAMPLE x + 2 y - z = - 5 - 3x + 4y - 2 z = 5 3x + 6 y - 3 z = - 15 - 3x + 4y – 2z = 5 10y – 5z = -10 Mult. 1st eqn by 3 and add to 2nd

  14. EXAMPLE x + 2 y - z = - 5 5x + 6y + 3 z = 3 -5x - 10 y + 5 z = 25 5x + 6y + 3z = 3 - 4y + 8z = 28 Mult. 1st eqn by -5 and add to 3rd

  15. EXAMPLE Now we have 2 equations in only y & z: 10y - 5 z = -10 - 4y + 8 z = 28 Divide 1st equation by 5 Divide 2nd equation by 4

  16. EXAMPLE 2y - z = - 2 - y + 2 z = 7 2y – z = -2 -2y + 4z = 14 3z = 12 z = 4 Multiply 2nd equation by 2 & add to 1st

  17. EXAMPLE 2y – z = -2 2y – 4 = -2 2y = 2 y = 1 x + 2y – z = - 5 x + 2 (1) - 4 = - 5 x - 2 = - 5 x = - 3 (-3, 1, 4)

  18. EXAMPLE DO EXAMPLES 9, 10, 11

  19. CONCLUSION OF • SECTION 6.1

More Related