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Systems of Linear Equations (9/7/05)

Systems of Linear Equations (9/7/05). Linear equations 4 x + 3 y = 13; 1.7 x 1 - .5 x 2 + 6.3 x 3 = 3.7 Not linear (why??) 4 x 2 + 6 y = 5; x 1 x 2 + 5 x 1 = 8

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Systems of Linear Equations (9/7/05)

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  1. Systems of Linear Equations (9/7/05) • Linear equations • 4x + 3y = 13; 1.7x1 - .5x2 + 6.3x3 = 3.7 • Not linear (why??) • 4x 2 + 6y = 5; x1 x2 + 5x1 = 8 • A system of linear equa tions (or linear system) is two or more linear equations in unknowns x1 , x2 , … , xn .

  2. Solutions to Linear Systems • A solution of a linear system is a list of numbers (s1, s2, … , sn ) which satisfy all the equations simultaneously. • Example: Find a solution to the system3x + 4y = 5; -x + 2y = 6 • Is this solution unique? • What is the geometric meaning of this system and its solution?

  3. More on solutions • A given linear system has either: • One unique solution (“consistent ”), or • No solutions (“inconsistent ”), or • Infinitely many solutions (also “consistent”) • Give examples of the latter two possibilities. What is the geometric meaning in two variables? In three variables?

  4. Matrix Notation • A linear system is much more easily written down as a matrix , i.e., a rectangular array of numbers. • Example: Write down the coefficient matrix and the augmented matrix forx1 - 2x2 + x3 = 0 2x2 - 8x3 = 8-4x1 +5x2 + 9x3 = 9

  5. Solving a Linear System • A system can be solved by replacing it with a simpler equivalent system (meaning a system which has the same solution(s)) whose solution(s) are obvious. • We do this using elementary row operations: replacement, interchange, and scaling. Try this on the system given on the previous slide.

  6. Assignment for Friday • Get the textbook and look over. Read the Note to Students. • Read the Introductory Example and Section 1.1. • Do the 4 Practice Problems on page 10. (Note: Practice Problems are completely solved right after the Exercises.)

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