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7.3 Multivariable Linear Systems

7.3 Multivariable Linear Systems. We will be looking at equations with 3 variables like Equations generate a plane in the x-y-z coordinate plane. Graph using the intercepts (when the other variables are 0). (0, 0, 3). (0, 6, 0). (4, 0, 0). Types of Solutions 2 variable systems:.

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7.3 Multivariable Linear Systems

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  1. 7.3Multivariable Linear Systems

  2. We will be looking at equations with 3 variables like Equations generate a plane in the x-y-z coordinate plane Graph using the intercepts (when the other variables are 0) (0, 0, 3) (0, 6, 0) (4, 0, 0)

  3. Types of Solutions 2 variable systems: 1 solution infinite solutions no solution Consistent systems Inconsistent system

  4. Multivariable systems: Consistent Systems Solution: 1 point Solution: 1 line Solution: 1 plane Inconsistent Systems N o S o l u t i o n s

  5. Row Echelon Form: creating “stair-step” pattern with leading coefficients of 1 • Elementary Row Operations (things you may do): • Switch two equations • Multiply an equation by a constant (not zero) • Add a multiple of one equation to another (without changing the original) Use these to obtain Row Echelon Form

  6. Solving Multivariable Systems: Gaussian Elimination • Big Picture: • Obtain Row Echelon Form • Back Substitution Initial System: Row Echelon Form: From here you could “back” substitute z into the 2nd equation, solve for y. Then “back” substitute y and z into the 1st equation to solve for x.

  7. Example: The 1st equation is fine. It has x, y, and z The 2nd equation should not have an x term Step → add equations 1 and 2, replace 2 The 2nd equation is now fine. It has y and z The 3rd equation should not have y or z Start by getting rid of the x term Step → add -2 times the first to the 3rd Now try to eliminate the y term Step → add equations 2 and 3, replace 3 You could multiply equation 3 by ½ or just realize z must be 2.

  8. Obtain your answer by back-substituting z = 2 into the 2nd equation Then back-substituting y = -1 and z = 2 into the 1st equation Answer: (1, -1, 2) → a single point

  9. Answer Example (your turn)

  10. Partial Fractions Be able to break up a rational expression like Basic set up: Get rid of denominators:

  11. Set coefficient of like terms equal to each other: Solve the system:

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