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Understanding Systems of Equations: RREF Methodology

Learn to solve systems of equations using Reduced Row Echelon Form (RREF) and Inverse Multiplication techniques. Discover types of solutions and avoid errors. Access matrix setup guidelines and practical examples.

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Understanding Systems of Equations: RREF Methodology

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Presentation Transcript

  1. Warm-Up • Solve the following system of equations:

  2. Reduced Row Echelon Form (RREF)

  3. Learning Targets • Possible solutions for a system • The differences between RREF and Inverse Multiplication • Using Reduced Row Echelon Form to solve systems

  4. Quick Recap • In order to setup a matrix our data must be: • Setup in the Standard Form • All similar variables must be in the same order • Any missing variables in the equations must be represented with a zero • Our matrix system using inverse multiplication has three matrices: • Coefficient, Variable and Constant

  5. Recap Cont. • Matrices cannot divide one another • Multiplying the coefficient matrix by its inverse isolates the variable matrix • Multiplying the constant matrix by our inverse as well will solve for the variables

  6. How many types of solutions can we have? • Pause and Ponder, SILENTLY! • One Solution • No Solution • Infinite Solutions

  7. Example Solve using Inverse Multiplication: What Does it Mean?!?

  8. Example Solve using Inverse Multiplication: AGAIN!?!

  9. Problems with Inverse Multiplication • Inverse multiplication only produces a solution when there is only one solution. • If we have no solution or infinite solutions then we will get an ERROR • So how do we know if it is infinite solutions or no solution?

  10. RREF • Reduced Row Echelon Form • This form allows us to consolidate the coefficient and constant matrices into one matrix • We can then perform row operations that will clearly state the exact answer

  11. How to set it up: Old Way: Coefficient Variable Constant

  12. How to set it up: ***Notice there is no more variable matrix and we add the constant as an additional column. RREF Coefficient Constant

  13. Now to perform the math… • In our calculators we can go under the matrix screen and select RREF • We can then choose the matrix to perform this operation on.

  14. Solution • The solution will be in a matrix that is the same dimensions. • We can then read the results as variables and their solutions. X-Variable Y-Variable Z-Variable

  15. No Solution: Enter as a 3X4 matrix Last row: 0 0 0 1 No Solution.

  16. Infinite Solutions: Enter as a 3X4 matrix Last row: 0 0 0 0 Infinitely Solutions

  17. You Try using RREF:

  18. Website to Visualize the Solutions • http://www.cpm.org/flash/technology/3dsystems.swf

  19. For the weekend… • Worksheet

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