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Lecture #9:Introduction to System of Linear Equations

Lecture #9:Introduction to System of Linear Equations. Lecture #7: Particle Assumption Modeling of Problem Free Body Diagram: FBD Newton’s Law of Motion Lecture #8: Linear Equations: y=mx +b Solution System: N.S., U.S., I.S. Augmented Matrix Solving a System of Linear Equations.

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Lecture #9:Introduction to System of Linear Equations

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  1. Lecture #9:Introduction to System of Linear Equations • Lecture #7: • Particle Assumption • Modeling of Problem • Free Body Diagram:FBD • Newton’s Law of Motion • Lecture #8: • Linear Equations: y=mx +b • Solution System: N.S., U.S., I.S. • Augmented Matrix • Solving a System of Linear Equations

  2. Example Problem • Problem 3.12: A body with a mass of 300kg is supported by flexible cable system. Determine TA, TB, TC, TD, TE. A B C D E

  3. Maple Solution of Linear Equations • Suppose we would like to find x & y for 1x + 2y = 10 2x – 3y = 21 • Lets use MAPLE commands > eq2:= 2*x – 3*y – 21; > eq1:= 1*x + 2*y-10; > sol:= solve({eq1,eq2},{x,y}); > soln:=evalf(sol);

  4. Linear Equations • Y=mx + b : (linear) • A more general form: a1x + a2y = b • a1x1 + a2x2 + a3x3+……..anxn = b y x

  5. Linear Equations: 4x + 2y =1 3x1 – 2x2 + 5x3 = 9 7x + 8y –9z-10w=20 3x1-x3+x4=sin(Pi/2) 7xy +6xz = 2x Non-linear Equations 4x2 + 2y = 9 3x1 – 2x2x3 + 5x3 = 9 7x + 8y –9z-10/w=20 3x1-x3+x4=sin(x1) 7xy +6z = 2x Examples Linear and Non-linear

  6. Solutions No-solution, unique sol. and infinite sol. y y y1 & y2 y y1 y2 x x x y1 y2

  7. System of linear equations • The general form: A11x1+A12x2+A13x3+…..A1nxn=B1 Rx=0 A21x1+A22x2+A23x3+…..A2nxn=B2 Ry=0 A31x1+A32x2+A33x3+…..A3nxn=B3 Rz=0 . . . . . . . . . . Am1x1+Am2x2+Am3x3+…Amnxn=Bm

  8. Matrix Form: • General Matrix Form: The Coefficients ROW # Column #

  9. System of linear eqns. 1x + y + 2z = 9 2x + 4y – 3z = 1 3x + 6y –5z = 0 Remember: Rx=0 Ry=0 Rz=0 Augmented Matrix: (array of numbers of the system of eqns) Augmented Matrix:

  10. Activity:#1 • For Problem in example #1 write down the augmented matrix. • FBD at CDE: • TD = 1699 N • TC = 3398 N • FBD at ABC: • -TA (0.766) + TB (0.866) = 1699 • TA (0.643) + TB (0.500) = 2943

  11. Activity:#2 • Also, for problem 3.22 write down the augmented matrix.

  12. Solving a System of Linear Eqns. • GOAL • FIND the solution for x, y,z (TA, TB, TC, TD, TE) • The idea is to replace a given system by a system which has the same solution set, but it is easier to solve.

  13. Basic Operations to find Unknown • Multiply a row by a nonzero constant. (the row you multiply by a number after adding the two rows will not change) • Interchange two rows. • Add a multiple of one row to another row.

  14. Gauss-Jordan Elimination • Goal: to reduce the augmented matrix into a form simple enough such that system of equation are solved by inspection.

  15. Reduced row-echelom form • If row does not consist entirely of zeros, then the first non-zero number in row is 1. • If a row consist of zeros, then they are moved to the bottom of matrix. • In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occur farther to the right of above row. • Each column that contains a leading 1 has zero everywhere.

  16. Reduced Row echelom Must have zeros above and below each leading 1. Row-echelom form Must have zeros below each leading 1. IMPORTANT

  17. Gauss-Jordan Elimination Method • Step1: Locate the leftmost column that does not consist entirely of zero. • Step 2: Interchange the top row with another row, if necessary, to bring a nonzero entry to the top from step 1. • Step 3: If the entry that is now at the top is a constant, divide entire row by it. • Step 4: Add multiples to top row to the rows below such that all entries have 1 as leading term. • Step 5: Cover top row and begin with step 1 applied to submatrix.

  18. Activity:#3 • For Problem in example #1 solve using Gauss-Jordan Method. • FBD at CDE: • TD = 1699 N • TC = 3398 N • FBD at ABC: • -TA (0.766) + TB (0.866) = 1699 • TA (0.643) + TB (0.500) = 2943

  19. Example: Find x, y, z for this system of linear equations: 1x + y + 2z = 9 2x + 4y – 3z = 1 3x + 6y –5z = 0

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