Simultneous Liner Equtions Simultneous Liner Equtions nd Liner Algebr Simultneous liner equtions (SLE s) occur frequently in Sttics, Dynmics, Circuits nd other engineering clsses Need to be ble to, nd SLE s Note tht methods for solving SLE s do not pply to which re covered in lter clsses in ME 349 Are these Simultneous Liner Equtions? A Y A X T sin θ 0 W T cos θ 0 ( 10 ft) T sinθ ( 6 ft) W 0 Wht is Liner Algebr? Liner Algebr is the methods used to solve nd nlyze. Theory of mtrices nd determinnts rise from this effort.
Liner Eqution A is n eqution of the form n x n + n-1 x n-1 +... + 1 x 1 b. Here i nd b re. A to liner eqution is n ssignment of vlues to the vribles (x i s) tht mke the eqution. System of Liner Equtions A is simply set of liner equtions. i.e. 11 x 1 + 12 x 2 +... + 1n x n b 1 21 x 1 + 22 x 2 +... + 2n x n b 2... m1 x 1 + m2 x 2 +... + mn x n b m where coefficients ij re. How do you know you hve SLE? Must be ble to write the equtions in mtrix formt: 11 21 m1 21 22 m2. 1 n x1 b1 2n x2 b2 mn xn bm System of Liner Equtions A solution to system of equtions is simply n ssignment of vlues to the vribles tht stisfies (is solution to) in the system. If system of equtions hs solution we sy it is consistent. could hve If system does not hve we sy tht it is inconsistent.
Consistent or inconsistent? Consistent or inconsistent? x + 3y 4 x + 3y 4 2x y 1 2x + 6y 8 3x + 2y 6 3x + 2y 12 How Mny Solutions Could System Hve? As we sw system of liner equtions cn hve solutions, solution or solutions. Exmple 2 Solution of liner equtions Consider two liner equtions However this is difficult to visulize in more thn three vribles.
Exmple 2 (cont d) Liner Independent Equtions Options for lines Prllel - No solution - Coincident -, solutions Equtions Intersect -, one unique solution An eqution in set of equtions is if it cnnot be generted by ny combintion of the other equtions. In order to hve unique solution the number of equtions must be to the number of unknowns. Equtions Liner Dependent Equtions If n eqution in set of equtions cn be generted by liner combintion of the other equtions then it is clled eqution. Dependent or Independent? Exmples:
How to Find Solutions? Solving Systems Algebriclly Liner Equtions - Methods of Solution: Substitution (or bck-substitution) Steps: Pick one of the equtions nd solve for one of the vribles in terms of others Substitute into remining equtions Repet until equtions reduced to one unknown Solve for vribles Bck substitute to solve for other vribles Difficulty with substitution? More thn two vribles Frctions or non-integer coefficients Consider the following system 2 x + 3 y z 5 y z 1 z 3 Wht re its solution(s)? Exmple #3 Additionl Resources Consistent or inconsistent? Dependent or independent? Find solution(s) by substitution. http://numericlmethods.eng.usf.edu/topics/primer_sle.html see A Textbook Chpter on System of Equtions http://en.wikibooks.org/wiki/liner_algebr/introduction An entire college-level textbook on liner lgebr vilble from this site Covers everything we will cover in ME 349 plus lot more much of this mteril is covered in grdute level clsses like GES 551 nd ME 577 We will not do forml proofs see website for detils