Multivariable Linear Systems Skill 21

Objectives Use Back-Substitution to solve systems in Row-Echelon Form Use Gaussian Elimination to solve systems of linear equations Solve non-square systems of linear equations Find partial fraction decomposition

Row-Echelon A form to put systems in so you can quickly use substitution to solve. Modify the system until the first equation has three terms. Modify the system until the second equation has two terms. Modify the system until the third equation has one term. Substitute backwards one at a time.

Example; Solve Row-Echelon Form 2𝑥−𝑦+3𝑧=17 −7𝑦−4𝑧=−9 𝑧=4 −7𝑦−4 𝑧 =−9 2𝑥−𝑦+3𝑧=17 −7𝑦−4 4 =−9 2𝑥− −1 +3 4 =17 −7𝑦=7 2𝑥=4 𝒚=−𝟏 𝒙=𝟐 𝟐,−𝟏,𝟒

Example; Solve Row-Echelon Form −3𝑥+2𝑦−4𝑧=15 5𝑦−2𝑧=27 𝑧=−6 5𝑦−2 𝑧 =27 −3𝑥+2𝑦−4𝑧=15 5𝑦−2 −6 =27 −3𝑥− 3 +3 −6 =15 5𝑦=15 −3𝑥=−15 𝒚=𝟑 𝒙=𝟓 𝟓,𝟑,−𝟔

Gaussian Elimination Elementary Row Operations for Systems of Equations 1) Interchange two equations. 2) Multiply one equation by a non-zero constant. 3) Add a multiple of one equation to another equation.

Example; Using Gaussian Elimination 𝑥−2𝑦+3𝑧=9 𝑦+4𝑧=7 2𝑥−5𝑦+5𝑧=17 𝑹 𝟏 + 𝑹 𝟐 𝑥−2𝑦+3𝑧=9 𝑦+4𝑧=7 −𝑦−𝑧=−1 𝒙−𝟐𝒚+𝟑𝒛=𝟗 −𝒙+𝟑𝒚+𝒛=−𝟐 𝟐𝒙−𝟓𝒚+𝟓𝒛=𝟏𝟕 𝟐 𝑹 𝟏 + 𝑹 𝟑 𝑥−2𝑦+3𝑧=9 𝑦+4𝑧=7 3𝑧=6 𝑹 𝟐 + 𝑹 𝟑 𝒙−𝟐𝒚+𝟑𝒛=𝟗 𝒚+𝟒𝒛=𝟕 𝒛=𝟐 𝟏 𝟑 𝑹 𝟑

Example; Using Gaussian Elimination, Continued 𝑥−2𝑦+3𝑧=9 𝑦+4𝑧=7 𝑧=2 𝑦+4 𝑧 =7 𝑥−2𝑦+3𝑧=9 𝑦+4 2 =7 𝑥−2 −1 +3 2 =9 𝑦+8=7 𝑥+2+6=9 𝒙=𝟏 𝒚=−𝟏 𝟏,−𝟏,𝟐

Example; Using Gaussian Elimination 𝑥+𝑦+𝑧=6 −3𝑦−𝑧=−9 −2𝑦−4𝑧=−16 −𝟐 𝑹 𝟏 + 𝑹 𝟐 −𝟑 𝑹 𝟏 + 𝑹 𝟑 𝑥+𝑦+𝑧=6 𝑦+2𝑧=8 −3𝑦−𝑧=−9 − 𝟏 𝟐 𝑹 𝟑 ↔ 𝑹 𝟐 𝒙+𝒚+𝒛=𝟔 𝟐𝒙−𝒚+𝒛=𝟑 𝟑𝒙+𝒚−𝒛=𝟐 𝑥+𝑦+𝑧=6 𝑦+2𝑧=8 5𝑧=15 𝟑 𝑹 𝟐 + 𝑹 𝟑 𝒙+𝒚+𝒛=𝟔 𝒚+𝟐𝒛=𝟖 𝒛=𝟑 𝟏 𝟓 𝑹 𝟑

Example; Using Gaussian Elimination, Continued 𝑥+𝑦+𝑧=6 𝑦+2𝑧=8 𝑧=3 𝑦+2 𝑧 =8 𝑥+𝑦+𝑧=6 𝑦+2 3 =8 𝑥+ 2 + 3 =6 𝑦+6=8 𝑥+5=6 𝒙=𝟏 𝒚=𝟐 𝟏,𝟐,𝟑

Example; Systems with fewer equations than variables 𝑥−2𝑦+𝑧=2 3𝑦−3𝑧=−3 −𝟐 𝑹 𝟏 + 𝑹 𝟐 𝒙−𝟐𝒚+𝒛=𝟐 𝟐𝒙−𝒚−𝒛=𝟏 𝑥−2𝑦+𝑧=2 𝑦−𝑧=−1 𝟏 𝟑 𝑹 𝟐 𝑥−2𝑦+𝑧=2 𝑦=𝑧−1 𝑥−2𝑦+𝑧=2 𝑥−2 𝑧−1 +𝑧=2 Let 𝒛=𝒂 𝑥−2𝑧+2+𝑧=2 So, 𝒙=𝒂 𝒙=𝒛 and, 𝒚=𝒂−𝟏 𝒂,𝒂−𝟏,𝒂

Example; Systems with fewer equations than variables 𝒙−𝒚+𝟒𝒛=𝟑 𝟒𝒙−𝒛=𝟎 Let 𝒙=𝒂 𝑥−𝑦+4𝑧=2 𝑧=4𝑥 So, 𝒛=𝟒𝒂 and, 𝒚=𝟓𝒂−𝟑 𝑥−𝑦+4𝑧=3 𝑥−𝑦+4𝑥=3 5𝑥−𝑦=3 𝒂,𝟓𝒂−𝟑,𝟒𝒂 𝒚=𝟓𝒙−𝟑

Partial Fraction Decomposition 1) Divide if improper 2) Factor Denominator 3) Linear Factors 4) Quadratic Factors

Example; Partial Fraction Decomposition 𝑥+7 𝑥 2 −𝑥−6 = 𝐴 𝑥−3 + 𝐵 𝑥+2 𝑥+7=𝐴 𝑥+2 +𝐵 𝑥−3 𝑥+7=𝐴𝑥+2𝐴+𝐵𝑥−3𝐵 1=𝐴+𝐵 7=2𝐴−3𝐵 −𝟐 1=𝐴+𝐵 −2=−2𝐴−2𝐵 7=2𝐴−3𝐵 1=𝐴−1 + 𝑨=𝟐 𝟐 𝒙−𝟑 − 𝟏 𝒙+𝟐 5=−5𝐵 𝑩=−𝟏

Example; Partial Fraction Decomposition 𝑥+8 𝑥 2 +6𝑥+8 = 𝐴 𝑥+4 + 𝐵 𝑥+2 𝑥+8=𝐴 𝑥+2 +𝐵 𝑥+4 𝑥+8=𝐴𝑥+2𝐴+𝐵𝑥+4𝐵 1=𝐴+𝐵 8=2𝐴+4𝐵 −𝟐 1=𝐴+𝐵 −2=−2𝐴−2𝐵 8=2𝐴+4𝐵 1=𝐴+3 + 𝑨=−𝟐 6=2𝐵 −𝟐 𝒙+𝟒 + 𝟑 𝒙+𝟐 𝑩=𝟑

21: Multivariable Linear Systems Summarize your notes Questions? Homework Worksheet Quiz