1. Multivariable Linear Systems
Skill 21

2. 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

3. 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.

4. 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− − =17
−7𝑦=7
2𝑥=4
𝒚=−𝟏
𝒙=𝟐
𝟐,−𝟏,𝟒

5. 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

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.

7. 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

8. 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 − =9
𝑦+8=7
𝑥+2+6=9
𝒙=𝟏
𝒚=−𝟏
𝟏,−𝟏,𝟐

9. 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

10. 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 =6
𝑦+6=8
𝑥+5=6
𝒙=𝟏
𝒚=𝟐
𝟏,𝟐,𝟑

11. 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

12. 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

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

14. 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

15. 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

16. 21: Multivariable Linear Systems
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