Lesson 7.3 Multivariable Linear Systems Essential Question: How do you solve systems of linear equations in more than two variables?

What is a multivariable system? This is a system that contains more than 2 variables. 3𝑥+2𝑦−𝑧=8 𝑥+7𝑦−4𝑧=12 −3𝑥+𝑦+2𝑧=10

What does a solution look like? A solution of a system of three linear equations is an ordered triple (𝑥, 𝑦, 𝑧) whose coordinates make each equation true. The graph of this solution is the intersection of three planes.

How do you solve a system with multivariables? Substitution Elimination Augmented Matrices We are going to study the elimination method.

Elimination Method The goal of elimination with a multivariable system is to rewrite the system in a form to which back-substitution can be applied.

System of Linear Equations in Three Variables System with 3 Variables Equivalent System in Row-Echelon Form 𝑥−2𝑦+3𝑧=9 −𝑥+3𝑦+𝑧=−2 2𝑥−5𝑦+5𝑧=17 𝑥−2𝑦+3𝑧=9 𝑦+4𝑧=7 𝑧=2

What is row-echelon form? It has a “stair-step” pattern with leading coefficients of 1

Solve 𝑥−2𝑦+3𝑧=9 𝑦+4𝑧=7 𝑧=2

Solve 2𝑥−𝑦+5𝑧=22 𝑦+3𝑧=6 𝑧=3

How do you solve systems of linear equations in more than two variables? Choose one variable to eliminate. Pick 1 equation to use twice. Pair this equation with the other two. Eliminate the chosen variable from each new system. Now pair the new equations to form a new system. Solve this system for either variable left. Substitute your answers into the equations.

Solve 𝑥−2𝑦+3𝑧=9 −𝑥+3𝑦+𝑧=−2 2𝑥−5𝑦+5𝑧=17

Solve 𝑥+𝑦+𝑧=6 2𝑥−𝑦+𝑧=3 3𝑥+𝑦−𝑧=2

Solve 𝑥−3𝑦+𝑧=1 2𝑥−𝑦−2𝑧=2 𝑥+2𝑦−3𝑧=−1

Solve 𝑥+𝑦−3𝑧=−1 𝑦−𝑧=0 −𝑥+2𝑦=1

Gaussian Elimination You rewrite the system in row-echelon form to solve by using elementary row operations. You want the coefficients for each variable to be 1 when you create the equations. You’ll have one equation that is x, one that is y and one that is z as the leading term.

Elementary Row Operations Interchange two equations. Multiply one of the equations by a nonzero constant. Add a multiple of one equation to another equation.

Solve with Gaussian elimination 𝑥−2𝑦+3𝑧=9 −𝑥+3𝑦+𝑧=−2 2𝑥−5𝑦+5𝑧=17

Solve with Gaussian elimination 𝑥+𝑦+𝑧=6 2𝑥−𝑦+𝑧=3 3𝑥+𝑦−𝑧=2

Solve with Gaussian elimination 𝑥+2𝑦+𝑧=1 𝑥−2𝑦+3𝑧=−3 2𝑥+𝑦+𝑧=−1

Solve with Gaussian elimination 𝑥+𝑦−3𝑧=−1 𝑦−𝑧=0 −𝑥+2𝑦=1

Nonsquare Systems A system where the number of equations differs from the number of variables. This type of system cannot have a unique solution because it has less equations than variables.

Solve 𝑥−2𝑦+𝑧=2 2𝑥−𝑦−𝑧=1

Solve 𝑥−𝑦+4𝑧=3 4𝑥−𝑧=0

Partial Fraction Decomposition A rational expression can often be written as the sum or two or more simpler rational expressions. 𝑥+7 𝑥 2 −𝑥−6 = 2 𝑥−3 + −1 𝑥+2

Decomposition of 𝑁 𝑥 𝐷 𝑥 into Partial Fractions Divide if improper (degree of 𝑁 𝑥 ≥ degree of 𝐷 𝑥 ) and apply steps 2 – 4 to the proper rational expression. Factor denominator – completely factor into linear factors 𝑝𝑥+𝑞 𝑚 and quadratic factors 𝑎 𝑥 2 +𝑏𝑥+𝑐 𝑛 where they are irreducible over the reals. Linear factors Quadratic factors

Linear Factors For each factor of the form: 𝑝𝑥+𝑞 𝑚 the partial fraction decomposition must include the following sum of m fractions 𝐴 1 𝑝𝑥+𝑞 + 𝐴 2 𝑝𝑥+𝑞 2 +…+ 𝐴 𝑚 𝑝𝑥+𝑞 𝑚

Quadratic Factors For each factor of the form: 𝑎 𝑥 2 +𝑏𝑥+𝑐 𝑛 the partial fraction decomposition must include the following sum of n fractions. 𝐵 1 𝑥+ 𝐶 1 𝑎 𝑥 2 +𝑏𝑥+𝑐 + 𝐵 2 𝑥+ 𝐶 2 𝑎 𝑥 2 +𝑏𝑥+𝑐 2 +…+ 𝐵 𝑛 𝑥+ 𝐶 𝑛 𝑎 𝑥 2 +𝑏𝑥+𝑐 𝑛

Write the partial fraction decomposition of 𝑥+7 𝑥 2 −𝑥−6

Write the partial fraction decomposition of 𝑥+8 𝑥 2 +6𝑥+8

Write the partial fraction decomposition of 𝑥+11 𝑥 2 −2𝑥−15

Write the partial fraction decomposition of 5𝑥+7 𝑥 3 +2 𝑥 2 −𝑥−2

Write the partial fraction decomposition of 𝑥 2 +1 𝑥 𝑥−1 3

Write the partial fraction decomposition of 𝑥 2 +2𝑥+7 𝑥 𝑥−1 2

Write the partial fraction decomposition of 5 𝑥 2 +20𝑥+6 𝑥 3 +2 𝑥 2 +𝑥

Write the partial fraction decomposition of 3 𝑥 2 −𝑥+5 𝑥 3 −2 𝑥 2 +𝑥

Write the partial fraction decomposition of 𝑥 3 −4 𝑥 2 −19𝑥−35 𝑥 2 −7𝑥

How do you solve systems of linear equations in more than two variables?

Ticket Out the Door Solve: 2𝑥−5𝑦+3𝑧=−18 3𝑥+2𝑦−𝑧=−12 𝑥−3𝑦−4𝑧=−4