1. Multivariable LINEAR systems
Ch 7.3
Multivariable LINEAR systems

2. EX 1
Solve x – 2y + 3z = 9 y + 4z = 7 z = 2

4. Planes!

5. The 3D world

6. EX 2 (Using Gaussian Elimination)
Solve x -2y + 3z = 9 -x + 3y + z = -2 2x -5y + 5z = 17

7. Practice!

8. Ex 3 Solving a system with infinite solutions
x + y – 3z = -1 -x + 2y = 1 y – z = 0
Let z = a
Then back sub!

9. Solving non-square systems when given two equations
Step 3: Substitute z=a into the new equation “3” and solve for the remaining variable in terms of a.
Step 4: Substitute z = a and the answer from step 3 into one of the original equations to get your remaining variable in terms of a.
X – 2y + z = 2 2x – y – z = 1
Step 1: Let z = a
Step 2: Eliminate another variable other than z from two equations. Call this equation “3”.

10. You Try! X – 3y + 2z = 18 5x – 13y +12 z = 80 Step 1: Let z = a
Step 3: Substitute z=a into the new equation “3” and solve for the remaining variable in terms of a.
Step 4: Substitute z = a and the answer from step 3 into one of the original equations to get your remaining variable in terms of a.
X – 3y + 2z = 18
5x – 13y +12 z = 80
Step 1: Let z = a
Step 2: Eliminate another variable other than z from two equations. Call this equation “3”.
(-5a+3, -a-5,a)

11. “Warm Up”
Recall your knowledge of rational expressions. Add Add 2 𝑥−3 + −1 𝑥+2

12. Partial Fractions
A Rational Expression can be often written as the sum of two or more simpler rational expressions. For example, 𝑥+7 𝑥 2 −𝑥 −6 = 2 𝑥−3 + −1 𝑥+2 One of the most important applications of partial fractions is in calculus
“Partial Fractions”

13. EX 1” Write the partial fraction decomposition of 𝑥+7 𝑥 2 −𝑥 −6
Step 1: Factor the denominator to determine what each denominator of your partial fractions will be Step 2: Write the expression Step 4: Equate Coefficients and Solve Step 3: Multiply each side of eq. by LCD
(x -3)(x+2)
A + B = 1
2A – 3B = 7
A = 2 and back-sub to get B = -1
So answer is 2 𝑥−3 + −1 𝑥+2
𝑥+7 𝑥 2 −𝑥 −6 = 𝐴 𝑥−3 + 𝐵 𝑥+2
X + 7 = A (x+2) + B(x-3)
= Ax + 2A + Bx -3B
= (A + B)x + 2A – 3B

14. You Try! Write the partial fraction decomposition
a) 5 𝑥 2 +𝑥− b) 𝑥−2 𝑥 2 +4𝑥+3

15. EX 2: Partial Fraction Decomposition with Repeated Linear Factors
Write the partial fraction decomposition of 5 𝑥 2 +20𝑥+6 𝑥 3 +2 𝑥 2 +𝑥

16. You Try!
2𝑥−3 (𝑥−1)(𝑥−1)

17. Practice Test!