7. Systems of Equations & Matrices

Write the partial fraction decomposition of each rational expression. (x³-4x²+9x-5)/(x² -2x+3)²

Write the partial fraction decomposition of each rational expression. (7x-4)/(x²-x-12)

In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\(\begin{cases}\)w + 2x + 3y - z = 7 \\2x - 3y + z = 4 \\(\w\) - 4x + y = 3\(\end{cases}\)$

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B

Solve each problem. See Examples 5 and 9. Solve the system of equations (4), (5), and (6) from Example 9.

$25x+40y+20z=2200$ (4)

$4x+2y+3z=280$ (5)

$3x+2y+z=180$ (6)

In Exercises 16–24, write the partial fraction decomposition of each rational expression. x/(x - 3)(x + 2)

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (3x - 2)/((x + 4)(3x² + 1))

Determine if the given ordered triple is a solution of the system. $(4,1,2)$

$\(\begin{cases}\)x - 2y = 2 \\2x + 3y = 11 \\(\y\) - 4z = -7\(\end{cases}\)$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

x + y - 5z = -18

3x - 3y + z = 6

x + 3y - 2z = -13

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

2x - y = 6

4x - 2y = 0

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\(\begin{cases}\)x + y - z = -2 \\2x - y + z = 5 \\-x + 2y + 2z = 1\(\end{cases}\)$

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

A(BC)

Write the partial fraction decomposition of each rational expression. 5x²+6x+3/(x + 1)(x² + 2x + 2)

Solve each system, using the method indicated.

x - z = -3

y + z = 6

2x - 3z = -9 (Gauss-Jordan)