7. Systems of Equations & Matrices

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

$\(\begin{cases}\)2x + 2y + 7z = -1 \\2x + y + 2z = 2 \\4x + 6y + z = 15\(\end{cases}\)$

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. D-A

Solve each system in Exercises 5–18. $\(\begin{cases}\)2x + y = 2 \\(\x\) + y - z = 4 \\3x + 2y + z = 0\(\end{cases}\)$

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.

(3/8)x - (1/2)y = 7/8

-6x + 8y = -14

Solve the system: (Hint: Let A = ln w, B = ln x, C = ln y, and D = ln z. Solve the system for A, B, C, and D. Then use the logarithmic equations to find w, x, y, and z.)

$\(\begin{cases}\)2 \(\ln\) w + \(\ln\) x + 3 \(\ln\) y - 2 \(\ln\) z = -6 \\4 \(\ln\) w + 3 \(\ln\) x + \(\ln\) y - \(\ln\) z = -2 \\\(\ln\) w + \(\ln\) x + \(\ln\) y + \(\ln\) z = -5 \\\(\ln\) w + \(\ln\) x - \(\ln\) y - \(\ln\) z = 5\(\end{cases}\)$

Write the partial fraction decomposition of each rational expression. x²+2x+7/x(x − 1)²

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

$\(\begin{cases}\)3w - 4x + y + z = 9 \\(\w\) + x - y - z = 0 \\2w + x + 4y - 2z = 3 \\-w + 2x + y - 3z = 3\(\end{cases}\)$

Solve each system in Exercises 5–18. $\(\begin{cases}\)x + y = -4 \\(\y\) - z = 1 \\2x + y + 3z = -21\(\end{cases}\)$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $\(\frac{x^3 + x^2}{(x^2 + 4)^2}\)$

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x³ + 4)/(9x³ - 4x)

In Exercises 9 - 16, find the following matrices: b. A - B

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 A = [1 2 3 4], B = 3 4

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.

6x - 3y - 4 = 0

3x + 6y - 7= 0

Solve each problem. See Examples 5 and 9. A cashier has a total of 30 bills, made up of ones, fives, and twenties. The number of twenties is 9 more than the number of ones. The total value of the money is \$351. How many of each denomination of bill are there?