Stewart Precalc 6e Chapter 10 Test

A system of equations is given. (a) Determine whether the system is linear or nonlinear. \(\begin{cases} x + 3 y = 7 \\ 5 x + 2 y = -4 \end{cases}\)

A system of equations is given. (a) Determine whether the system is linear or nonlinear. \(\begin{cases} 6 x + y^2 = 10 \\ 3 x - y = 5 \end{cases}\)

Use a graphing device to find all solutions of the system rounded to two decimal places. \(\begin{cases} x - 2 y = 1 \\ y = x^3 - 2 x^2 \end{cases}\)

In 2.5 hours an airplane travels 600 km against the wind. It takes 50 min to travel 300 km with the wind. Find the speed of the wind and the speed of the airplane in still air.

Determine whether each matrix is in reduced row-echelon form, row-echelon form, or neither. (a) \(\begin{pmatrix} 1 & 2 & 4 & -6 \\ 0 & 1 & -3 & 0 \end{pmatrix}\) (b) \(\begin{pmatrix} 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & 3 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}\) (c) \(\begin{pmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 3 \end{pmatrix}\)

Use Gaussian elimination to find the complete solution of the system, or show that no solution exists. (a) \(\begin{cases} x - y + 2 z = 0 \\ 2 x - 4 y + 5 z = -5 \\ 2 y - 3 z = 5 \end{cases}\) (b) \(\begin{cases} 2 x - 3 y + z = 3 \\ x + 2 y + 2 z = -1 \\ 4 x + y + 5 z = 4 \end{cases}\)

Use Gauss-Jordan elimination to find the complete solution of the system. \(\begin{cases} x + 3 y - z = 0 \\ 3 x + 4 y - 2 z = -1 \\ -x + 2 y = 1 \end{cases}\)

Anne, Barry, and Cathy enter a coffee shop. Anne orders two coffees, one juice, and two doughnuts and pays \$6.25. Barry orders one coffee and three doughnuts and pays \$3.75. Cathy orders three coffees, one juice, and four doughnuts and pays \$9.25. Find the price of coffee, juice, and doughnuts at this coffee shop.

Let \(A = \begin{pmatrix} 2 & 3 \\ 2 & 4 \end{pmatrix}\), \(B = \begin{pmatrix} 2 & 4 \\ -1 & 1 \\ 3 & 0 \end{pmatrix}\), and \(C = \begin{pmatrix} 1 & 0 & 4 \\ -1 & 1 & 2 \\ 0 & 1 & 3 \end{pmatrix}\). Carry out the indicated operation, or explain why it cannot be performed. (a) \(A + B\) (b) \(A B\) (c) \(B A - 3 B\) (d) \(C B A\) (e) \(A^{-1}\) (f) \(B^{-1}\) (g) \(\det(B)\) (h) \(\det(C)\)

(a) Write a matrix equation equivalent to the following system. \(\begin{cases} 4 x - 3 y = 10 \\ 3 x - 2 y = 30 \end{cases}\) (b) Find the inverse of the coefficient matrix, and use it to solve the system.

Only one of the following matrices has an inverse. Find the determinant of each matrix, and use the determinants to identify the one that has an inverse. Then find the inverse. \(A = \begin{pmatrix} 1 & 4 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 4 & 0 \\ 0 & 2 & 0 \\ -3 & 0 & 1 \end{pmatrix}\)

Solve using Cramer's Rule: \(\begin{cases} 2 x - z = 14 \\ 3 x - y + 5 z = 0 \\ 4 x + 2 y + 3 z = -2 \end{cases}\)

Find the partial fraction decomposition of the rational function. (a) \(\dfrac{4 x - 1}{(x - 1)^2 (x + 2)}\) (b) \(\dfrac{2 x - 3}{x^3 + 3 x}\)

Graph the solution set of the system of inequalities. Label the vertices with their coordinates. (a) \(\begin{cases} 2 x + y \leq 8 \\ x - y \geq -2 \\ x + 2 y \geq 4 \end{cases}\) (b) \(\begin{cases} x^2 + y \leq 5 \\ y \leq 2 x + 5 \end{cases}\)