91Ó°ÊÓ

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use \(x, y ;\) or \(x, y, z ;\) or \(x_{1}, x_{2}, x_{3}, x_{4}\) as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. $$ \left[\begin{array}{llll|l} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right] $$

The sum of two numbers is \(16 .\) If the larger number is 4 less than 3 times the smaller number, find the two numbers.

Solve each system of equations using Cramer's Rule if is applicable. If Cramer's Rule is not applicable, write, "Not applicable. \(\left\\{\begin{array}{r}x+2 y-z=0 \\ 2 x-4 y+z=0 \\ -2 x+2 y-3 z=0\end{array}\right.\)

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{l} 3 x-6 y=7 \\ 5 x-2 y=5 \end{array}\right. $$

Graph each system of linear inequalities. State whether the graph is bounded or unbounded, and label the corner points. $$\left\\{\begin{array}{r}x \geq 0 \\\y \geq 0 \\\2 x+y \leq 6 \\\x+2 y \leq 6\end{array}\right.$$

Use properties of determinants to find the value of each determinant if it is known that \(\left|\begin{array}{lll}x & y & z \\ u & v & w \\ 1 & 2 & 3\end{array}\right|=4\). \(\left|\begin{array}{rrr}x & y & z \\ -3 & -6 & -9 \\ u & v & w\end{array}\right|\)

Mixed Practice In Problems \(51-58\), use the division algorithm to rewrite each improper rational expression as the sum of a polynomial and a proper rational expression. Find the partial fraction decomposition of the proper rational expression. Finally, express the improper rational expression as the sum of a polynomial and the partial fraction decomposition.. $$ \frac{x^{3}+x^{2}-3}{x^{2}+3 x-4} $$

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. $$ \left\\{\begin{array}{r} 2 x-3 y-z=0 \\ -x+2 y+z=5 \\ 3 x-4 y-z=1 \end{array}\right. $$

Use the division algorithm to rewrite each improper rational expression as the sum of a polynomial and a proper rational expression. Find the partial fraction decomposition of the proper rational expression. Finally, express the improper rational expression as the sum of a polynomial and the partial fraction decomposition. $$ \frac{x^{4}-5 x^{2}+x-4}{x^{2}+4 x+4} $$

Bill's Coffee House, a store that specializes in coffee, has available 75 pounds (lb) of \(A\) grade coffee and \(120 \mathrm{lb}\) of \(B\) grade coffee. These will be blended into 1-lb packages as follows: an economy blend that contains 4 ounces (oz) of \(A\) grade coffee and 12 oz of \(B\) grade coffee, and a superior blend that contains 8 oz of \(A\) grade coffee and 8 oz of \(B\) grade coffee. (a) Using \(x\) to denote the number of packages of the economy blend and \(y\) to denote the number of packages of the superior blend, write a system of linear inequalities that describes the possible numbers of packages of each kind of blend. (b) Graph the system and label the corner points.