Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $\frac{1}{x^{2}+x}$

(a) write the system of equations as a matrix equation AX = B and (b) use Gauss-Jordan elimination on the augmented matrix $[A \vdots B]$ to solve for the matrix X. Use a graphing utility to check your solution.

$$\left\{\begin{aligned} 2 x_{1}+3 x_{2} &=5 \\ x_{1}+4 x_{2} &=10 \end{aligned}\right.$$

You and a friend agree to meet at your favorite fast-food restaurant between 5:00 and 6:00 P.M. The one who arrives first will wait 15 minutes for the other, after which the first person will leave. What is the probability that the two of you will actually meet, assuming that your arrival times are random within the hour?

Consider a person who invests in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 4.5% on AAA bonds, 5% on A bonds, and 7% on B bonds. The person invests twice as much in B bonds as in A bonds. Let x, y, and z represent the amounts (in dollars) invested in AAA, A, and B bonds, respectively.

$$\left\{\begin{aligned} x+y+& z &=(\text { total investment }) \\ 0.045 x+0.05 y+0.07 z &=(\text { annual return }) & \\ 2 y-& z &=0 \end{aligned}\right.$$

Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. Total Investment $500,000 Annual Return$28,000

Find the equation of the parabola $y=a x^{2}+b x+c$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. $(−1, 0) (1, 4), (2, 3)$

Solve the system by the method of elimination.

$$\left\{\begin{aligned} x-4 y-z &=3 \\ 2 x-5 y+z &=0 \\ 3 x-3 y+2 z &=-1 \end{aligned}\right.$$

In given exercise, the matrix product A X performs the translation of the point (x, y) to the point (x+h, y+k), when

$$A=\left[\begin{array}{lll} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{array}\right] \text { and } X=\left[\begin{array}{c} x \\ y \\ 1\end{array}\right]$$

Find $A^{-1} B$. What does $A^{-1} B$ represent?

$$A=\left[\begin{array}{rrr}1 & 0 & -2 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{array}\right], \quad X=\left[\begin{array}{r}1 \\ -2 \\ 1\end{array}\right]$$

Solve for x.

$$\left|\begin{array}{rrr}{1} & {2} & {x} \\ {-1} & {3} & {2} \\ {3} & {-2} & {1}\end{array}\right|=0$$

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $\frac{x+4}{x^{2}(3 x-1)^{2}}$

Show that B is the inverse of A .

$$A=\left[\begin{array}{rr} -4 & 3 \\ 3 & -2 \end{array}\right], \quad B=\left[\begin{array}{ll} 2 & 3 \\ 3 & 4 \end{array}\right]$$

In given exercise, the matrix product A X performs the translation of the point (x, y) to the point (x+h, y+k), when

$$A=\left[\begin{array}{lll} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{array}\right] \text { and } X=\left[\begin{array}{c} x \\ y \\ 1\end{array}\right]$$

Find $A^{-1}$ .

$$A=\left[\begin{array}{rrr}1 & 0 & -2 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{array}\right], \quad X=\left[\begin{array}{r}1 \\ -2 \\ 1\end{array}\right]$$

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations.

$$\left\{\begin{aligned} 2 x+5 y +w=11 \\ x+4 y+2 z-2 w =-7 \\ 2 x-2 y+5 z+ w=3 \\ x -3 w=-1 \end{aligned}\right.$$

Solve the system graphically or algebraically. Explain your choice of method.

$$\left\{\begin{array}{l} 3 x-7 y=-6 \\ x^{2}-y^{2}=4 \end{array}\right.$$

In given exercise, the matrix product A X performs the translation of the point (x, y) to the point (x+h, y+k), when

$$A=\left[\begin{array}{lll} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{array}\right] \text { and } X=\left[\begin{array}{c} x \\ y \\ 1\end{array}\right]$$

Find B=A X .

$$A=\left[\begin{array}{rrr}1 & 0 & -2 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{array}\right], \quad X=\left[\begin{array}{r}1 \\ -2 \\ 1\end{array}\right]$$