방정식과 부등식 전체 문제

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

$\begin{array}{c}\frac{1}{2}x-\frac{1}{3}y=4 \\ \frac{3}{2}x-y=0\end{array}$

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

$\begin{array}{c} \\ -\frac{1}{2}x-4y=4 \\ 2x+16y=2\end{array}$

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

$\begin{array}{c}5x-y=1 \\ -10x+2y=-2\end{array}$

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

$\begin{array}{c}4x-6y-2z=\frac{1}{10} \\ x-7y+5z=-\frac{1}{4} \\ 3x+6y-9z=\frac{6}{5}\end{array}$

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

$\begin{array}{c}x+z=20 \\ x+y+z=20 \\ x+2y+z=10\end{array}$

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

$\begin{array}{c}5x-4y-3z=0 \\ 2x+y+2z=0 \\ x-6y-7z=0\end{array}$

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

$\begin{array}{c}{y}^{2}+{x}^{2}=25 \\ {y}^{2}-2{x}^{2}=1\end{array}$

For the following exercises, graph the following inequalities.

$y<{x}^{2}+9$

For the following exercises, graph the following inequalities.

$\begin{array}{c}{x}^{2}+{y}^{2}>4 \\ y<{x}^{2}+1\end{array}$

For the following exercises, write the partial fraction decomposition.

$\frac{-8x-30}{{x}^{2}+10x+25}$

For the following exercises, write the partial fraction decomposition.

$\frac{13x+2}{{(3x+1)}^{2}}$

For the following exercises, write the partial fraction decomposition.

$\frac{{x}^{4}-{x}^{3}+2x-1}{x{({x}^{2}+1)}^{2}}$

For the following exercises, perform the given matrix operations.

$5\left[\begin{array}{cc}4 & 9 \\ -2 & 3\end{array}\right]+\frac{1}{2}\left[\begin{array}{cc}-6 & 12 \\ 4 & -8\end{array}\right]$

For the following exercises, perform the given matrix operations.

$\left[\begin{array}{ccc}1 & 4 & -7 \\ -2 & 9 & 5 \\ 12 & 0 & -4\end{array}\right]\,\left[\begin{array}{cc}3 & -4 \\ 1 & 3 \\ 5 & 10\end{array}\right]$

For the following exercises, perform the given matrix operations.

${\left[\begin{array}{cc}\frac{1}{2} & \frac{1}{3} \\ \frac{1}{4} & \frac{1}{5}\end{array}\right]}^{-1}$

For the following exercises, perform the given matrix operations.

$\det |\begin{array}{cc}0 & 0 \\ 400 & 4,000\end{array}|$

For the following exercises, perform the given matrix operations.

$\det |\begin{array}{ccc}\frac{1}{2} & -\frac{1}{2} & 0 \\ -\frac{1}{2} & 0 & \frac{1}{2} \\ 0 & \frac{1}{2} & 0\end{array}|$

For the following exercises, perform the given matrix operations.

If $\det (A)=-6,$ what would be the determinant if you switched rows 1 and 3, multiplied the second row by 12, and took the inverse?

For the following exercises, perform the given matrix operations.

Rewrite the system of linear equations as an augmented matrix.

$$\begin{array}{c}14x-2y+13z=140 \\ -2x+3y-6z=-1 \\ x-5y+12z=11\end{array}$$

For the following exercises, perform the given matrix operations.

Rewrite the augmented matrix as a system of linear equations.

$$\left[\begin{array}{ccc}1 & 0 & 3 \\ -2 & 4 & 9 \\ -6 & 1 & 2\end{array}|\begin{array}{c}12 \\ -5 \\ 8\end{array}\right]$$