Why does entry occur?

For a fixed element $a \in R$ define

$$C(a)=\{r \in R | r a=a r\}$$

. Prove that C(a) is a subring of R containing a. Prove that the center of R is the intersection of the subrings C(a) over all $a \in R$.

Determine whether the ordered pair $(3,4)$ is a solution of the system.

$$x+2 y=11$$

$2 x-y=2$

When is a correcting entry necessary?

Simplify the expression. Assume that all letters denote positive numbers.

$$\sqrt[3]{x^4}+\sqrt[3]{8 x}$$

Decide whether the given matrix is an elementary matrix or not.

$$\left[\begin{array}{ll}2 & 1 \\ 1 & 0\end{array}\right]$$

For each pair of points, find the slope of the line containing them. $(-4,-5)$ and $(-8,3)$

Determine the slope of the line that passes through the pair of points. $(9,-4),(7,-1)$

Use the clues to complete the crossword puzzle. The answers consists of vocabulary words.

Down 11. not causing any harm

Show that the function satisfies the equation.

$$\begin{matrix} \text{Function} & \text{Equation}\\ \ {y=\frac{10-\cos x}{x}} & \ {x y^{\prime}+y=\sin x}\\ \end{matrix}$$

In this exercise, find the equation of the line passing through the given pair of points.

$$(10,0),(1,9)$$

Find a matrix P that diagonalizes the matrix A, and determine $P^{-1} A P$.

$$A=\left[\begin{array}{rr}{1} & {0} \\ {6} & {-1}\end{array}\right]$$

.

Determine the inverse of the matrix or determine that the matrix is not invertible.

$$\left[\begin{array}{rr}2 & -6 \\ -3 & 9\end{array}\right]$$

In your own words, define the term. hibernation