Recognizing Interjections. Rewrite each of the following sentences, using an appropriate interjection in place of the feeling shown in parentheses. (Joy) We're all thrilled you came.

Solve the system of linear equations below by elimination. Check your solution.

2x - 5y = 8

3x + 5y = -13

In triangle $A B C, C=42^{\circ}, a=20$, and $b=12$. Find side $c$

Simplify by adding like terms:

$$x a-\frac{3 x^2 a^3}{x a^2}+\frac{2 x}{a^{-1}}$$

If gcd(a .30)=1. show that 60 divides $a^4+59$.

Show that

$$f ( x ) = \frac { x ^ { 2 } } { x ^ { 7 } + 1 }$$

takes on the value 0.4.

Approximate $f(x)$ to four decimal places.

$$f(x)=-2.1 e^{-0.71 x}, \quad x=1.9$$

A die is rolled and a spinner like the one shown is spun. Find the probability.

$P(2$ or 5 and $B$ or $D)$

If $(a, c) = 1$ and $(b, c) = 1$, prove that $(ab, c) = 1$.

Determine whether the statement is true and give an explanation or counterexample. If $f(x)=\sum_{k=0}^{\infty} c_k x^k$ on the interval $|x|<1$, then $f\left(x^2\right)=\sum c_k x^{2 k}$ on the interval $|x|<1$.

Determine whether the equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x.

$$y ^ { 4 } = x$$

Simplify by combining like terms. $4 t-(t+3 t)$

Use a calculator to approximate each pair of expressions to the nearest thousandth. Then state which property of logarithms this calculation illustrates.

$$\ln 12+\ln 5, \quad \ln 60$$

(a) Let $\mathbf{C}$ be an $n \times n$ matrix that satisfies $\mathbf{C}^2+\mathbf{C}=\mathbf{I}$. Show that $\mathbf{C}^{-1}=\mathbf{I}+\mathbf{C}$.

(b) Show that $\mathbf{C}^3=-\mathrm{I}+2 \mathrm{C}$ and $\mathrm{C}^4=2 \mathrm{I}-3 \mathrm{C}$.