Q: Write the augmented matrix for the system: $$6x + 6y + 2z = 34$$, $$4x + 6y + 6z = 20$$, $$2x + 8y - 2z = 20.$$

$$\begin{bmatrix} 6 & 6 & 2 & 34 \\ 4 & 6 & 6 & 20 \\ 2 & 8 & -2 & 20 \end{bmatrix}$$

Question 1

Expand the logarithmic expression $$\log_3$ (7 \cdot 11)$$ using properties of logarithms.

Accepted Answer

$$\log_3$ 7 + \log_3$ 11$$

Question 2

Condense the expression $$\ln x + 9 \ln y$$ into a single logarithm.

Accepted Answer

$$\ln (xy^9$)$$

Question 3

Solve the exponential equation $$4^x = 256 by $$expressing both sides as powers of the same base.

Accepted Answer

$$x = 4$$

Question 4

Solve for $$x in $$the logarithmic equation $$\log_6$ (x - 2) = 3.$$

Accepted Answer

$$x = 218$$

Question 5

A student needs $14,000 in 5 years for college. If the money is invested at 11% compounded continuously, how much should be invested now? (Round to the nearest dollar.)

Accepted Answer

$$8,077$$

Question 6

The population of a country was 21 million in 1983 and 29 million in 1992. Using the exponential growth function $$A = 21e$$^{kt}$$, what is the value of k$ to three decimal places?

Accepted Answer

$$k = 0.036$$

Question 7

The half-life of silicon-32 is 710 years. If 100 grams is present now, how much will be present in 800 years? (Round to three decimal places.)

Accepted Answer

$$45.794$$	ext{ grams}$

Question 8

Solve the system of equations by the addition method: $$6x + 6y = 6$$ and $$-2x + 4y = -8. $$What is the solution?

Accepted Answer

$$(2, -1)$$

Question 9

Solve the exponential equation $$e^{5x} = 2$. Express the solution in terms of natural logarithms.

Accepted Answer

$$x = \frac{\ln 2}{5}$$

Question 10

Solve the logarithmic equation $$\log_2$ (x - 4) + \log_2$ (x - 10) = 4.$$

Accepted Answer

$$x = 12$$

Question 11

An endangered fish population decreases exponentially: $$A = A_0$ e$$^{kt}$$. Seven years ago, there were 1,500 fish; today, there are 800. When will the population drop below 100?

Accepted Answer

23 years from today

Question 12

Solve the system by substitution: $$5x - 2y = -162$$, $$x - 4y = 0. $$What is the solution?

Accepted Answer

$$(-36, -9)$$

Question 13

Write the augmented matrix for the system: $$6x + 6y + 2z = 34$$, $$4x + 6y + 6z = 20$$, $$2x + 8y - 2z = 20.$$

Accepted Answer

$$\begin{bmatrix} 6 & 6 & 2 & 34 \ 4 & 6 & 6 & 20 \ 2 & 8 & -2 & 20 \end{bmatrix}$$

Question 14

Solve the system using matrices (Gauss-Jordan elimination): $$-4x - y + 2z = -18$$, $$2x + 4y + 5z = 91$$, $$-4x - 2y + z = -35. $$What is the solution?

Accepted Answer

$$(7, 8, 9)$$

Question 15

Evaluate $$\log_{10} 19 to $$four decimal places.

Accepted Answer

$$1.2788$$

Question 16

Solve for $$x in$$ $$\log_4$ (x + 1) - \log_4$ (x - 4) = 3.$$

Accepted Answer

$$x = 257/63$$

Question 17

Solve for $$x in$$ $$\log (x + 2) = \log (5x - 3).$$

Accepted Answer

$$x = \frac{5}{4}$$

Question 18

Larry has $$2,700 to $$invest and needs $$3,200 in$$ $$14$$ years. What annual rate of return is needed if interest is compounded continuously? (Round to two decimals.)

Accepted Answer

$$1.21\%$$

Question 19

Solve for $$x in$$ $$\log_5$ x + \log_5$ (x - 24) = 2.$$

Accepted Answer

$$x = 25$$

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