I have a few questions I need help with. They are to help review for our exam.

4) Without graphing, is each system independent, dependent, or inconsistent?
-x-3y=0
x-y=4
(i believe dependent)
dependent
independent
inconsistent

6) Which matrix represents the system of equations below?
-2x-7y=3
7x-6y=12
(not sure)
-2 -7 3
7 -6 12

-2 -7 -3
7 -6 12

7 -6 12
-2 -7 3

-2 -7 -3
7 -6 12

7) Which steps translate the graph of y=x^2 to y=-(x-3)^3+2
(my choice translate 3 unite to right translate up 2 units)

translate 3 units to the right, translate up 2 units

reflect across the x-axis, translate 3 units to the left, translate down 2 units

translate 3 units to the left, translate down 2 units

reflect across the x-axis, translate 3 units to the right, translate up 2 units

15) The half-life of a certain radioactive material is 68 hours. An initial amount of the material has a mass of 641kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 4 hours. Round your answer to the nearest thousandth.

y=641(1/2)^68x;0kg

y=1/2(1/641)^1/68x;0.342kg

y=641(1/2)^1/68x;615.390kg

y=2(1/641)^1/68x;1.367kg

18) Name the property of real numbers illustrated by the equation

5*(sqrt 9*4)=(5*sqrt9)*4
(my choice commutative property of multiplication)

commutative property of multiplication
associative property of multiplication
distributive property
associative property of addition

20) The table shows the median home prices in Florida. What is the equation of a trend line that models a relationship between time and home prices? Use the equation to predict the median home price in 2020.

year 1940 1950 1960 1970 1980 2000
median price($) 20,600 38,300 54,000 60,500 85,300 100,300 111,100
(my choice y=1508.33x+20,600;$141,300)

y=45.25x+20,600;111,100
y=1508.33x+20,600;141,300
y=1508.33+20,600;111,100
y=45.25x+20,600;141,300

I know this may seem like alot but it is review, so if someone might beable to help me out so I can be ready to take exam. Thank you in advance

4) already did this:

independent
dependent if the lines are parallel

6)
look at the coefficients!
1st set matches

7)
already did this:
reflect across the x-axis, translate 3 units to the right, translate up 2 units

15)
think about it: every 68 hours it divides in half. So, the exponential will be (1/2)^(x/68)

y=641(1/2)^1/68x;615.390kg

18)
associative
You have a*(b*c) = (a*b)*c

commutative would be a*(b*c) = a*(c*b) or (b*c)*a

20)
If x represents the years since 1940, then I agree.

Thank you Steve this helps alot studying for exam

4) Without graphing, the system -x-3y=0 and x-y=4 is consistent and dependent. Both equations represent the same line, so they have infinitely many solutions.

6) The matrix that represents the system of equations -2x-7y=3 and 7x-6y=12 is:
-2 -7 3
7 -6 12

7) The correct steps to translate the graph of y=x^2 to y=-(x-3)^3+2 are to translate 3 units to the right and translate up 2 units.

15) The exponential function that models the decay of this material is: y=641(1/2)^(x/68). To find how much radioactive material remains after 4 hours, substitute x=4 into the equation: y=641(1/2)^(4/68) = 615.390kg (rounded to the nearest thousandth).

18) The property of real numbers illustrated by the equation 5*(sqrt 9*4)=(5*sqrt9)*4 is the commutative property of multiplication.

20) The equation of a trend line that models the relationship between time (years) and home prices is y=1508.33x+20,600. To predict the median home price in 2020, substitute x=2020 into the equation: y=1508.33(2020)+20,600 = $3,045,226.60 (rounded to the nearest dollar).

4) The given system of equations is:

-x - 3y = 0
x - y = 4

To determine if the system is independent, dependent, or inconsistent, we can check the coefficients of x and y in both equations.

For the first equation, the coefficient of x is -1 and the coefficient of y is -3.
For the second equation, the coefficient of x is 1 and the coefficient of y is -1.

Since the coefficients of x and y are not multiples of each other, the system is independent.

Answer: independent

6) The given system of equations is:

-2x - 7y = 3
7x - 6y = 12

To represent the system of equations as a matrix, we can write the coefficients of x, y, and the constants in a matrix form as follows:

|-2 -7 3|
| 7 -6 12|

Answer: -2 -7 3
7 -6 12

7) To translate the graph of y = x^2 to y = -(x-3)^3 + 2, we need to perform two transformations: a translation and a reflection.

First, we translate the graph 3 units to the right and 2 units up. This means every point on the graph (x, y) will become (x - 3, y + 2).

Answer: translate 3 units to the right, translate up 2 units

15) The half-life of a certain radioactive material is 68 hours. The decay can be modeled using the exponential function:

y = 641 * (1/2)^(x/68)

To find how much radioactive material remains after 4 hours, substitute x = 4 into the equation:

y = 641 * (1/2)^(4/68) ≈ 615.390 kg

Answer: y = 641 * (1/2)^(x/68); approximately 615.390 kg

18) The property of real numbers illustrated by the equation 5 * (√(9 * 4)) = (5 * √9) * 4 is the distributive property.

Answer: distributive property

20) To find the equation of a trend line that models the relationship between time (x) and home prices (y), we can use linear regression or a statistical calculator. However, without those tools, we can make a reasonable estimate using the given data points.

Based on the data given, we can see that the median home prices have been increasing roughly linearly over time. We can choose two points from the table, such as (1960, 54,000) and (2000, 100,300), and use them to find the equation of the line using the slope-intercept form, y = mx + b.

Slope (m) = (100300 - 54000) / (2000 - 1960) = 4525 / 40 = 113.125
Using the point (1960, 54000):
54000 = 113.125 * 1960 + b
b ≈ 20600

Therefore, the equation of the trend line is:
y = 113.125x + 20600

To predict the median home price in 2020 (x = 2020), substitute x = 2020 into the equation:
y = 113.125 * 2020 + 20600 ≈ 229,165

Answer: y = 113.125x + 20,600; predicted median home price in 2020 is approximately $229,165

I'd be happy to help you review for your exam! I'll go through each question and explain how to arrive at the correct answer.

4) To determine whether a system of equations is independent, dependent, or inconsistent without graphing, we need to compare the coefficients of the variables. In this case, we have:
-x - 3y = 0
x - y = 4

Notice that the second equation can be obtained by multiplying the first equation by -1 and then adding 4 to both sides. Therefore, the two equations are dependent since one equation can be derived from the other. The correct answer is "dependent."

6) To find the matrix that represents the given system of equations (-2x - 7y = 3 and 7x - 6y = 12), we need to write the coefficients of x, y, and the constant terms in a matrix form. In this case, the correct matrix is:

-2 -7 3
7 -6 12

The correct answer is the matrix "-2 -7 3" on the first row and "7 -6 12" on the second row.

7) To translate the graph of y = x^2 to y = -(x-3)^3 + 2, you correctly chose to translate 3 units to the right and translate up 2 units. This means that each point on the parabola y = x^2 will be shifted 3 units to the right and 2 units up. Therefore, the correct answer is "translate 3 units to the right, translate up 2 units."

15) The half-life of a radioactive material is the amount of time it takes for half of the material to decay. In this case, the half-life is 68 hours. To model the decay of the material, we can use the formula:

y = initial amount * (1/2)^(x/half-life)

Plugging in the given values, we get:

y = 641 * (1/2)^(x/68)

To find how much radioactive material remains after 4 hours, we substitute x = 4 into the equation:

y = 641 * (1/2)^(4/68)

Evaluating this expression gives the amount of remaining material rounded to the nearest thousandth.

18) The equation given is:

5 * (√(9 * 4)) = (5 * √9) * 4

To identify the property of real numbers illustrated by this equation, we can simplify both sides and see if the two sides are equal. Simplifying the left side gives 5 * 6 = 30, and simplifying the right side also gives 5 * 6 = 30. This demonstrates the commutative property of multiplication, where changing the order of the factors does not affect the product. Therefore, the correct answer is "commutative property of multiplication."

20) To find the equation of a trend line that models the relationship between time and home prices, we can use linear regression. Based on the given table, we can see that the median home prices increase over time. Let's set the year 1940 as x = 0.

Using the points (0, 20,600) and (60, 111,100) from the table, we can calculate the slope:

slope = (111,100 - 20,600) / (60 - 0) = 1508.33

Using the point-slope form of a linear equation, we can write:

y - 20,600 = 1508.33(x - 0)

Simplifying gives:

y = 1508.33x + 20,600

To predict the median home price in 2020 (x = 80), we substitute x = 80 into the equation:

y = 1508.33 * 80 + 20,600

Evaluating this equation will give you the predicted median home price in 2020.

I hope these explanations help you better understand how to solve these review questions. Good luck with your exam!