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Write the equation of the following circles​

The calculated value of k on the line is 9

From the question, we have the following parameters that can be used in our computation:

XZ is the perpendicular bisector of segment WY

This means that

WX = XY

substitute the known values in the above equation, so, we have the following representation

3k - 4 = 2k + 5

So, we have

3k - 2k = 4 + 5

Evaluate

k = 9

Hence, the value of k is 9

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The expression log 3x⁹y⁴ can be written as a sum of logarithms:

log(3x⁹y⁴) = log(3) + 9log(x) + 4log(y)

To express log(3x⁹y⁴) as a sum and/or difference of logarithms, we use the logarithmic properties.

Using the power rule of logarithms, we can write the expression as:

log(3x⁹y⁴) = log(3) + log(x⁹) + log(y⁴)

we can further simplify so that the variables will be to the first degree as follows;

log(3x⁹y⁴) = log(3) + 9log(x) + 4log(y).

Therefore, the expression log(3x⁹y⁴) can be written as a sum of logarithms:

log(3x⁹y⁴) = log(3) + 9log(x) + 4log(y)

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The measure of the indicated internal angle in the circle is 113 degrees.

The inscribed angle theorem states that an angle x inscribed in a circle is half of the central angle 2x that subtends the same arc on the circle.

It is expressed as:

Internal angle = 1/2 × ( x + y )

From the image:

Intercepted arc TU = x = 125 degrees

Intercepted arc SP = y = 101 degrees

Internal angle = ?

Plug the given values into the above formula and solve for the internal angle:

Internal angle = 1/2 × ( x + y )

Internal angle = 1/2 × ( 125 + 101 )

Internal angle = 1/2 × 226

Internal angle = 113°

Therefore, the internal angle is 113 degrees.

Option B) 113° is the correct answer.

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Answer:

a) For an unsubsidized loan of $15,000 at 6.8% interest for 4 years, the simple interest accumulated while you are still in school is **$4,080**.

b) If you capitalized the loan, the balance when you graduate would be **$20,322.68**.

c) After you graduate, you will begin making payments on your loan at 6.8% compounded monthly for 10 years. The monthly payment for this loan is **$174.30**.

d) The total amount of interest paid for this loan is **$10,118.00**.

e) If the loan is not capitalized, it means that the interest accrued during the in-school period is not added to the principal balance of the loan. Instead, it is paid off separately after graduation.

f) For a subsidized loan of $15,000 at 6.8% compounded monthly for 10 years, the monthly payment for this loan is **$163.19**.

g) The total amount of interest paid for this subsidized loan is **$9,782.80**.

h) The difference in savings for the subsidized loan is **$335.20**.

We know that 20 miles is the distance to tbe airport. So, we should start by substituting the x on each expression with 20.

This way we can find the cost of each ride.

Dave

C=1.8x

C=1.8(20)

C=36

Dave's ride would cost 36 pounds .

Pete

C=1.5x+5

C=1.5(20)+5

C= 30+5

C=35

Pete's ride would cost 35 pounds.

Rick

C=1.9x-3.5

C=1.9(20)-3.5

C=38-3.5

C=34.5

Rick's ride would cost 34.5 pounds.

The cheapest ride would be Rick's Rickshaw.

Now, we just figure out the change .

Jenny has 40 pounds

So, subtracting with the cheapest rate of 34.5:

40-34.5=5.5

Answer. Jenny would get 5.5 pounds in change if she took the cheapest ride.

Answer:

Equation of the line in slope-intercept form:  y = 2x - 3

Step-by-step explanation:

The general equation of the sloe-intercept form of a line is given by:

y = mx + b, where

Step 1:  Find m, the slope:

We can find m, the slope of the line using the slope formula, which is given by:
m = (y2 - y1) / (x2 - x1), where:

Thus, we can plug in (2, 1) for (x1, y1) and (-1, -5) for (x2, y2) to find m, the slope of the line:

m = (-5 - 1) / (-1 - 2)

m = (-6) / (-3)

m = 2

Thus, the slope of the line is 2.

Step 2:  Find b, the y-intercept of the line:

We can find b, the y-intercept of the line by plugging in any of the two points for (x, y) and 2 for m in the slope-intercept form.  Let's use (2, 1) for (x, y):

1 = 2(2) + b

1 = 4 + b

-3 = b

Thus, the y-intercept of the line is -3.

Therefore, the equation of a line in slope-intercept form passing through the points (2, 1) and (-1, -5) is y = 2x - 3

The size of angle DAB in the quadrilateral is 48°.

The sum of the interior angles of a quadrilateral is 360°. We can say:

[tex]\angle \text{A} +\angle\text{B} + \angle\text{C} + \angle\text{D} = 360^\circ[/tex]

[tex]\angle \text{A} +38^\circ + \angle\text{C} + 226^\circ = 360^\circ[/tex]

[tex]\angle \text{A} + \angle\text{C} + 264^\circ = 360^\circ[/tex]

[tex]\angle \text{A} + \angle\text{C} = 360^\circ- 264^\circ[/tex]

[tex]\angle \text{A} + \angle\text{C} = 96[/tex]

Since angles DAB and BCD are the same size. This implies ∠A  = ∠C. Thus:

[tex]\angle\text{A} + \angle\text{A} = 96[/tex]

[tex]2\angle\text{A} = 96[/tex]

[tex]\angle\text{A} = \dfrac{96}{2}[/tex]

[tex]\angle\text{A} = 48^\circ[/tex]

Therefore, the size of angle DAB is 48°.

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The expression (1 - x)(125 + 80y) gives us the final cost for Hector after applying the coupon, taking into account the surfboard rental fee and the fee per hour for the surfing classes with the instructor.

(1 - x): This represents the remaining percentage after applying the coupon. If the coupon offers a discount of x percent, then (1 - x) represents the percentage of the total cost that Hector has to pay.

(125 + 80y): This represents the original cost without any discounts or coupons. The term 125 represents the surfboard rental fee, and 80y represents the fee per hour that Hector works with the instructor (where y is the number of hours).

(1 - x)(125 + 80y): Multiplying the two terms together gives us the cost after applying the coupon. The expression (1 - x)(125 + 80y) represents the product of the remaining percentage after the coupon and the original cost.

Therefore, the expression (1 - x)(125 + 80y) gives us the final cost for Hector after applying the coupon, taking into account the surfboard rental fee and the fee per hour for the surfing classes with the instructor.

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The experimental design described is using a sample survey to gather results, providing a cost-effective and efficient way to estimate the likelihood of winning for different candidates in the student council election. B Sample survey Option B

The experimental design described in the scenario, where 100 students from the school are randomly asked who they intend to vote for in the student council election, is using a Sample Survey to gather its results.

A sample survey involves selecting a subset, or sample, of individuals from a larger population and collecting data from that sample to make inferences about the entire population. In this case, the 100 students randomly chosen from the school represent the sample, while the larger population would be all the students in the school.

The use of a sample survey is a common and practical approach when the population size is large, as it would be impractical or time-consuming to survey every individual in the population (which would be a census). By randomly selecting a sample, the goal is to obtain a representative subset that reflects the characteristics and opinions of the larger population.

The purpose of this sample survey is to gauge the voting intentions of the students and make predictions about the likely winner of the student council election based on the responses obtained from the sample. The results from the 100 students surveyed would be analyzed to estimate the proportion of the larger population that supports each candidate, and the candidate with the highest estimated proportion of support would be considered the most likely to win the election.

Therefore, the experimental design in question is using a Sample Survey to gather its results and make predictions about the likely winner of the student council election.

Option b

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The combination that will successfully tile a flat surface is regular pentagons and regular hexagons (option C).

The correct combination that will tile a flat surface is option C: regular pentagons and regular hexagons.

To understand why this combination works, we need to consider the properties of regular polygons and their ability to tile a plane without gaps or overlaps.

Regular polygons are shapes with equal side lengths and equal angles. For a combination of polygons to tile a flat surface, the polygons must fit together in a way that their sides align perfectly without leaving any gaps or overlaps.

Option A, which suggests using regular heptagons (7-sided polygons) and equilateral triangles, will not tile a flat surface. This is because the angles of the heptagons (internal angles of 128.57 degrees) and the angles of the equilateral triangles (internal angles of 60 degrees) do not add up to a full 360 degrees, which is necessary for tiling a plane.

Option B, proposing squares and regular octagons, will not tile a flat surface either. The internal angles of squares are 90 degrees, while the internal angles of regular octagons are 135 degrees. These angles do not match up, preventing a proper tiling.

Option D, combining regular hexagons and squares, does not work either. Although both regular hexagons and squares have internal angles of 120 degrees and 90 degrees, respectively, they cannot fit together without leaving gaps or overlaps. Attempting to tile with just these two shapes will result in a non-periodic pattern.

Option C, however, presents regular pentagons and regular hexagons. Both polygons have internal angles that add up to 360 degrees, allowing them to fit together perfectly. Specifically, a regular hexagon has internal angles of 120 degrees, while a regular pentagon has internal angles of 108 degrees. By alternating these two shapes, they can fit together to tile a flat surface without any gaps or overlaps.

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The slope of the line which is 1/7, and the points (-8, -8), (6, v) indicates that value of v is -6

The slope of a line is the ratio of the rise to the run of the line. The slope of a line that passes through the points (x₁, y₁), and (x₂, y₂), can be presented as follows;

Where;

(x₁, y₁), are the x- and y-coordinate of  one of the points on the line

(x₂, y₂) is the x- and y-coordinate of the other point on the line

The points through which the line passes are (-8, -8), (6, v)

The slope of the line is; 1/7

Therefore, we get;

(x₁, y₁) = (-8, -8)

(x₂, y₂) = (6, v)

Slope of the line = (v - (-8))/(6 - (-8)) = 1/7

(v + 8)/14 = 1/7

7 × (v + 8) = 14 × 1 = 14

7·v + 56 = 14

7·v = 14 - 56 = -42

v = -42/7 = -6

v = -6

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Answer: My guess is 0

Step-by-step explanation:

Checked with my math teacher

The graph of the inequality -2y + 3 < 5 is a shaded region below the line y = -1.

1. Start by rewriting the inequality with y on the left side: -2y + 3 < 5.

2. Subtract 3 from both sides of the inequality: -2y < 2.

3. Divide both sides of the inequality by -2. Since we are dividing by a negative number, the inequality sign flips: y > -1.

4. This means that y must be greater than -1 to satisfy the inequality.

5. To graph this inequality, draw a number line and mark -1 with an open circle (since y is not equal to -1).

6. Shade the region to the right of -1, since y must be greater than -1.

7. Since there are no other variables or coefficients in the inequality, the line y = -1 is a horizontal line passing through -1 on the y-axis.

8. Indicate on the graph that the line itself is not part of the solution by making it a dashed line instead of a solid line.

9. Therefore, the graph of the inequality -2y + 3 < 5 is a shaded region below the line y = -1.

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The y-intercept of g(x) is less than the y-intercept of f(x). The y-intercept of g(x) is -1/2, which is greater than the y-intercept of f(x), which is -4. So, the correct answer is A. The y-intercept of g(x) is less than the y-intercept of f(x).

To determine the y-intercept and x-intercept of the function g(x) = 4/(x - 8), we need to find the values of x for which g(x) equals zero and the value of g(x) when x equals zero.

First, let's find the x-intercept of g(x), which is the value of x for which g(x) equals zero. Setting g(x) = 0 and solving for x, we have:

0 = 4/(x - 8)

This equation has no solutions because a fraction can only be zero if its numerator is zero, but in this case, the numerator (4) is always non-zero. Therefore, g(x) does not have an x-intercept.

Next, let's find the y-intercept of g(x), which is the value of g(x) when x equals zero:

g(0) = 4/(0 - 8)

g(0) = 4/(-8)

g(0) = -1/2

Therefore, the y-intercept of g(x) is -1/2.

Now, let's compare the y-intercepts of g(x) and f(x). From the given table, we see that the y-intercept of f(x) is -4.

Since -4 is not equal to -1/2, we can conclude that statement B, "The y-intercept of g(x) is equal to the y-intercept of f(x)," is false.

To summarize, the correct answer is: A. The y-intercept of g(x) is less than the y-intercept of f(x).

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The standard form which are equivalent to the specified factored form of the quadratic equations are;

f(x) = (x + 2)·(x - 6) → B. f(x) = x² - 4·x -12

f(x) = (x - 4)·(x + 3) → D. f(x) = x² - x - 12

f(x) = (x - 12)·(x + 1) → A. f(x) = x² - 11·x - 12

f(x) = (x - 3)·(x + 4) → C. f(x) = x² + x - 12

The factored form of a quadratic function is the expression of the function as the product of the linear factors, which is the form; y = (x + p)·(x + q)

The factored form of the quadratic equations can be expanded to find the equivalent standard form as follows;

f(x) = (x + 2) × (x - 6) = x² - 6·x + 2·x - 12 = x² - 4·x - 12

f(x) = (x - 4) × (x + 3) = x² - 4·x +3·x - 12 = x² - x - 12

f(x) = (x - 12) × (x + 1) = x² - 12·x + x - 12 = x² - 11·x - 12

f(x) = (x - 3) × (x + 4) = x² - 3·x + 4·x - 12 = x² + x - 12

Therefore, we get;

A. f(x) = x² - 11·x - 12 ⇒ f(x) = (x - 12) × (x + 1)

B. f(x) = x² - 4·x - 12 ⇒ f(x) = (x + 2) × (x - 6)

C. f(x) =  x² + x - 12 ⇒ f(x) =  (x - 3) × (x + 4)

D. f(x) = x² - x - 12 ⇒ f(x) = (x - 4) × (x + 3)

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The value of x is $6,666.67.Assume that x equals the amount Seema invested at 3% interest. Also, let us designate the interest from Seema's $4,000 investment at 5% by I1 and the interest from Seema's x dollars investment at 3% by I2.

Simple interest is calculated as Simple Interest = (Principal * Rate * Time)/100.

Because we want to compute the return at the end of the year, we know the time for both investments is one year.

We may write two equations using this formula: I₁ = (4000 * 5 * 1) / 100 = 200I₂ = (x * 3 * 1) / 100 = 0.03x

We also know that Seema's return on investment is the same for both investments. As a result, we may write: I1 = I2 200 = 0.03x

We may now solve for x: x = 200/0.03x = 6666.67.

Therefore, the value of x is $6,666.67.

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The x-intercepts and y-intercepts of the graph are x = 5 and y = 2

From the question, we have the following parameters that can be used in our computation:

The graph (see attachment)

The x-intercepts are when y = 0

In this case, it is

x = 5

The y-intercepts are when x = 0

In this case, it is

y = 2

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Answer:

To determine the scale factor used, we can compare the corresponding side lengths of the two triangles.

In triangle D, the side lengths are 6, 8, and 10.

In triangle D', the corresponding side lengths are 18, 24, and 30.

To find the scale factor, we can divide the corresponding side lengths of triangle D' by the corresponding side lengths of triangle D.

Side length ratio:

18/6 = 3

24/8 = 3

30/10 = 3

The ratio of corresponding side lengths is consistent at 3 for each pair. Therefore, the scale factor used is 3.

So, the correct answer is 3.

Step-by-step explanation:

Step 1: Identify the corresponding side lengths of the two triangles.

In triangle D, the side lengths are given as 6, 8, and 10.

In triangle D', the corresponding side lengths are given as 18, 24, and 30.

Step 2: Calculate the ratio of each corresponding side length of D' to D.

For the first pair of corresponding sides:

18/6 = 3

For the second pair of corresponding sides:

24/8 = 3

For the third pair of corresponding sides:

30/10 = 3

Step 3: Analyze the ratios obtained.

Since all three ratios are equal to 3, it indicates a consistent scale factor between the corresponding side lengths of the two triangles.

Step 4: Determine the scale factor.

The scale factor is the constant ratio by which each corresponding side length is multiplied to obtain the corresponding side length of the larger triangle. In this case, since the ratio is 3 for all pairs, the scale factor used is 3.

So, the step-by-step explanation confirms that the scale factor used is 3.

Answer:

The scale factor is 3

Step-by-step explanation:

For triangle D to be dilated into triagle D', each side of D is multiplied by 3,i.e,

6 * 3 = 18

8 * 3 = 24

10 * 3 = 30

Therefor, the scale factor to turn triangle D to D' is 3

In approximately the year 2001 (or the 1.373rd year since 2000), the approximate number of oil barrels imported from Country B per day will be 1.93 million.

To solve the equation y = 0.069x + 1.24 for x, we need to isolate x on one side of the equation. Let's rearrange the equation:

y = 0.069x + 1.24

Subtract 1.24 from both sides:

y - 1.24 = 0.069x

Divide both sides by 0.069:

(y - 1.24) / 0.069 = x

Now we have x isolated on one side of the equation. We can use this equation to determine the year in which the approximate number of oil barrels imported from Country B per day will be 1.93 million.

Let's substitute y = 1.93 into the equation:

(x - 1.24) / 0.069 = 1.93

Multiply both sides by 0.069:

x - 1.24 = 0.069 * 1.93

x - 1.24 = 0.13317

Add 1.24 to both sides:

x = 0.13317 + 1.24

x = 1.37317

Now we have the value of x, which represents the number of years since 2000. To determine the year, we add the value of x to 2000:

Year = 2000 + 1.37317

Year ≈ 2001.373

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The original mass of the red giant was approximately 13.67 solar masses (MSun).

After 0.8 million years (0.8 * 10⁶ years), the red giant's mass is 11 MSun. This means that it has lost a certain amount of mass over this time period.

The mass loss can be calculated by multiplying the mass loss rate by the time period:

Mass loss = Mass loss rate * Time

Mass loss = (1 MSun / 300,000 years) * (0.8 * 10^6 years)

Mass loss = 2.67 MSun

Now, we can find the original mass by subtracting the mass loss from the current mass:

M original = M current + Mass loss

M original = 11 MSun + 2.67 MSun

M original = 13.67 MSun

Therefore, the original mass of the red giant was approximately 13.67 solar masses (MSun).

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A red giant loses a solar mass in 300,000 years via a superwind. After 0.8 million years, it has a mass of 11MSun. What was its original mass?

Answer:

To determine the number of ways a committee can be formed with one representative from each team, you can multiply the number of choices for each team.

Number of choices for the football team = 53

Number of choices for the basketball team = 15

Number of choices for the baseball team = 12

To find the total number of ways, you multiply these numbers together:

53 choices * 15 choices * 12 choices = 9,540 ways

Therefore, there are 9,540 ways to form a committee with one representative from each team.

Step-by-step explanation:

Answer:

The total is 9540

Step-by-step explanation:

Do 128 x 22 say for that it is the 53 and 15 players Then do 127 x 21 which it would be 15 x 14 but you are doing the first number honestly  then 127 x 21 and that would be 12 x 11 so therefore you will multiply it by 53 x 15 x 12 = 9,540

The GDP for the first quarter of 2020 is $120 billion.

To calculate the GDP (Gross Domestic Product) for the first quarter of 2020, we need to consider the components of GDP: consumption (C), investment (I), government spending (G), and net exports (NX). The formula for GDP is:

GDP = C + I + G + NX

Given the following statistics for the first quarter of 2020:

Consumers purchased $50 billion of goods and services (C = $50 billion).

Businesses invested $10 billion back into their businesses and held $5 billion of goods produced during the year in their inventories (I = $10 billion).

Federal, state, and local governments spent $70 billion throughout the year (G = $70 billion).

The nation exported $90 billion worth of goods and services while importing $100 billion.

To calculate net exports (NX), we subtract the value of imports from the value of exports:

NX = Exports - Imports

NX = $90 billion - $100 billion

NX = -$10 billion (negative value indicates a trade deficit)

Now, we can substitute the given values into the GDP formula:

GDP = C + I + G + NX

GDP = $50 billion + $10 billion + $70 billion + (-$10 billion)

GDP = $120 billion.

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The sky appears blue because of Rayleigh scattering while water appears blue due to selective absorption and scattering of light.

The blue color of the sky is a result of Rayleigh scattering which occurs when the Earth's atmosphere scatters shorter wavelengths of light (blue and violet) more efficiently than longer wavelengths (red and orange).

When sunlight enters the atmosphere, the blue light is scattered in all directions by the molecules and tiny particles present in the air creating the blue appearance of the sky.

Similarly, water appear blue due to selective absorption and scattering of light. Water molecules selectively absorb colors from the visible light spectrum and absorb longer wavelengths (such as red) more effectively than shorter wavelengths (such as blue).

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The value of ninth term (a₉) is equal to: B. 8/729.

In Mathematics and Geometry, the nth term of a geometric sequence can be calculated by using this mathematical equation (formula):

aₙ = a₁rⁿ⁻¹

Where:

From the second term of this geometric sequence, we have:

-24 = a₁r

a₁ = -24/r

From the fifth term of this geometric sequence, we have:

8/9 = a₁r⁴

8/9 = a₂ × r³

8/9 = -24 × r³

r³ = -8/216

r = -∛1/27

r = -1/3

Now, we can determine ninth term as follows;

a₁ = -24/(-1/3)

a₁ = 72

a₉ = a₁r⁸

a₉ = 72(-1/3)⁸

a₉ = 72/6,561

a₉ = 8/729

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Answer:

The first blank is the less than symbol (<)

The second blank is the greater than symbol (>)

Step-by-step explanation:

It looks like you are comparing the left side of the equation to the right side.

If we look at the first unknown, we can multiply out the numbers on the left side to see if it's equal to the right side.

[tex]2^2 * 4^2 = 64[/tex]

while the right side is this:

[tex]8^4 = 4096[/tex]

we can see that one side is clearly bigger than the other side. Thus, we would use the less than symbol in this case as 64 is less than 4096:

[tex]2^2 * 4^2 < 8^4[/tex]

[tex]64 < 4096[/tex]

We can do the same for the second unknown as well:

[tex]2^4 * 4^2 = 256\\8^2 = 64[/tex]

In this case, we can see that the left side is bigger than the right side. Thus, we will use the greater than symbol as 256 is greater than 64:

[tex]2^4*4*2 > 8^2\\256 > 64[/tex]

Answer:

Slope: 1/2

y-intercept: (0, 4.5)

Slope Intercept = y = mx + b

Slope Intercept = y = 1/2x + 4.5

Step-by-step explanation:

The slope is 1/2 based on our rule, "Rise over Run" Based on the two points listed on the graph, we can say that we "rise" once and "run" twice. This makes our slope 1/2.

The y-intercept is where the line crosses the y-axis. If use our slope, we can determine that if we "rise" half of our slope in the opposite direction (0.5) and run half of our run in the opposite direction (1) we will intersect at (0, 4.5).

With this information, we can construct our slope-intercept equation:

The slope-intercept formula is as follows: y = mx + b

m is our slope, and b is out y-intercept.

y = 1/2x + 4.5

Answer:

Rental cost for each movie = $3.25

Rental cost for each video game = $5.25

Step-by-step explanation:

We can find the rental cost for each movie and each video game using a system of equations where:

First equation:

Since Jina rented 6 movies and 2 video games for a total of $30, our first equation is given by:

6M + 2V = 30

Second equation:

Since Jina also rented 3 movies and 5 video games for a total of $36, our second equation is given by:

3M + 5V = 36

Method to solve:  Elimination:

We can solve this with eliminate.  First, we need to multiply the second equation by -2.  Then, we must add the two equations to eliminate the Ms and solve for V:

Multiplying the entire second equation by -2:

-2(3M + 5V = 36)

-6M - 10V = -72

Adding the two equations:

    6M + 2V = 30

+

   -6M - 10V = -72

(6M - 6M) + (2V - 10V) = (30 - 72)

-8V = -42

Solving for V:

(-8V = -42) / -8

V = 5.25

Thus, the rental cost for each video game is $5.25.

Solving for M:

We can now find M by plugging in 5.25 for V in any of the two equations in our system.  

Let's use the first equation:

Plugging in 5.25 for V in 6M + 2V = 30 to solve for M:

6M + 2(5.25) = 30

(6M + 10.50 = 30) - 10.50

(6M = 19.50) / 6

M = 3.25

Thus, the rental cost for each movie is $3.25.

The line segment is calcilated by  the formula  represented mathematically as (x1+x2/2, y1+y2/2), The midpoint is (5,5).

The midpoint of a line segment is the point on the line that is halfway between the endpoints of the line.

To find the midpoint of a line segment, you need to average the x-coordinates of the endpoints and the y-coordinates of the endpoints.

In other words, the x-coordinate of the midpoint is the average of the x-coordinates of the endpoints, and the y-coordinate of the midpoint is the average of the y-coordinates of the endpoints.

The midpoint formula can be represented mathematically as (x1+x2/2, y1+y2/2)

Where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment.

Let's illustrate this with an example:

Suppose we have a line segment with endpoints (2,3) and (8,7).

To find the midpoint of the line segment, we use the formula (x1+x2/2, y1+y2/2)(2+8/2, 3+7/2)(10/2, 10/2)

The midpoint is (5,5).

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The equation of the line passing through the point (2, 3) and making an angle of 45 degrees with the line x - 3y = 5 is

y = -3x + 9.

The equation of a line passing through a given point and making an angle of 45 degrees with another line can be found using the concept of slope.

Let's start by finding the slope of the given line, x - 3y = 5. To do this, we need to rearrange the equation into slope-intercept form, which is y = mx + b, where m represents the slope.

Rearranging the equation, we get:

x - 3y = 5
-3y = -x + 5
y = (1/3)x - (5/3)

So, the slope of the given line is 1/3.

Since the line we want to find makes an angle of 45 degrees with the given line, the slope of the line we want to find will be the negative reciprocal of the slope of the given line. The negative reciprocal of 1/3 is -3.

Now, let's use the point-slope form of a linear equation to find the equation of the line passing through the point (2, 3) with a slope of -3. The point-slope form is given by:

y - y1 = m(x - x1)

Substituting the values into the formula, we get:

y - 3 = -3(x - 2)

Expanding the equation:

y - 3 = -3x + 6

Simplifying:

y = -3x + 9

Therefore, the equation of the line passing through the point (2, 3) and making an angle of 45 degrees with the line x - 3y = 5 is y = -3x + 9.

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