A circle of radius 6 is centred at the origin, as shown.
The tangent to the circle at point P crosses the y-axis at (0, -14).
Work out the coordinates of point P.
Give any decimals in your answer to 1 d.p.

The image of point P (6, 1) under a 180° counterclockwise rotation about the origin is (-6, -1).

To find the image of point P (6, 1) under a 180° counterclockwise rotation about the origin, we can use the rotation formula for 2D coordinates.

The formula for rotating a point (x, y) counterclockwise by θ degrees about the origin is:

x' = x [tex]\times[/tex] cos(θ) - y [tex]\times[/tex] sin(θ)

y' = x [tex]\times[/tex] sin(θ) + y [tex]\times[/tex] cos(θ)

In this case, θ is 180°.

So, let's substitute the values of x and y from point P into the rotation formula:

x' = 6 [tex]\times[/tex] cos(180°) - 1 [tex]\times[/tex] sin(180°)

y' = 6 [tex]\times[/tex] sin(180°) + 1 [tex]\times[/tex] cos(180°)

Now, let's simplify these equations using the trigonometric values for 180°:

[tex]x' = 6 \times (-1) - 1 \times 0[/tex]

[tex]y' = 6 \times 0 + 1 \times (-1)[/tex]

Simplifying further:

x' = -6

y' = -1

Therefore, the image of point P (6, 1) under a 180° counterclockwise rotation about the origin is (-6, -1).

Please note that the rotation formula assumes angles are measured in radians.

However, for simplicity, we used degrees in this explanation.

The trigonometric functions (cos and sin) can be evaluated in radians using their corresponding values for 180°.

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The walls of the archway are approximately 8.9 apart.

To find the distance between the walls of the archway, we need to find the horizontal distance where the function f(x) intersects the x-axis. This is because the archway's walls are vertical, and their distance apart is the same as the horizontal distance between the points where the archway meets them.

To find the x-intercepts of the function f(x) = -0.5x^2 + 10, we need to set f(x) = 0 and solve for x:

0 = -0.5x^2 + 10

0.5x^2 = 10

x^2 = 20

x = ±√20

Since the archway is a physical object, we can discard the negative value for x, which means the archway meets the walls at x = √20 feet.

To find the distance between the walls of the archway, we can double this value:

2√20 ≈ 8.94

Then it's concluded that the walls of the archway are approximately 8.9 feet apart.

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I think it might be C

The probability P(2 or fewer) is 0.678.

To find the probability of 2 or fewer successes in a binomial experiment with 10 trials and a success probability of 0.2, we can use the binomial probability formula:

P(2 or fewer) = P(0) + P(1) + P(2)

where P(0), P(1), and P(2) represent the probabilities of getting 0, 1, or 2 successes, respectively.

P(0) = (10 choose 0) * 0.2^0 * 0.8^10 = 0.1074
P(1) = (10 choose 1) * 0.2^1 * 0.8^9 = 0.2684
P(2) = (10 choose 2) * 0.2^2 * 0.8^8 = 0.3020

Therefore,

P(2 or fewer) = 0.1074 + 0.2684 + 0.3020 = 0.6778

Rounded to at least three decimal places, the probability P(2 or fewer) is 0.678.

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Where every student offer at least one of the subjects, there are 4 students who offer all three subjects.

To find the number of students who offer only one subject, we can use the formula:

number of students offering only one subject = total number of students offering the subject - number of students offering both subjects

For math, there are 22 students offering math and 5 offering both math and biology, so the number of students offering math only is:

22 - 5 = 17

Similarly, for physics and biology, the numbers of students offering the subjects only are:

14 - 7 = 7 (physics)
16 - 5 = 11 (biology)

To find the number of students offering all three subjects, we can use the formula:

number of students offering all three subjects = total number of students offering at least one subject - number of students offering only one subject in each subject + number of students offering no subject

We know that there are 7 students offering at least one subject. To find the number of students offering no subject, we can subtract this from the total number of students:

39 - 7 = 32

Now we can plug in the numbers:

number of students offering all three subjects = 7 - 17 - 7 - 11 + 32
= 4

Therefore, there are 4 students who offer all three subjects.

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The value of measure of arc QI is,

⇒ m QI = 94°

We have to given that;

⇒ m YS = 180°

⇒ m ∠QBI = 137°

Hence, We can formulate;

⇒ m ∠QBI = 1/2 (m YS + m QI)

⇒ 137 = 1/2 (180 + m QI)

⇒ 274 = 180 + m QI

⇒ m QI = 274 - 180

⇒ m QI = 94°

Thus, The value of measure of arc QI is,

⇒ m QI = 94°

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The number of cells found in an average Koala is 1.30 x 10¹², under the condition that a cell is a sphere with radius 10 or 0. 001 centimeter.

Then the volume of a sphere with radius 10 cm is considered to be 4/3π(10)³ cubic cm that is approximately 4,188.79 cubic cm.
The evaluated volume of a sphere with radius 0.001 cm is 4/3π(0.001)³ cubic cm that is approximately 0.00000419 cubic cm.

Then the evaluated mass of a single cell is  found by applying the formula
mass = density x volume
In case of larger cell, the mass will be
mass = 1.1 g/cm³ x 4,188.79 cubic cm
= 4,607.67 grams

In case of  smaller cell, the mass will be

mass = 1.1 g/cm³ x 0.00000419 cubic cm
= 0.00000461 grams

As koalas measure an average of 6 kilograms or 6,000 grams², we can evaluate the number of cells in an average koala using division of the weight of the koala by the mass of a single cell

In case of larger cells
number of cells = weight of koala / mass of single cell
number of cells = 6,000 grams / 4,607.67 grams
≈ 1.30 x 10⁶ cells

For smaller cells:
number of cells = weight of koala / mass of single cell
number of cells = 6,000 grams / 0.00000461 grams

≈ 1.30 x 10¹² cells
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⁶

The slope of the line that divides the region into two equal parts is 8/7.

We begin by finding the x-coordinates of the points where the parabola intersects the x-axis. Setting y = 0, we get:

[tex]2x - 7x^2 = 0[/tex]

x(2 − 7x) = 0

x = 0 or x = 2/7

Thus, the parabola intersects the x-axis at x = 0 and x = 2/7.

We want to find the slope of the line through the origin that divides the region bounded by the parabola and the x-axis into two regions with equal area.

Let's call this slope m.

We know that the area under the parabola from x = 0 to x = 2/7 is:

A = ∫[0,2/7] (2x − 7[tex]x^2[/tex]) dx

A = [[tex]x^2[/tex] − (7/3)[tex]x^3[/tex]] from 0 to 2/7

A = (4/21)

Since we want the line to divide this area into two equal parts, the area to the left of the line must be (2/21).

Let's call the x-intercept of the line h. Then the equation of the line is y = mx, and the area to the left of the line is:

(1/2)h(mx) = (1/2)mhx

We want this to be equal to (2/21), so we can solve for h:

(1/2)mhx = (2/21)

h = (4/21m)

The x-coordinate of the point of intersection of the line and the parabola is given by:

2x − 7[tex]x^2[/tex] = mx

Simplifying, we get:

[tex]7x^2 - (2 + m)x = 0[/tex]

Using the quadratic formula, we get:

[tex]x = [(2 + m) \pm \sqrt((2 + m)^2 - 4(7)(0))]/(2(7))[/tex]

x = [(2 + m) ± √(4 + 4m + [tex]m^2[/tex])]/14

x = [(2 + m) ± (2 + m)]/14

x = 1/7 or x = −(2/7)

Since we want the line to pass through the origin, we must choose x = 1/7, and we can solve for m:

[tex]2(1/7) - 7(1/7)^2 = m(1/7)[/tex]

m = 8/7

Therefore, the slope of the line that divides the region into two equal parts is 8/7.

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The total length of the string in feet is 6 feet, and the combined capacity of both water jugs in cups is 75 cups.

Step 1: To find the total length of the string in inches, Mara needs to add the lengths of the three pieces of string:

30 inches + 22 inches + 20 inches = 72 inches

So the total length of the string in inches is 72 inches.

Step 2: To convert inches to feet, we need to divide the number of inches by 12 (since there are 12 inches in a foot):

72 inches ÷ 12 = 6 feet

Therefore, once Mara puts together the three pieces of string, the total length will be 6 feet.

Step 1: To find out how many ounces of water both water jugs hold altogether, we need to add the capacity of the two jugs:

300 ounces + 300 ounces = 600 ounces

So both water jugs together can hold 600 ounces of water.

Step 2: To convert ounces to cups, we need to divide the number of ounces by 8 (since there are 8 ounces in a cup):

600 ounces ÷ 8 = 75 cups

Therefore, both water jugs together can hold 75 cups of water.

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The situation in which Shawn's activity will result in a final value of zero is Shawn travels east ten feet and then travels south ten feet. The correct option is D.

This is because when Shawn travels east ten feet, he moves horizontally to the right of his starting point. When he travels south ten feet after that, he moves vertically downwards from his previous position, cancelling out the horizontal movement he made earlier.

The displacement caused by Shawn's movement in the east direction is equal in magnitude but opposite in direction to the displacement caused by his movement in the south direction.

The net displacement of Shawn's movement is zero, and he ends up back at his starting point. Options A, B, and C do not involve any movements that result in a net displacement of zero.

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The evaluated missing length in right triangle by using the Pythagorean theorem is  9 yards under the condition given the triangle is a right triangle.

The Pythagoras theorem projects that in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides,
It is given to us that in a right triangle,
Hypotenuse = 15 yd
Perpendicular = 12 yd

Therefore, applying Pythagoras theorem;
Base² = 15² - 12²

Base² = 225 - 144

Base² = 81

Base = √81

Base = 9 yards

Hence, The missing length present in the right triangle by applying the Pythagorean theorem is,
9 yards

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The complete question is

Determine the missing length in each right triangle using the Pythagorean theorem. Round the answer to the nearest tenth, if necessary

Loan H's monthly payment will be $42.46 smaller than loan I's (rounded to the nearest cent). The correct option is a.

To determine which loan will have the smaller monthly payment, we need to calculate the monthly payments for both loans using the given information.

For loan H, the monthly interest rate is 12.24%/12 = 1.02%, and the number of payments is 3 years x 12 months/year = 36. Using the formula for the monthly payment on a loan with monthly compounding, we have:

P = (r(PV))/(1 - (1+r[tex])^{(-n)})[/tex]

where P is the monthly payment, r is the monthly interest rate, PV is the principal value of the loan, and n is the total number of payments.

Plugging in the values given for loan H, we get:

P = (0.0102 x $5,650) / (1 - (1+0.0102)⁻³⁶) = $186.25

For loan I, the monthly interest rate is 10.97%/12 = 0.9142%, and the number of payments is 4 years x 12 months/year = 48. Using the same formula as above, we have:

P = (0.009142 x $6,830) / (1 - (1+0.009142)⁻⁴⁸) = $227.04

Therefore, the monthly payment for loan H is $186.25 and the monthly payment for loan I is $227.04.

To find the difference between the monthly payments, we subtract the monthly payment for loan H from the monthly payment for loan I:

$227.04 - $186.25 = $40.79

Therefore, the answer is (a) loan H's monthly payment will be $42.46 smaller than loan I's (rounded to the nearest cent).

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she can add each number one by one , and whenever she needs to carry numbers over. She can add them to the already existing numbers. she cna check with a calculator

Answer: $532,000

Step-by-step explanation:

If the company is making $140,000 and they get 20% each year, we just multiply it by 20%, or 0.20, and get $28,000. So, we would multiply that by 14, the years the company operated, and then add it to the original $140,000.

28,000 x 14 = 392,000

392,000 + 140,000 = 532,000

So, over the course of 14 years, the company made a profit of $532,000.

The equation that best represents the situation is y = 3/4x. The correct answer is A.

The given problem involves two people, Jamie and Anabel, who jogged a certain distance. Let's say Jamie jogged x km. Anabel jogged 1/4 less than Jamie, which means she jogged 3/4 of x km (since 1 - 1/4 = 3/4).

To represent this situation in an equation, we need to find the relationship between the distance jogged by Jamie and the distance jogged by Anabel. Since Anabel jogged 3/4 of the distance jogged by Jamie, we can write:

distance jogged by Anabel = 3/4(distance jogged by Jamie)

Using the given variable x for the distance jogged by Jamie, we can rewrite the equation as:

distance jogged by Anabel = 3/4x

And since the question is asking for an equation that best represents the situation, the correct answer is:

Cy = 3/4x

Therefore, Cy = 3/4x is the equation that best represents the situation where Jamie jogged x km and Anabel jogged 1/4 less than Jamie. The correct answer is A.

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The maximum profit will be realized by producing and selling 2.5 kilograms of whatchamacallits.

The maximum profit, in hundreds of dollars, will be $500.

To find the maximum profit, we need to first calculate the profit function P(x), which is the difference between the revenue and the cost functions:

P(x) = R(x) - C(x)
P(x) = 450x - (50x + 1000)
P(x) = 400x - 1000

To find the amount of kilograms that should be produced and sold to realize a maximum profit, we need to find the value of x that maximizes the profit function.

We can do this by taking the derivative of the profit function and setting it equal to zero:

P'(x) = 400
400 = 0

Since the derivative is a constant value, there is no critical point or inflection point. Therefore, the profit function is increasing at a constant rate, and the maximum profit will be achieved at the highest possible value of x.

To find that value, we can set the profit function equal to zero and solve for x:

P(x) = 400x - 1000 = 0
400x = 1000
x = 2.5

Therefore, the maximum profit will be realized by producing and selling 2.5 kilograms of whatchamacallits.

To find the maximum profit itself, we can substitute this value of x into the profit function:

P(2.5) = 400(2.5) - 1000 = 500

So the maximum profit, in hundreds of dollars, will be $500.

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The pet boarder can accommodate 70 dogs.

Let's calculate how many dogs and cats the pet boarding facility can hold using the ratio of dogs to cats, which is 5:2.

First, we can figure out how many parts there are in the ratio: 5 + 2 = 7.

This indicates that there are 7 equal components in the ratio, 5 of which are dogs and 2 of which are cats.

We need to multiply the result by the number of dog components (5) after dividing the total number of parts (7) into the 98 available locations to get the number of dogs:

Number of dogs = (5 / 7) * 98

Number of dogs = 70

Therefore, the pet boarder can accommodate 70 dogs.

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The value of the car after 8 years is given as follows:

$10,836.76.

An exponential function has the definition presented as follows:

[tex]y = ab^\frac{x}{n}[/tex]

In which the parameters are given as follows:

The parameter values for this problem are given as follows:

a = 86000, n = 5, b = 1 - 0.726 = 0.274.

Hence the function for the value of the car after x years is given as follows:

[tex]y = ab^\frac{x}{n}[/tex]

[tex]y = 86000(0.274)^\frac{x}{5}[/tex]

The value of the car after 8 years is then given as follows:

[tex]y = 86000(0.274)^\frac{8}{5}[/tex]

y = $10,836.76.

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Answer:

4 or [tex]\frac{4}{1}[/tex]

Step-by-step explanation:

To solve this we need to remember SOH-CAH-TOA. With SOH being Sine, CAH being Cosine, and TOA being Tangent. In the last term (TOA), the O means opposite and the A is adjacent. This means the segment opposite of angle H you have to divide that by the segment adjacent to H.

In this case, the opposite is 28 and the adjacent is 7. So we have to do [tex]\frac{28}{7}[/tex]. This is tan(H). Now we have to simplify this. Now we get our tangent of H to be [tex]\frac{4}{1}[/tex] or 4. So 4/1 or 4 is our answer

We can start by integrating both sides of the differential equation to obtain:

∫f '(x) dx = ∫([tex]5x^2 - x^2/5[/tex]) dx

f(x) = (5/3)[tex]x^3[/tex] - (1/15) [tex]x^5[/tex] + C

where C is the constant of integration.

To find the value of C, we can use the initial condition f(1) = 0:

f(1) = (5/3)[tex](1)^3[/tex] - (1/15) [tex](1)^5[/tex] + C = 0

Simplifying this equation gives:

C = (1/15) - (5/3)

C = -2/9

Therefore, the solution to the initial value problem f '(x) = 5[tex]x^2[/tex] − [tex]x^2[/tex]/5 , f(1) = 0 is:

f(x) = (5/3) [tex]x^3[/tex] - (1/15) [tex]x^5[/tex] - (2/9)

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The height of the building is 16 meters.

What is right triangle?

A right triangle is a type of triangle that has one of its angles measuring 90 degrees (π/2 radians). The side which is opposite to the right angle is the hypotenuse, while the other two sides are called the legs.

We can solve this problem using the Pythagorean theorem, which relates the sides of a right triangle. Let h be the height of the building. Then we can draw a right triangle with one leg of length h and the other leg of length 12m, representing the height and length of the shadow, respectively. The hypotenuse of this triangle is the distance from the top of the building to the tip of the shadow, which is 20m. So we have:

h² + 12² = 20²

Simplifying and solving for h, we get:

h² = 20² - 12²

h² = 256

h = sqrt(256)

h = 16

Therefore, the height of the building is 16 meters.

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The probability that a randomly selected 10th grade boy exceeds 70 in is approximately 0.0127 or 1.27%.

The height of a 16-year-old boy in the 96th percentile is approximately 73 inches.

For the first question, we need to find the z-score for a height of 70 inches using the formula:

z = (x - μ) / σ

where x is the height of 70 inches, μ is the mean of 63.5 inches, and σ is the standard deviation of 2.9 inches.

z = (70 - 63.5) / 2.9 = 2.241

Using a standard normal table, we can find the area to the right of this z-score, which represents the probability that a randomly selected 10th grade boy exceeds 70 inches. The area to the right of 2.24 is 0.0127. Therefore, the probability is approximately 0.0127 or 1.27%.

For the second question, we need to find the z-score associated with the 96th percentile using a standard normal table. The 96th percentile is the point below which 96% of the data falls and above which 4% of the data falls. This corresponds to a z-score of approximately 1.75.

To find the height of a 16-year-old boy in the 96th percentile, we can use the formula:

x = μ + z * σ

where x is the height we want to find, μ is the mean of 68.3 inches, σ is the standard deviation of 2.9 inches, and z is the z-score we just found.

x = 68.3 + 1.75 * 2.9 = 73.28

Therefore, the height of a 16-year-old boy in the 96th percentile is approximately 73 inches.

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The instantaneous rate of change of cost with respect to x when x = 100 is 4.

We can begin by calculating C(100+h) and C(100):

C(100+h) = 0.000002(100+h)^3 + 4(100+h) + 300

C(100+h) = 0.000002(1,000,000 + 300h^2 + 30h^2 + h^3) + 400 + 4h + 300

C(100+h) = 0.000002h^3 + 0.0006h^2 + 4h + 700

C(100) = 0.000002(100)^3 + 4(100) + 300

C(100) = 2 + 400 + 300

C(100) = 702

Therefore,

C(100+h) - C(100) = (0.000002h^3 + 0.0006h^2 + 4h + 700) - 702

C(100+h) - C(100) = 0.000002h^3 + 0.0006h^2 + 4h - 2

Now, we can find the rate of change of cost with respect to x by dividing this expression by h and taking the limit as h approaches 0:

(C(100+h) - C(100))/h = (0.000002h^3 + 0.0006h^2 + 4h - 2)/h

(C(100+h) - C(100))/h = 0.000002h^2 + 0.0006h + 4 - (2/h)

As h approaches 0, the term 2/h approaches infinity, which means the rate of change of cost with respect to x is undefined. However, we can calculate the limit of the expression as h approaches 0 from the left and from the right to see if it has a finite value:

limit (h->0+) ((C(100+h) - C(100))/h) = 4

limit (h->0-) ((C(100+h) - C(100))/h) = 4

Since the left and right limits are equal, the overall limit exists and equals 4. Therefore, the instantaneous rate of change of cost with respect to x when x = 100 is 4.

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Answer: 832 plants

Step-by-step explanation:

If there are 4 plants for every 1 square ft the ratio is 4:1.

This tells us to multiply 208x4 giving us 832.

A frequency table was created using data points representing the number of holes in each of Grandma Marion's dresses. The table shows the number of dresses with 5, 7, and 8 holes, respectively.

To create a frequency table for the given data, first, the unique values in the data set are identified, which are 5, 7, and 8. Then, the number of occurrences of each unique value is counted, resulting in the frequencies 1 for 5, 2 for 7, and 3 for 8.

Count the frequency of each data point

5: 1 dress

7: 2 dresses

8: 3 dresses

Finally, these values are organized into a table with two columns, one for the unique values and another for their corresponding frequencies. The resulting frequency table shows the number of dresses with each number of holes eaten by moths.

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The answer is 96 inches, which is equivalent to 8 feet.

To find the length of the top edge of the yield sign, we need to use the formula for the area of a trapezoid:

A = (b1 + b2)h/2

where A is the area of the trapezoid, h is the height, b1, and b2 are the lengths of the two parallel bases of the trapezoid.

In this case, we are given the area of the sign (216 square inches) and the height (18 inches), but we don't know the length of either base. However, we do know that the shape of a yield sign is that of a regular octagon, which means it has eight equal sides and eight equal angles.

If we draw a line from the top of the sign to the midpoint of one of the sides, we will form a right triangle with the height of the sign as one leg, half the length of the top edge as the other leg, and the length of one of the sides as the hypotenuse. We can use the Pythagorean theorem to find the length of the side:

a^2 + b^2 = c^2

where a is the height of the sign (18 inches), b is half the length of the top edge (what we are trying to find), and c is the length of one of the sides.

Since the sign has eight sides, we can divide the total area by 8 to get the area of one of the eight triangles that make up the sign. We can then use this area to find the length of one of the sides:

A = (bh)/2

216 sq. in. = (bh)/2

432 sq. in. = bh

Since the sign is a regular octagon, each of the eight triangles has the same base (the side of the octagon) and height (half the length of the top edge), so we can use this equation to solve for b:

432 sq. in. = b(18 in.)/2

b = 48 in.

Now we know that half the length of the top edge is 48 inches, so the full length of the top edge is:

2(48 in.) = 96 in.

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The function table that matches the given rule output = input - 3 is

Input = -2, 1, 6 and output = -5, -2, 3

A) first function table

Output = Input - 3

Value of input:- -2

Putting the value of the input

Output = -2 -3

The value of output we get

Output = -5

Value of input:- 1

Putting the value of the input

Output = 1 -3

The value of output we get

Output = -2

Value of input:- 6

Putting the value of the input

Output = 6 -3

The value of output we get

Output = 3

Hence function table A is correct match

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According to recent studies, there is a notable difference in credit scores between those over the age of 55 and those between the ages of 18-24.

On average, individuals over the age of 55 tend to have higher credit scores than those in the 18-24 age range. This is primarily due to the fact that older individuals have had more time to establish and build their credit history, whereas younger individuals are just starting out and may not have had the opportunity to establish credit yet.

Additionally, older individuals tend to have more stable financial situations and may have less debt compared to younger individuals who may be dealing with student loans or other types of debt.

However, it's important to note that credit scores can vary greatly between individuals, regardless of age, and there are many factors that contribute to credit score, such as payment history, credit utilization, and length of credit history.

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Answer:

(2,-4)

Step-by-step explanation:

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

2x+3y=−8,3x+y=2

To make 2x and 3x equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by 2. Then simplify

6x+9y=−24,6x+2y=4

Add 6x to −6x. Terms 6x and −6x cancel out, leaving an equation with only one variable that can be solved, add 9y to −2y, add −24 to −4, and divide both sides by 7.

y=−4

Substitute −4 for y in 3x+y=2. Because the resulting equation contains only one variable, you can solve for x directly. Add 4 to both sides of the equation and divide both sides by 3.

x=2

Answer: B

Step-by-step explanation: 250 x 140% = 350