Form an equation of line given these coordinates (2,4) and (-2,-8)​

By using Fermat's Little Theorem, we were able to prove through a series of mathematical steps that 431 divides 2⁴³⁻¹

In mathematics, proving that a number divides another number means finding out if it can evenly divide the other number with no remainder.

Here we need to prove that 431 divides 2⁴³⁻¹, we can use the mathematical property of modular arithmetic.

In modular arithmetic, we are interested in finding the remainder when a number is divided by another number.

In this case, we want to find the remainder when 2⁴³⁻¹ is divided by 431.

If the result is 0, then we can conclude that 431 divides 2^43 -1.

In order to do this, we can use Fermat's Little Theorem.

Since 431 is a prime number, we can use Fermat's Little Theorem to calculate

=> 2⁴³⁰ mod 431.

If the result is 1, then we know that

=> 2⁴³ mod 431

=> 2^(43 mod 430) * 2⁴³⁰ mod 431 = 2³ mod 431.

Finally, we can calculate 2³ mod 431 and if the result is 1, we can conclude that

=> 2⁴³⁻¹ mod 431 = 1.

This means that 2 is divisible by 431 with no remainder, so we can conclude that 431 divides 2⁴³⁻¹.

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If Caron gives $75, he will get $21 as a change, when he buys a constellation display.

To estimate the amount of change Caron will get back, we need to calculate the total cost of the two-sided tape and the star stickers. The two-sided tape costs $12 and Caron wants to buy 3 packs of star stickers which cost $14 each, so the total cost of the star stickers is 3 * $14 = $42. The total cost of both the two-sided tape and the star stickers is $12 + $42 = $54.

If Caron gives $75, he will get $75 - $54 = $21 as change. This is a rough estimate of the amount of change Caron will get back, and it's always a good idea to have a little extra money in case the actual cost is slightly higher.

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c = log_10(25) is the value of c that makes y(1) = 5.

What is differential ?

A differential is a gear train with three drive shafts that has the property that the rotational speed of one shaft is the average of the speeds of the others, or a fixed multiple of that average.

y'(x) = c * x^(c-1) * x^(-2) + (-2) * x^c * x^(-2) = (c - 2) * x^(c-3)

Now, we can substitute y(x) and y'(x) into the differential equation:

xy' + 2y = 3x

x * (c - 2) * x^(c-3) + 2 * x^c * x^(-2) = 3x

(c - 2) * x^(c) + 2 * x^c * x^(-2) = 3x

(c - 2) * x^c + 2 * x^(-2) = 3x

This confirms that y(x) = x^c * x^(-2) is indeed a solution of the differential equation xy' + 2y = 3x.

To find the value of c so that y(1) = 5, we need to substitute x = 1 and y = 5 into the solution y(x) = x^c * x^(-2):

y(1) = 1^c * 1^(-2) = 5

1^c = 5 * 1^2 = 25

c = log_1(25)

c = log_10(25) / log_10(1) = log_10(25)

So, c = log_10(25) is the value of c that makes y(1) = 5.

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Number of the 61 days in 2009 the Dow decreased by more than 1% is 0.514.

A probability distribution that is symmetric about the mean is the normal distribution, sometimes referred to as the Gaussian distribution. It demonstrates that data that are close to the mean occur more frequently than data that are far from the mean.

Given:

61 days in 2009 the Dow decreased by more than 1%.

In 2009

z = (1- (-0.198))/2.331

z = 0.514

In 2010

z = (1-0.078)/0.821

z = 1.123

Thus, number of the 61 days in 2009 the Dow decreased by more than 1% is 0.514.

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Step-by-step explanation: 1 kilometer is equal to 0.621371 miles. To convert 13 kilometers to miles, you can multiply 13 by 0.621371.

13 kilometers * 0.621371 miles/kilometer = 8.042443 miles

So 13 kilometers is equal to 8.042443 miles.

In the simultaneous equations the value of m is 3 and the equation have many solution.

The term called equation in math is known as a condition on a variable (or variables) such that two expressions in the variable have equal value.

Here we have know that the general solution is calculated as,

=> x = (m-5)/(m-3)

and

=> y = 2/(m-3)

When we equate the equations then we get,

=> (m - 5) / (m -3) = 2/(m - 3)

=> m - 5 = 2

=> m = 3

So this dependent system has infinitely many solutions.

When m=3 the equations become look like 3x+3y=5 and 5x+5y=3.

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The probability to win the world series by cubs as per given probability of each game with no ties is equal to 29% ( in percent nearest whole number).

Total number of games played by the teams 'n' = 7

Number of games required to win the world series 'x' = 4

Probability of winning a game by cubs 'p' ( success) = 0.60

1 - p = 1 - 0.60

       = 0.40

Probability of winning the world series by Cubs

= ⁿCₓ pˣ ( 1 - p )ⁿ⁻ˣ

= ⁷C₄ ( 0.60 )⁴ ( 0.40 )⁷⁻⁴

= ( 7! / (4!) ( 7 -4 )! )  ( 0.60 )⁴ ( 0.40 )³

= ( 7 × 5 ) ( 0.0082944 )

= 0.29034

= 29% ( in percent )(nearest whole number)

Therefore, the probability of winning the world series by cubs is equal to 29%.

The above question is incomplete , the complete question is:

In the World Series, the team that wins 4 games out of seven is the champion. Suppose that the Cubs are playing the red Sox (and the world is about to end), and the Cubs have a probability of 3/5 of winning a game over the Sox and there is no ties. What is the probability that the Cubs will win the world series ? Express your answer as a percent rounded to the nearest whole percent.

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The true statement of the inequality is that g is greater than 9 and as a false statement, g is less than 9.

Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.

The symbol a < b indicates that a is smaller than b.

When a > b is used, it indicates that a is bigger than b.

a is less than or equal to b when a notation like a ≤ b.

a is bigger or equal value of an is indicated by the notation a ≥ b.

Given, An inequality of 5g > 45.

Dividing both sides by 5 we have (5/5)g > 45/5.

g > 9.

Now, The true statement is 'g' is greater than 9 and as a false statement 'g' is less than 9.

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1. 9 cans of white paint will she need to mix with 6 cans of blue

2. The rate of white to blue is 9 : 6.

3. The unit rate is 3/2.

Given:

Anne must add 3 cans of white paint to every 2 cans of blue paint.

So, the ratio of white to blue = 3:2

and, for 6 cans of blue paint the white paint needed

= 6 x 3/2

= 3x 3

= 9 cans

2. The rate of white to blue is 9 : 6.

3. The unit rate is 3/2.

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The value of x of the parallelogram is 7 units

The product of the base and height gives the total area of the parallelogram.

The area of a parallelogram can be calculated using the formula:

Area = base * height

A = b * h

where the base is the length of one side of the parallelogram, and height is the perpendicular distance between the two parallel sides.

Given that A = 24 square units, b = 8

A = b * h

28 = 8 * h

28 = 8h

h = 28/8

h = 3.5 units

sin 30 °= h/x (opposite / hypotenuse)

0.5 = 3.5/x

0.5x = 3.5

x = 3.5/0.5

x = 7 units

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Within 5 hours, Trey covered 315 miles. He would need to travel 504 miles in 11 hours at the same speed.

You must ascertain the unit rate. Divide the miles by the hours to arrive at that.

The unit rate should be found to be 63 miles per hour.

You need to double that by 8 (because it takes 8 hours),

Which is 504 miles.

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[tex] \frac{3}{1} = \frac{d}{9} \\ [/tex]

cross multiply...

[tex]d = 27[/tex]

Value of the following expressions for a = 3 + 4j and b  = 3 - 4j , where      j = √−1 are as follow :

1. a + b = 6

2.  a × b = 25

Calculation of the value for the required expression by substituting the given values are:

1. a + b

= ( 3 + 4 j ) + ( 3 - 4j )

Take like terms together we get,

= ( 3 + 3 ) + (  4j - 4j )

= 6 + 0j

= 6

To get the value of second expression substitute the value of 'a' and 'b' we get :

2. a × b

= ( 3 + 4 j ) × ( 3 - 4j )

Apply the formula : ( x + y ) (x - y ) = x ² - y²

= ( 3 )² - ( 4j )²

= 9 - 16j²

Here j = √-1 ⇒ j² = -1

= 9 - 16 (-1 )

= 9 + 16

= 25

Therefore, the value of 1. a + b = 6 and 2. a × b = 25.

The above question is incomplete, the complete question is:

Given a = 3 + 4j and b  = 3 - 4j , where j = √−1 , find (use just pencil and paper, not software or calculators) Find the value of the following :

1. a + b    2. a × b

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A hypothesis is a) prediction of results and b) tentative statement that something maybe true.

A hypothesis is a statement or assumption made about a phenomenon or relationship, used as a starting point for further investigation. It is a tentative explanation for an observation, phenomenon or problem that can be tested through experimentation or additional data collection.

So a hypothesis can be true of false. A hypothesis is not certainly a fact too since it need to be tested.

If a hypothesis is proven to be true through experimentation and data analysis, it becomes a confirmed explanation or prediction. This means that the evidence supports the hypothesis and it can be used as a basis for further study or explanation. However, a true hypothesis is always subject to revision or rejection if new information or evidence becomes available.

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The multiple of 2⅓, add 2¼. is 2 7/12.

A multiple in mathematics is created by multiplying any number by an integer. In other words, if b = na for some integer n, known as the multiplier, it can be said that b is a multiple of a given two numbers, a and b. This is equivalent to stating that b/a is an integer if an is not zero.

In other words, the product of 4 and any natural integer is the multiple of 4. For instance, the result of multiplying 4 by 4 is 16, making 16 a multiple of 4. Examples of multiples of four are 4, 12, 20, 24, and so forth.

Start at 2⅓, add 2¼. I need the first 4 multiples​?

→2 1/3+2 1/4

⇒2 3+4/12

⇒2 7/12

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The quotient of 10 and a number b is the number you get when you divide 10 by b.

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A circular pie cut in 8 pieces and sold at a fractional form 4(2/8) and 5(1/2), yields a total sale of 78 pieces of pie.

What is a circle?

A circle is a closed, two-dimensional object where every point in the plane is equally spaced from a central point. The line of reflection symmetry is formed by all lines that traverse the circle. Additionally, every angle has rotational symmetry around the centre.

A pie is made in a circular shape.

Number of pieces in one pie = 8 pieces

Number of pieces of pie sold by first class of fifth grades = 4(2/8)

Number of pieces of pie sold by second class of fifth grades = 5(1/2)

The total number of pieces sold is = Pieces sold by first class + Pieces sold by second class

To find the pieces sold by first class multiply the fractional form by number of pieces -

4(2/8) × 8

34/8 × 8

34

The first class sold 34 pieces of pie.

To find the pieces sold by second class multiply the fractional form by number of pieces -

5(1/2) × 8

11/2 × 8

11 × 4

44

The second class sold 44 pieces of pie.

The total number of pies sold is -

The total number of pieces sold is = 34 + 44

The total number of pieces sold is = 78

Therefore, 78 pies were sold.

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The dimensions of the square are √2n by √2n

Let's call the number of 1-blocks required to form a square "n".

The dimensions of the square can be represented as x^2 + x * (2 * n / x)

This is so because:

Since the square is a perfect square, both sides of the equation must be equal:

x^2 = x * (2 * n / x)

Solving for x:

x^2 = 2n

Take the square root of both sides

x = √2n

So the dimensions of the square are √2n by √2n

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Using properties of right angle, The value of a will be 32 units.

Particular Right Triangle Formula

The right triangle formula, which states that the square of a right-angled triangle's right triangle is equal to the product of its other two squares, is the fundamental Pythagoras theorem formula.

How long are the other two legs if a hypotenuse of a triangle with a 45°, 45°, and 90° angle is 32 units.

We are aware that Leg: Ankle: Length x = x: x: x2 is the formula for a triangle with a 45°, 45°, and 90° angle.

We can state that x2 = 32 because it is a known that the hypotenuse equals 32. As a result, x = 3 is determined.

After replacing this number of x = 3, the lengths of the empty sides may now be determined.

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Complete question -

Find values of a and b -

The value of [tex]a_{4} = 37[/tex]

We need to find value of [tex]a_{4}[/tex].

A sequence is a collection of elements that are arranged according to a specific pattern.

Arithmetic sequence is a sequence where each consequtive term has a common constant difference.

Now,

Given that value of [tex]a_{1} = 3[/tex]

[tex]a_{n} = 4 a_{n-1} -1[/tex]

Then to find value of [tex]a_{4}[/tex], we can find [tex]a_{2} , a_{3}[/tex]as;

[tex]a_{2} = -2(a_{1} ) - 1\\a_{2} = -2(4) - 1\\a_{2} = -9[/tex]

[tex]a_{3} = -2(a_{2} ) - 1\\a_{3} = -2(9) - 1\\a_{3} = -19[/tex]

Hence, the value of [tex]a_{4}[/tex] as;

[tex]a_{4} = -2(a_{3} ) - 1\\a_{4} = -2(-19) - 1\\a_{4} = 37[/tex]

Therefore, The value of [tex]a_{4}[/tex]= 37.

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Answer:

  A.  44.5w + 106.37 = 239.87; w = 3

Step-by-step explanation:

You want an equation and solution for finding the number of weeks (w) required to earn enough for a $239.87 gaming chair if you have saved $106.37 and earn $44.50 per week.

The total amount saved will be the sum of the amount already saved ($106.37) and the amount of earnings. We want that total to be $239.87.

Earnings are $44.50 per week. After w weeks, they total 44.50w. Then the amount saved will be ...

  earnings + already saved = total savings

  44.50w + 106.37 = 239.87

We can find the value of w by subtracting 106.37 from this equation:

  44.50w = 133.50

  w = 3 . . . . . . . . . . . . . . . divide by 44.50

The equation and solution are ...

  44.5w +106.37 = 239.87; w = 3 . . . . . . . matches choice A

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The substitution that should best be used in solving the recurrence T(n) = 5T(4n/3) + 2 is "n = (4/3)^k".

The idea behind choosing a substitution for a recurrence relation is to simplify the relation and make it easier to solve. In this case, we want to simplify T(4n/3) to something involving T(n). By setting n = (4/3)^k, we can rewrite T(4n/3) as T((4/3)^(k+1)).

This substitution allows us to express T(4n/3) in terms of T(n) and simplify the relation.

T(n) = 5T(4n/3) + 2

becomes

T(n) = 5T((4/3)^(k+1)) + 2

We can then use this substitution to solve the relation using, for example, the Master Theorem.

In conclusion, the substitution "n = (4/3)^k" is the best option because it simplifies the relation and makes it easier to solve.

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If the Heptagon  has been dissected into five triangle , the the proof that sum of the interior angle of any heptagon is always 900° is shown below .

To prove that the sum of the interior angles of a heptagon is 900°, we can use the five triangles that have been dissected from the Heptagon, as follows: let the 5 triangles be A , B , C , D and E .

we know that the sum of interior angle of triangle is always 180° ;

Triangle A: The sum of the angles of triangle A is 180°.

Triangle B: The sum of the angles of triangle B is 180°.

Triangle C: The sum of the angles of triangle C is 180°.

Triangle D: The sum of the angles of triangle D is 180°.

Triangle E: The sum of the angles of triangle E is 180°.

So , the sum of all the angles in the five triangles is 180°×5 = 900°, which is also equal to the sum of the interior angles of the heptagon.

Therefore , we can conclude that the sum of the interior angles of any heptagon is always 900°.

The given question is incomplete , the complete question is

The Heptagon  has been dissected into five triangle with angle labeled , use the five triangle to prove that the sum of the interior angle of any heptagon is always 900° .

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The length of the other side of the triangle is 30 cm. Thus, option b is correct.

The hypotenuse's square is equal to the sum of the squares of the other two sides if a triangle has a straight angle (90 degrees), according to the Pythagoras theorem. Keep in mind that BC² = AB² + AC² in the triangle ABC signifies this. Base AB, height AC, and hypotenuse BC are all used in this equation. The longest side of a right-angled triangle is its hypotenuse, it should be emphasized.

Here given that  hypotenuse = 32 and one side = 16

Using Pythagoras theorem

32² = 16² + x²

x = [tex]\sqrt{32^2 - 16^2}[/tex]

x = 30

Hence, the length of the other side of the triangle is 30 cm.

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the inequality model given by eva is wrong, the correct answer is 5/3(y-1) = x.

What is inequality?

Inequalities are mathematical expressions where neither side is equal. In inequality, as opposed to equations, we compare two values. Less than (or less than or equal to), greater than (or greater than or equal to), or not equal to signs are used in place of the equal sign in between. Sometimes it can be about a "not equal to" relationship, where one thing is more than the other or less than. In mathematics, an inequality is a relationship that results in a non-equal comparison between two numbers or other mathematical expressions.

The given inequality is y =3/5x +1

Now subtract 1 from both sides y-1 = 3/5x

further taking 5/3 in both side we should have

5/3(y-1) = x

Hence, the inequality model given by eva is wrong, the correct answer is 5/3(y-1) = x.

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To display the distribution of grades (A, B, C, D, F) in a course, it would be correct to use histogram

a) We note that the given histogram contains a

very large number of bars and that most bars are quite low while also very narrow.

The display is then not appropriate, as the histogram should contain less bars to make the shape in the distribution more obvious.

(b) The distribution of grades is skewed to the left as the highest bars in the histogram are to the right in the graph.

The grades appear to be centered roughly at 170, while the graders vary from about 75 to 190.

Moreover, there appear to be a few outliers in the distribution, as there are gaps between the bars of the histogram.

since Histogram contains too many bars and  Left-skewed histogram can be considered to display.

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The fraction by which the income decreased in August is given as follows:

1/5.

The total income is given by a proportional relationship of the cost per item and the number of items sold, hence:

y = kx.

In which:

For August, the parameters are given as follows:

Hence the August expression is given as follows:

y = 4k/3 x 3x/5

y = 12kx/15

y = 4kx/5.

Hence the decrease is given as follows:

1 - 4/5 = 5/5 - 4/5 = 1/5.

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The number of years taken for the tree to reach the same height found using mathematical expression is 2 years.

What is an expression?

Mathematical statements are called expressions if they have at least two terms that are related by an operator and contain either numbers, variables, or both. Addition, subtraction, multiplication, and division are all possible mathematical operations.

Let x be the number of years taken to reach the same height.

Height of the first tree after x years,

12 + 2.5(x)

Height of the second tree after x years,

15 + 1(x)

Given that both heights are the same after x years.

So,

12 + 2.5x = 15 + x

1.5x = 3

x = 2

Therefore the number of years taken for the tree to reach the same height is 2 years.

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In this particular case, the p-value is 3.2578x10⁻¹³, which is much smaller than the level of significance of 0.05.

A chi-square test of goodness of fit is a statistical test used to compare the observed data to a theoretical distribution.

In mathematical terms, the null hypothesis states that the observed data follows the theoretical distribution, and is represented by

H0= The observed data follows the theoretical distribution. The alternative hypothesis, which is the opposite of the null hypothesis, states that the observed data does not follow the theoretical distribution, and is represented by

HA= The observed data does not follow the theoretical distribution.

Based on the p-value of 3.2578x10⁻¹³ and the level of significance of 0.05, we can make the following statistical decision:

p-value < α (3.2578x10-13 < 0.05),

Therefore reject H0 and accept HA.

n other words, we can conclude that the observed data does not follow the theoretical distribution, and there is strong evidence to support this conclusion. The chi-square test of goodness of fit is a useful tool for making statistical decisions about the distribution of data.

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