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The slope of the line can be calculated as  [tex]m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}[/tex].

Correct options are (A) and (D).

What is the slope of the line?

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.

As the definition of the slope says that it is the ratio of rise by run.

so we can make the formula as

[tex]m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}[/tex]

Hence, the slope of the line can be calculated as  [tex]m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}[/tex].

for example lets have two points (9,1) and (8,4) so the value of the slope can be calculated as

4-1 / 8-9

3/-1

= -3

Therefore, the slope of the line passing through the given points is -3

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Answer:

slope of line w = - [tex]\frac{8}{4}[/tex]

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

y = [tex]\frac{5}{8}[/tex] x + [tex]\frac{9}{4}[/tex] ← is in slope- intercept form

with slope m = [tex]\frac{5}{8}[/tex]

given a line with slope m then the slope of a line perpendicular to it is

[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{\frac{5}{8} }[/tex] = - [tex]\frac{8}{5}[/tex]

The volume of the rectangular prism can be calculated using the formula , l x b x h , where l = length , b = base and h is the height.

The SI unit for rectangular prism volume is m³.

A polyhedron which has two congruent and a parallel bases  is known as a  rectangular prism. It also known as a cuboid.

A rectangular prism has 6 faces and 12 edges. It is referred to as a prism because of the length of its cross-section.

We study prisms under the domain of geometry .

They are of two different types,

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Answer:

4 nautical miles per hour

Step-by-step explanation:

Speed = Distance / Time

Distance = Speed x time

D1 = D2

S1 = 16 nautical miles per hour

t1  =10 hours

We know S2 is unknown and the t2 = 8 hours

So, we take

S1 x T1 = S2 x T2

16 x 10 = S2 x 8

S2 = 16 x 10 / 8

S2 = 20 nautical miles per hour

20 - 16 = 4 nautical miles per hour

So, it has to increase its speed by 4 nautical miles per hour

Answer: 4 nautical miles per hour

It is given that in the first case, the ship takes 10 hours to travel at the speed of 16 nautical miles per hour. distance=speed × time=16×10=160 nautical miles. Therefore, the answer to the above question is 4 nautical miles per hour.

Step-by-step explanation:

To solve for x in an angle, you first need to identify what type of angle you are dealing with. Depending on the type of angle, you will use different formulas to solve for x. For example, if you are dealing with a right angle, you can use the Pythagorean Theorem to solve for x.

1. Identify the given information:-

First, identify the type of triangle and the angles that are given.

2. Use the appropriate formula:-

If it is a right triangle, use the Pythagorean Theorem to solve for the missing side. If it is an isosceles triangle, use the law of sines to solve for the missing angle.

3. Substitute the given values into the appropriate formula:-

Substitute the given values into the appropriate formula and simplify.

4. Solve for the unknown:-

Solve the equation for the unknown value (x).

5. Check your answer:-

Check your answer by plugging it back into the original equation to make sure it is correct.

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Two ratios are referred to as being in proportion if they are the same. If the four elements are a, b, c, and d in that order, then a : b :: c : d = a/b = c/d.

The following formulas also relate to proportion:

a : b :: c : d = a/b = c/d

In which,

The ratio of averages to extremes is 1:1. You can write this as ad = bc.

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Answer:

C. All real number

Step-by-step explanation:

Most of the f(x) functions have the domain of (negative infinity, positive infinity) and the function in the picture is the same. So, the answer is C

The 4 types of algebra are  Elementary Algebra,  Linear Algebra, Abstract Algebra, Boolean Algebra.

1. Elementary Algebra - This type of algebra deals with basic operations such as addition, subtraction, multiplication, and division. An example of elementary algebra is solving for x in the equation 2x + 3 = 7. In this equation, the variable x has a value of 4.

2. Linear Algebra - This type of algebra deals with linear equations and linear transformation. An example of linear algebra is solving for the vector x in the equation Ax = b, where A is a matrix and b is a vector.

3. Abstract Algebra - This type of algebra deals with abstract concepts such as groups, rings, and fields. An example of abstract algebra is finding the inverse of a matrix A. The inverse of A can be found by solving the equation A x A-1 = I, where I is the identity matrix.

4. Boolean Algebra - This type of algebra deals with logical operations such as OR, AND, NOT, and XOR. An example of Boolean algebra is finding the logical complement of a statement P. The logical complement of P can be found by solving the equation P OR NOT P = TRUE.

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The Fourier series of one period a(x) on [tex]-\pi[/tex] to [tex]\pi[/tex] is 23.096.

What is Fourier series ?

A fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier Series makes use of the orthogonality relationships of the sine and cosine functions. Expand the function f(x) = ex in the interval [ – π , π ] using Fourier series formula.

Have given,

a(x) on domain ([tex]-\pi ,\pi[/tex])

Let ,

I = ∫[tex]\pi[/tex] [tex]e^{x}[/tex] dx

on integration

I = [tex][e^{x}]^{\pi }_{-\pi }[/tex]

I = [tex]e^{\pi } - e^{-\pi }[/tex]

I = 23.14 - 0.043

Thus, I = 23.096

The Fourier series of one period a(x) on [tex]-\pi[/tex] to [tex]\pi[/tex] is 23.096.

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The domain and scope of the identity function are the same. Equation for the identity function is  f(x) = x, or y = x.

An statement, rule, or law in mathematics that specifies the connection between an independent variable and a dependent variable (the dependent variable). A relationship between a group of inputs and one output each is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. Y=X2 is an illustration of this. You only receive one output for y if you enter anything for x. The fact that x is the input value leads us to argue that y is a function of x.

identity function are the same.

Equation for the identity function is

f(x) = x, or y = x.

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0.707 is obtained by multiplying one by root 2 . Commonly used to represent the square root is the symbol '√'. Using this symbol, the number "x" can be represented as the square root of "x" by writing '√x'.

A mathematical term for a factor that, when multiplied by itself, equals the original value of the original integer is the square root. Early in the second millennium BC, the Babylonians devised effective methods for approximating square roots.

here

=√2 / 2

= 1 / √2

=√2 / √2=1

=1√2 / √2√2

=√2 / 2

= 0.707

so , 0.707 is obtained by multiplying one by root 2.

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The graph of 4x7 40x6 100x5 is a straight line that decreases in slope as x increases.

The graph of 4x7 40x6 100x5 is a linear graph with a positive slope. The graph starts from the point (4,7) and increases in a linear fashion. At the point (40,6) the value of y is 6 and the value of x is 40. The graph then continues to increase as the value of x increases. At the point (100,5) the value of y is 5 and the value of x is 100. The graph of 4x7 40x6 100x5 is a linear graph with a positive slope, increasing from 4x7 to 100x5, and having a y-intercept of 0.The graph then continues to decrease in slope as x increases and reaches the point (40,6) and finally (100,5). This graph is a linear equation which can be written as

y = mx + b

where m is the slope and b is the y-intercept. . The y-intercept is the point where the graph crosses the y-axis and can be found by substituting 0 for x in the equation.The slope is calculated by subtracting the y-values and dividing by the x-values, which gives us a slope of -1/96. The y-intercept is calculated by substituting 0 for x in the equation, which gives us a y-intercept of 7.

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The number 119.43 when approximated to 1 significant figure is 100

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

Number = 119.43

The number is to be approximated to 1 significant figure

This means that we approximate all number leaving only the first digit

When this is done, we have

Approximation = 100

Hence, the number in 1 significant figure is 100

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The legs of a right isosceles triangle are the height and the base of the right isosceles triangle.

It is known that an isosceles triangle is a triangle which has two equal sides.

It is a right isosceles triangle when one side of the triangle is a perpendicular dropped from the vertex and forms a right angle with the base.

As the two sides are equal, the base also turns out to be perpendicular to the height forming a hypotenuse of the third side.

Hence, the two perpendiculars base and height are considered to be the legs of the right isosceles triangle.

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As the correlation coefficient moves from 0 to 1 or -1 to 0, the points would be closer to the line. Option A

The term correlation coefficient is used to show the level of the relationship between the variables. If the correlation coefficient is very close to 1 then it means that there is greater association between the variables. If it is far from one then it means that the variables are not associated at all.

We know that the regression lines show how variables can be associated and they can be used to show us the relationship between variables.

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The coefficients in a balanced equation compare quantities of two substances by the ratio of the substances' coefficients equals the ratio of their number of moles.

A mole is defined as 6.02214076 × 10²³ of some chemical unit, be it atoms, molecules, ions, or others. The mole is a convenient unit to use because of the great number of atoms, molecules, or others in any substance.

In many chemical situations, mole ratios are employed as conversion factors between products and reactants.

In a balanced chemical equation, the coefficients in front of the formulae can be used to determine the mole ratio.

Complete question:

How do the coefficients in a balanced equation compare quantities of two substances?

A. The ratio of the coefficients equals the ratio of the masses of the

substances

B. The sum of the coefficients of the reactants equals the sum of the

products

C. The ratio of the substances' coefficients equals the ratio of their

number of moles.

D. The coefficient of the reactant tells how many product molecules

will form.

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A mole is defined as 6.02214076 × 10²³ of some chemical unit, be it atoms, molecules, ions, or others. The mole is a convenient unit to use because of the great number of atoms, molecules, or others in any substance.

In many chemical situations, mole ratios are employed as conversion factors between products and reactants.

Complete question:

How do the coefficients in a balanced equation compare quantities of two substances?

A. The ratio of the coefficients equals the ratio of the masses of the

substances

B. The sum of the coefficients of the reactants equals the sum of the

products

C. The ratio of the substances' coefficients equals the ratio of their

number of moles.

D. The coefficient of the reactant tells how many product molecules

will form.

The value of the sine of angle D to the nearest hundredth will be 0.25.

The Law of Sines The law of trigonometric functions, sine law, sine formula, or sinusoidal rule is a trigonometric equation that relates the sizes of any triangle's edges to the sines of its orientations.

The sine law in the triangle ΔCDE is given as,

CD / sin E = DE / sin C = EC / sin D

In triangle ΔCED, the measure of angle ∠E = 90°, ED = 63, DC = 65, and CE = 16. Then we have

65 / sin 90° = 63 / sin C = 16 / sinD

Compare the first and the last term, then we have

65 / sin 90° = 16 / sin D

65 = 16 / sin D

sin D = 16 / 65

D = 14.25°

The worth of the sine of point D to the closest 100th will be 0.25.

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The x-intercept is the point where a line crosses the x-axis, and the y-intercept is the point where a line crosses the y-axis.

Given data :

Find x-intercept

The x-intercept is where the graph crosses the x-axis. To find the x-intercept, we set y1=0 and then solve for x.

-4x + y = 16

-4x + 1(0) = 16

x1 = -4    y1 = 0

Find y-intercept

The y-intercept is where the graph crosses the y-axis. To find the y-intercept, we set x2=0 and then solve for y.

-4x + y = 16

-4(0) + y = 16

y2 = 16    x2 = 0

Solve for x

To solve for x, we solve the equation so the variable x is by itself on the left side:

-4x + y = 16

x = -4 - 0.25y

Solve for y

To solve for y, we solve the equation so the variable y is by itself on the left side:

-4x + y = 16

y = 16 + 4x

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We can find the missing value of a triangle on a calculator in the following way,

If the value is a side of a triangle and the triangle is a right-angle triangle, we apply the Pythagoras Theorem. The formula of this theorem is a²+ b²= h², where h is the hypotenuse of the triangle, and a and b are the remaining two sides.

If the triangle is any other triangle, we use the law of sine, which is:

a/sin(A)= b/sin(B)= c/sin(C), where a, b, and c are sides of the triangle and A, B, and C are the angles opposite to the respective sides.

If the value is an angle of the triangle, we use the formula

x=180-(y+z), where x is the missing angle, and y and z are the remaining two angles of the triangle.

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To determine if a graph has a positive or negative leading coefficient, look at the sign of the coefficient of the first term of the equation for the graph. If the coefficient is positive, the graph has a positive leading coefficient. If the coefficient is negative, the graph has a negative leading coefficient.

To determine the sign of the leading coefficient of a graph, first look at the equation for the graph. The leading coefficient of a graph is the coefficient of the first term in the equation, so find the coefficient of the first term. If the coefficient of the first term is positive, then the leading coefficient is positive. If the coefficient of the first term is negative, then the leading coefficient is negative. For example, if the equation is y = 4x2 + 3x + 5, the coefficient of the first term is 4, and so the leading coefficient is positive. Conversely, if the equation is y = -2x2 + 3x + 5, the coefficient of the first term is -2, and so the leading coefficient is negative.

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In total, 15 to 16 gallons or 55 to 60 liters of paint were used to paint the interior of your 2000 square foot home.

It might be challenging to calculate or estimate the precise amount of paint that you will need when painting your home. Every paint has a varied coverage, which determines how many square feet may be painted with each gallon of paint.

According to a rule of thumb and general recommendations, you will need about 15 to 16 gallons, or 55 to 60 liters, of interior paint for a G+1/2 storey house with a 2000 square foot area, as well as 3 gallons of paint for interior trim for two to three coats, which includes painting the interior surface/wall area, overhangs, and soffits. In total, 15 to 16 gallons or 55 to 60 litres of paint were used to paint the interior of your 2000 square foot home.

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To graph a linear equation which is in the form of a straight line one easy method is by identifying two points on that line and drawing a straight line passing through the two points.

A linear equation of the form y = mx + c having slope m and y-intercept c is a straight line.

To find the two points, we can substitute a value for one variable and find the corresponding value of the other variable. In this case

For x = 0

y = -5×0 + 3

y = 3

So (0, 3) is a point on the line y = -5x + 3.

Similarly for y = 0

0 = -5x + 3

5x = 3

x = 3/5 = 0.6

So (0.6, 0) is a point on the line y = -5x + 3.

Now we can graph y = -5x + 3 by joining (0, 3) and (0.6, 0). Refer the attached figure for the graph.

--The question is incomplete, answering to the question below--

"How do you graph y = -5x + 3?"

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Can you explain the rest of the problem please?

Answer: (read the explanation)

Step-by-step explanation:

To graph the solution set to the given system of inequalities, we can begin by graphing the two inequalities separately. For the first inequality, y > x^2 – 2, we can graph the function y = x^2 – 2 on the coordinate plane and shade the region above the graph. This will represent the values of x and y that satisfy the inequality.

For the second inequality, y ≥ –x^2 + 5, we can graph the function y = –x^2 + 5 on the coordinate plane and shade the region below the graph. This will represent the values of x and y that satisfy the inequality.

Next, we can combine the two graphs by intersecting the shaded regions. The resulting graph will show the solution set to the system of inequalities. The solution set can be identified as the points on the coordinate plane that are contained within the shaded region on the combined graph.

Overall, to modify the graphs of f(x) and g(x) to graph the solution set to the given system of inequalities, we can graph the functions y = x^2 – 2 and y = –x^2 + 5 on the coordinate plane and shade the regions that satisfy the inequalities. The solution set can then be identified as the points within the shaded region on the resulting combined graph.

The coefficient of x^11 in the binomial expansion of (x - 3)^12 is -36.

What is binomial expansion?

The binomial theorem in basic algebra explains how a binomial's powers can be expanded algebraically.

Exponents of x and y added together always equal n. The binomial coefficients that are equally spaced from the start and end are equal, i.e., nC0 = nCn, nC1 = nCn-1, nC2 = nCn-2, etc.

Consider, the given expression (x - 3)^12

We have to find the coefficient of x^11 in the expansion of (x - 3)^12.

Applying the binomial theorem,

(a + b)^n =  Σ(i = 0 to n) nCi a^(n- i)b^i

Here a = x, b = -3.

⇒ (x - 3)^12 =  Σ(i = 0 to 12) 12Ci x^(12 - i)(-3)^i

[tex](x -3)^1^2 = x^ 1^2-36^1^1+59x^1^0 - 5940x^9 + 40095x^8 - 192456x^7+ 673596x^6 - 1732104x^5 + 3267695x^4-4330260x^3 + 3897234x^2- 2125764x+531441[/tex]

Hence, the coefficient of x^11 in the binomial expansion of (x - 3)^12 is -36.

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The graph that grows the fastest is green i.e., the graph of exponential function.

In this question we need to determine which  graph grows the fastest.

Given graph contains graph of quadratic function (blue colored graph), exponential function (green colored graph), cubic function (pink colored graph) and linear function (black straight  line)

We know that the exponential function grows faster because it grows by a factor that is multiplied by the previous y-value instead of being added

like the linear function.

In given graph the green function is an exponential function and it grows the fastest.

Therefore, the green graph grows the fastest.

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Consider following image.

First make the graph of the functions, then  if the two graphs are symmetric with respect to the line y = x (mirror images over y = x ), then they are inverse functions.

Symmetric

A figure or shape that can be divided into two equal parts by a line is called symmetric figures.

Inverse Functions

The inverse function of a function f  is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by F^-1.

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