Austin graphed a system of linear equations as shown. What x -value represents the solution to the given system of linear equations?

The provided collection of observations has a standard deviation of 2.8581.

The standard deviation in statistics is a measure of the degree of variation or dispersion in a set of values. A low standard deviation implies that the values are close to the set's mean, whereas a high standard deviation shows that the values are spread out over a larger range. A large standard deviation indicates that the data is widely dispersed (less dependable), whereas a low standard deviation indicates that the data is tightly grouped around the mean (more reliable).

Here,

n=9

sum=45

mean=5

variance=sum(xₙ-mean)/n

=(1-5)/9+.........+(9-5)/9

=60/9

=6.67

standard deviation=√(6.67)

=2.581

The standard deviation of given set of observation is 2.8581.

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The gradient of the vector field of function f(x, y, z) is (12x,2y,2z)

The term gradient of the vector in math is calculated as the vector whose components are given by the partial derivatives of the function and it can be evaluated at any given point, and the gradient represents the direction and magnitude of greatest increase of the function at that point.

Here we have given the function f(x, y, z) = 6x² + y² + z²

Now we have to calculate the gradient of the function by using the following step.

In the given question, we have identified the values of the following as,

=> x = 6x²

=> y = y²

=> z = z²

Now, in order to find the gradient of a function (which is a vector), differentiate the function with respect to each variable.

=> ∇f=(∂f/∂x, ∂f/∂y, ∂f/∂z)

Now, the value of

=> ∂f/∂x = 12x

=> ∂f/∂y = 2y

=> ∂f/∂z = 2z

Now, we have to plug in the point:

=> ∇f(x, y, z) = (12x,2y,2z)

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A graph is said to be infinite when it has an infinite number of points that could be connected, or an infinite number of lines, curves, or shapes that could be drawn on the graph.

A graph is a visual representation of data or information, typically plotted on an x-axis and a y-axis. A graph is said to be infinite when it has an infinite number of points that could be connected, or an infinite number of lines, curves, or shapes that could be drawn on the graph. For example, a line graph without an end point is considered infinite because it theoretically continues forever. Similarly, a circle graph with no beginning or end point could also be considered infinite. Other types of graphs that can be considered infinite include bar graphs, area graphs, scatter plots, and pie charts. When determining whether a graph is infinite, it is important to consider the purpose of the graph and its data. If the graph is meant to represent an infinite set of data points, then it can be considered infinite. Likewise, if the graph is meant to represent a function that goes on forever, then it should also be considered infinite. Ultimately, the definition of infinity can vary depending on the situation, so it is important to consider all aspects of the graph before making a judgement.

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Answer:

value of Johnny's investments today, at age 50, is approximately 18.39 dollars.....................

Step-by-step explanation:

To find the present value of Johnny's investments, you will need to use the formula for continuous compound interest:

where PV is the present value, A is the final amount, r is the annual interest rate, and t is the number of years.

Solving this equation gives a present value of approximately 18.39 dollars. This means that the value of Johnny's investments today, at age 50, is approximately 18.39 dollars, given the original investment amount and the interest rate.

It is important to note that this calculation assumes that the interest is compounded continuously, which means that the interest is compounded infinitely often over the course of the year. This results in a slightly higher present value than if the interest were compounded annually or quarterly.

The sample data does not provide strong support for the claim that the mean salary offer for mathematics and statistics graduates of this university is greater than the national average of $62,915.

At the null hypothesis, it is tested if there is not enough evidence that the mean is greater than 62915, that is:

[tex]H_0: \mu \leq 62915[/tex]

At the alternative hypothesis, it is tested if there is strong evidence that the mean is greater than 62915, that is:

[tex]H_1: \mu > 62915[/tex]

We have a right-tailed test, as we are testing if the mean is greater than a value.

The critical value for a right-tailed test, using the t-distribution, with a significance level of 0.05(standard) and 50 - 1 = 49 df, is of:

t = 1.6766.

Hence the decision is taken as follows:

The equation for the test statistic is presented as follows:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

In which:

The parameters for this problem are given as follows:

[tex]\overline{x} = 63400, \mu = 62915, s = 3900, n = 50[/tex]

Hence the test statistic is of:

t = (63400 - 62915)/(3900/square root(50))

t = 0.88.

0.88 < 1.6766, hence it does not provide strong evidence.

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Therre are many examples but let us take 6 as an example of arithmetic number.

The area of mathematics known as arithmetic deals with the study of numbers and the numerous operations that can be performed on them. Addition, subtraction, multiplication, and division are the fundamental mathematical operations.

For an integer to be considered arithmetic, its average positive divisors must also be integers.

Since 1,2,3 and 6 are the divisors of 6. The average of divisors of 6 is:

[tex]\frac{1+2+3+6}{4} =3[/tex]

where 3 is again an integer so 6 is an arithmetic number.

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You can find the missing number in an array without loop by counting the number of elements in the array, and then looping through the elements.

Here is a simple program to find the missing numbers in an integer array.

ArrayList<Integer> arr = new ArrayList<Integer>();

int a[] = { 1,3,4,5,6,7,10 };

int j = a[0];

for (int i=0;i<a.length;i++)

{

   if (j==a[i])

   {

       j++;

       continue;

   }

   else

   {

       arr.add(j);

       i--;

   j++;

   }

}

System.out.println("missing numbers are ");

for(int r : arr)

{

   System.out.println(" " + r);

}

You can find the missing number in an array without loop by counting the number of elements in the array, and then looping through the elements.

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There were 160 babies born in a Maine hospital in July, with 3/5 being girls. 70 babies weighed 8 pounds or more, with 50 of them being boys. 14 boys, 64 girls, 76 boys, 20 girls, and 90 babies weighed less than 8 pounds.

The below attached table shows the number of babies born in a hospital in Maine last July, broken down by gender and weight. The first column lists the gender categories: boys, girls, and total. The second column lists the number of babies who weighed less than 8 pounds (a, c, and e). The third column lists the number of babies who weighed 8 pounds or more (50, d, and 70). The fourth column lists the total number of babies (b, 96, and 160). According to the data, 14 boys weighed less than 8 pounds, 64 girls weighed less than 8 pounds, 76 boys weighed 8 pounds or more, 20 girls weighed 8 pounds or more, and 90 babies in total weighed less than 8 pounds.

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In a hospital in Maine, 160 infants were born in July, with 3/5 being females. 50 of the 70 infants weighed at least 8 pounds, and 70 of them were boys. Less than 8 pounds were carried by 14 boys, 64 girls, 76 boys, 20 girls, and 90 infants.

The data below, which is an attachment, details the number of infants born in a hospital in Maine in July of last year, broken down by weight and gender. Boys, girls, and all other genders are listed in the first column. The number of infants who weighed less than 8 pounds is shown in the second column (a, c, and e). The number of infants weighing 8 pounds or more is shown in the third column (50, d, and 70). The total number of infants is shown in the fourth column (b, 96, and 160). The statistics showed that 64 girls and 76 boys each weighed less than 8 pounds, 20 girls and 76 boys each weighed more than 8 pounds, and 90 newborns overall weighed less than 8 pounds.

In the given graph with the function, x = -9 if f(x) = 7

An illustration of a relation is a function graph. In set theory and modern mathematical foundations, a function is actually equal to its graph. It is often helpful to think of functions as mappings, which include the sets that constitute the domain and the codomain in addition to the relationship between input and output.

A codomain should be considered, for instance, when determining whether a function is onto (surjective) or not. The codomain is not determined by the graph of a function alone. The terms "function" and "graph of a function" are frequently used because, although they refer to the same thing, they express different viewpoints on it.

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As per the linear equation, the value of correlation coefficient we have identified that option (a) is correct.

The term linear equation refers in which the highest power of the variable is always 1 and it is also known as a one-degree equation.

Here we have given that Nadeem keeps track of the number of times he visits the library and the number of books he checks out.

And we need to find the correlation if you model his data with a linear equation

While we looking into the given question, we have identified that Nadeem keeps track of the number of times he visits the library and the number of books he checks out.

And based on the given details we have identified the following values from the given tables,

=> ∑xy = 285

=> ∑x = 33

=> ∑y = 51

=> y = 8.5

=> x = 5.5

=> ∑x² = 199

=> ∑x·∑y = 33×51

=> (∑x)² = 1,089

=> (∑y)² = 51

Therefore, the value of b is calculated as,

=> [(6 x 285) - (33 x 51)] / [6 x 199 - 1089]

=> 0.2571

And the value of a is calculated as,

=> 8.5 - 0.2575 × 5.5 = 7.08575

Then the correlation coefficient, r, is calculated as,

=> r = [(6 x 285) - (33 x 51)] / √[6 x 199 - 1089 x 6 x 471 - 51²]

When we simplify this one then we get,

=> r = 0.176

Based on the value of correlation coefficient we have identified that option (a) is correct.

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For solving algebra Determine the issue, Describe your knowledge, Plan beforehand, Execute the plan ,and Check to see whether the response makes sense.

One of the many different mathematical disciplines is algebra. Algebra, a common thread that runs through nearly all of mathematics, is, roughly speaking, the study of mathematical symbols and the rules for using these symbols in formulae. With the use of mathematical expressions, algebra is the area of mathematics that helps to depict situations or problems. We employ integers that have a fixed or definite value in algebra, such as 2, 7, 0.068, etc. In addition to using numbers, algebra also makes use of variables like x, y, and z.

to do algebra for beginners - Determine the issue, specify your knowledge, Plan, execute, and confirm that the solution makes sense.

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Our Median will be 37 when the Mode is 15 and the Mean is 48.

Relationship between Mean Median and Mode?

An empirically established link exists between the mean, median, and mode in statistics. The average difference between the mean and the mode is three times greater than the difference between the mean and the median, according to observations of innumerable data sets. The equation for this relationship is:

Mode - Mean = 3*(Mean – Median).

on solving this we get:

Mode = 3 * Median - 2 * Mean

In our question, it is given that the Mode is 15 and the Mean is 48 and we have to calculate the Median.

Putting these values in our formula, we get:

15 = 3 * Median - 2 * 48

3 * Median = 15 + 96

Median = 111/3 = 37.

So, finally, our result will be 37.

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We can determine the age by dividing the total number of half-lives by the half-life decay constant of the parent atom (again, this value is determined in a laboratory).

The amount of time needed for a quantity (of substance) to decrease to half of its initial value is known as the half-life (symbol t12).

In nuclear physics, the phrase is frequently used to indicate how rapidly unstable atoms decay radioactively or how long stable atoms last.

By dividing the total number of half-lives by the parent atom's half-life decay constant, we may compute the age (again, this value is determined in a laboratory).

The phrase is also used more broadly to describe any kind of exponential decay (or, very infrequently, nonexponential decay).

The biological half-life of medications and other compounds in the human body, for instance, is a term used in the medical sciences.

Half-opposite of life's exponential growth is time's doubling.

Therefore, we can determine the age by dividing the total number of half-lives by the half-life decay constant of the parent atom (again, this value is determined in a laboratory).

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When X and Y axes are rotated through an angle θ to form X' and Y' axes, then the relation between the axes and the angle is given by,

[tex]\rm{X=X'\cos\theta-Y'\sin\theta}[/tex]

[tex]\rm{Y=X'\sin\theta+Y'\cos\theta}[/tex]

Then x and y are replaced in terms of x' and y' as,

[tex]\rm{x=x'\cos\theta-y'\sin\theta}[/tex]

[tex]\rm{y=x'\sin\theta+y'\cos\theta}[/tex]

Here θ = π/6. Then,

[tex]\rm{x=x'\cos\left(\dfrac{\pi}{6}\right)-y'\sin\left(\dfrac{\pi}{6}\right)=\dfrac{x'\sqrt3-y'}{2}}[/tex]

[tex]\rm{y=x'\sin\left(\dfrac{\pi}{6}\right)+y'\cos\left(\dfrac{\pi}{6}\right)=\dfrac{x'+y'\sqrt3}{2}}[/tex]

We are asked to find the transformed equation for,

[tex]\longrightarrow\rm{P:x^2+2\sqrt3\,xy-y^2=2a^2}[/tex]

We will give substitutions for x and y to find it.

[tex]\small\text{$\longrightarrow\rm{P':\left(\dfrac{x'\sqrt3-y'}{2}\right)^2+2\sqrt3\left(\dfrac{x'\sqrt3-y'}{2}\right)\left(\dfrac{x'+y'\sqrt3}{2}\right)-\left(\dfrac{x'+y'\sqrt3}{2}\right)^2=2a^2}$}[/tex]

Simplify this equation to get,

[tex]\small\text{$\longrightarrow\rm{P':(x')^2-(y')^2=a^2}$}[/tex]

Hence the transformed equation is,

[tex]\longrightarrow\rm{\underline{\underline{x^2-y^2=a^2}}}[/tex]

Answer: 1/(16^13)

Step-by-step explanation: Another way of rewriting this would be

(8 - xy + x)^(2-3y). Plugging in numbers, this results to

[8 - (-2)(5) + (-2)]^[2 - 3(5)]

which can then be simplified to

(8+10-2)^(-13) = 16^(-13) = 1/(16^13) which has a pretty big number in the denominator

5 turning points are in the graph of the polynomial function given in the picture.

Given that,

We have to find how many turning points are in the graph of the polynomial function given in the picture.

We know that,

Turning points are places on a graph where the direction of the line switches from upward to downward and downward to upward

The graph's direction is changing at five separate points, and those points are:

(-5,6) ,(-3,-3) ,(0.5,0.5) ,(3,2.5) ,(5,5)

Therefore, 5 turning points are in the graph of the polynomial function given in the picture.

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Answer: -1324 spaces left in the car park.

Step-by-step explanation: There are 8 floors in the multi-storey car park and each floor has 5 rows of 20 cars, so there are 8 * 5 = <<85=40>>40 rows in the car park in total.

If each row has space for 20 cars, then the total number of parking spaces in the car park is 40 * 20 = <<4020=800>>800.

If 2124 cars have already parked, then there are 800 - 2124 = <<800-2124=-1324>>-1324 spaces left in the car park.

It is not possible for the car park to have negative spaces, so the answer must be 0. Therefore, if 2124 cars have already parked, there are 0 spaces left in the car park.

The measures of m and n are found to be m = 3 and n = 6 .

A more general definition of a diagonal of a polygon is a line segment that joins two vertices of the polygon which are not already joined by an edge of the polygon.

A diagonal of a rectangle divides it into two congruent right triangles. The diagonals of a rectangle are the same length. A quadrilateral whose diagonals bisect each other, intersect at right angles, and are congruent must be a square.

Top diagonals are congruent

9 = 3m

solve for m

m = 3

bottom diagonals are congruent

m + 8 = 2n -1

plug in 3 for m

3 + 8 = 2n -1

solve for n

11 +1 = 2n

12 = 2n

n=6

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Taking into account the rule of three, the vehicle cover a distance of 60 km in 45 minute.

The rule of three is a way of solving problems of proportionality between three known values and an unknown value, establishing a relationship of proportionality between all of them.

If the relationship between the magnitudes is direct, that is, when one magnitude increases, so does the other (or when one magnitude decreases, so does the other), the direct rule of three must be applied as follow, being a, b and c known data and x the variable to be calculated:

a ⇒ b

c ⇒ x

So: x= (c×b)÷a

You can apply the following rule of three: if the vehicle travels 80 km in 60 minutes, in 45 minutes it travels how much distance?

60 minutes ⇒ 80 km

45 minutes ⇒ distance

So:

distance= (45 minutes× 80 km)÷ 60 minutes

Solving:

distance= 60 km

In summary, the vehicle travel 60 km in 45 minute.

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The slope of function f(x) is of -1, while the slope of function g(x) is of zero. Thus, the function g(x) has a greater slope.

Given a table of values, the slope of a linear function is given by the rate of change of the output variable y relative to the input variable x.

From the table of function f(x), we have that when x increases by 2, y decays by 2, hence the slope of the function is given as follows:

m = -2/2 = -1.

From the graph, the slope is also given by the change in y divided by the change in x.

The graph at the end of the answer shows a constant line, meaning that y remains constant even when x changes, thus the change in y is of zero, meaning that the slope of function g(x) is of zero.

The graph of function g(x) is given by the image shown at the end of the answer.

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To find the number of integers less than 305 that are divisible by 3, we can divide 305 by 3 and round down to the nearest integer. This gives us 101. However, we need to subtract 1 to exclude the number 305 itself, so there are 100 numbers less than 305 that are divisible by 3.

A number is said to be divisible if it divides evenly (leaving no residue). 34, for instance, is divisible by 2 since 2 enters 34 equally. 34 is not divisible by 3 since it would result in a remainder.

According to the three-digit rule of division, a whole number is said to be divisible by three if the total of all its digits is precisely divided by three. A number's ability to be divided by three can be determined without executing division.

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AAA is a triangle similarity theory. Which means all the respective angles of two triangles are equal.

AAA Postulate: (Angle Angle Angle)

It states that if all three angles of a triangle are equal to all three respective angles of another triangle, then the two triangles will be similar.

Let's say the angles ABC , BCA ,CAB of a triangle ABC are equal to the sides XYZ, YZX, ZXY of triangle XYZ then we can say that according to AAA postulates both the sides are similar and as a result we can say that their respective sides will be also same.

AAA postulates sate that if the all three angles of two triangle is equal then they will be called as similar triangle.

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Answer: f(6) = 3

Step-by-step explanation:

If you divide any polynomial h(x) by a linear function (it is to the 1st power) l(x), then the remainder of h(x) when divided by l(x) must be the value of h(x) where x is when l(x) = 0, or the x-intercept.

For example,

x - 6 = 0, and its x-intercept is 6. Therefore, if you do f(6), the remainder will be its output.  

The required equation for the given case is obtained as y = -0.2x + 5.

The slope of a straight line is the tangent of the angle formed by it with the positive x axis as the reference. The negative slope indicates the rate of decrease while the positive shows the rate of increase.

The given problem can be solved as follows,

The general form of slope-intercept equation for a line is y = mx + c.    

From the given data the slope can be calculated as,

m = Difference of miles ÷ Difference of minutes  

   = (2 - 4) ÷ (15 - 5) = -1/5

Now, substitute m = -1/5 in y = mx + c to obtain,

y = -1/5x + c

Substitute y = 4 and x = 5 to get,

4 = -1/5 × 5 + c

=> c = 5

Thus, the required equation becomes as y = -1/5x + 5 or y = -0.2x + 5.

Hence, the equation that models Ye's distance from home, y, in terms of minutes, x is given as  y = -0.2x + 5.

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The multiplication of the given two matrices AC will be [tex]\left[\begin{array}{ccc}-15&12&17\\5&-9&1\end{array}\right][/tex].

matrix, a collection of numbers lined up in rows and columns to produce a rectangular array. The elements of the matrix, also known as the entry, are the numerals.

In another word, a matrix is a set of variables that are arranged in such a good way that we can create an algebraic equation from that.

As per the given matrices,

A = [tex]\left[\begin{array}{ccc}2&-1&3\\0&2&-1\end{array}\right][/tex]

C = [tex]\left[\begin{array}{ccc}-5&0&5\\2&-3&2\\-1&3&3\end{array}\right][/tex]

The number of columns in matrix A is the same as the number of rows in matrix C.

The multiplication will be as,

AC = [tex]\left[\begin{array}{ccc}2&-1&3\\0&2&-1\end{array}\right]\times\left[\begin{array}{ccc}-5&0&5\\2&-3&2\\-1&3&3\end{array}\right][/tex]

AC = [tex]\left[\begin{array}{ccc}(2(-5)-1(2)+3(-1)&(0+-1(-3)+3(3))&2(5)+-1(2)+3(3)\\(0+2(2)+(-1(-1))&(0+2(-3)+-1(3)&2(5)+0+2(2)+-1(3)\end{array}\right][/tex]

AC = [tex]\left[\begin{array}{ccc}-15&12&17\\5&-9&1\end{array}\right][/tex]

Hence "The multiplication of the given two matrices AC will be [tex]\left[\begin{array}{ccc}-15&12&17\\5&-9&1\end{array}\right][/tex]".

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Prime factorization is the process of expressing a number as the sum of all its prime factors. In prime factorization, each and every integer is a prime number.

It is a method for dividing a polynomial into pieces that will be multiplied together. At this moment, the polynomial's value will be zero.

A component is a number that completely divides another number. To put it another way, if adding two whole numbers results in a product, then the numbers we are adding are factors of the final since the product is divisible by them.

Factors can be found using either division or multiplication. Rules of divisibility can also be used.

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The area of the shaded region will be (15/16)π².

A graph is a diametrical representation of any function between the dependent and independent variables.

The graph is easy to understand the behavior of the graph.

As per the given function,

r = √θ

The θ goes from π/2 to 2π while r goes from 0 to √θ.

The area will be as,

Area =  [tex]\int\limits^{2\pi} _{\pi/2}\int\limits^{\sqrt{\theta} } _{0} {r} \, drd\theta[/tex]

A = [tex]\int\limits^{2\pi} _{\pi/2}[r^{2}/2]_{0}^{\sqrt{\theta}} d\theta[/tex]

A = 1/4[(2π)² - (π/2)²]

A = π²/4[4 - 1/4] = (15/16)π².

Hence "The area of the shaded region will be (15/16)π²".

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The complete question is below,

The ratio for the Illustration a+b/a-b=5/3 is 5:4.

Ratio demonstrates how many times one number can fit into another number. Ratios contrast two numbers by ordinarily dividing them. A/B will be the formula if one is comparing one data point (A) to another data point (B).

From the information given, the equation is written thus:

a + b / a - b = 5 / 3

Cross multiply

5(a - b) = 3 (a + b)

5a - 5b = 3a + 3b

Collect the like terms

5a - 3a = 3b + 5b

2a = 8b

Divide

a = 8b / 2

a = 4b

Therefore (a + b) / a will be:

= (4b + b) / 4b

= 5b / 4b

= 5/4

= 5 : 4

The ratio is 5:4.

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