The general solution to the system x(t) = c1 [tex]e^{-2t}[/tex] [-1/2, 1]T + c2 [tex]e^{-4t}[/tex] [-1, 1]T.
The given system of equations can be written in matrix form as:
x' = A x
where A is the coefficient matrix, and x = [x1 x2]T is the vector of dependent variables.
Substituting the values of A, we get:
x' = [(−3 1 )
(1,-3)] x
To find the general solution to this system, we first need to find the eigenvalues of the coefficient matrix A.
The characteristic equation of A is given by:
|A - λI| = 0
where λ is the eigenvalue and I is the identity matrix of order 2.
Substituting the values of A and I, we get:
|[(−3 1 )
(1,-3)] - λ[1 0
0 1]| = 0
Simplifying this expression, we get:
|(−3-λ) 1 | |-3-λ| |1 |
| 1 (-3-λ)| = | 1 | * |0 |
Expanding the determinant, we get:
(−3-λ)² - 1 = 0
Solving for λ, we get:
λ1 = -2
λ2 = -4
These are the eigenvalues of A.
To find the eigenvectors corresponding to each eigenvalue, we solve the following system of equations for each λ:
(A - λI)x = 0
Substituting the values of A, I and λ, we get:
[(-3+2) 1 | |-1| |1 |
1 (-3+2)] | 1 | * |0 |
Simplifying and solving for x, we get:
x1 = -1/2, x2 = 1
Therefore, the eigenvector corresponding to λ1 = -2 is:
v1 = [-1/2, 1]T
Similarly, we can find the eigenvector corresponding to λ2 = -4:
v2 = [-1, 1]T
Using the eigenvectors and eigenvalues, we can write the general solution to the system as:
x(t) = c1 [tex]e^{-2t}[/tex] [-1/2, 1]T + c2 [tex]e^{-4t}[/tex] [-1, 1]T
where c1 and c2 are arbitrary constants. This is the general solution in vector form.
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How many four-digit integers are even perfect squares and also perfect cubes?
The is 1 four-digit integers that is a perfect squares and perfect cube and it is 4096
What are perfect squares and perfect cubes?A perfect cube is the cube of a whole number, found by multiplying a whole number by itself three times. For example, the number 8 is a perfect cube because 2 × 2 × 2 = 8 .
A perfect square is a number that can be expressed as the product of an integer by itself or as the second exponent of an integer. For example, 25 is a perfect square because it is the product of integer 5 by itself, 5 × 5 = 25.
The only four-digit integers that is a perfect square and a perfect cube is 4096
i.e 64² = 4096
16³ = 4096
Therefore the number of four-digit number that is a perfect square and a perfect cube is 1
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Solve the quadratic equation numerically (using tables of x- and y- values). X squared + 7 x + 12 = 0.
The quadratic equation numerically for [tex]x^{2} + 7x +12 = 0[/tex] is when
x = -4 and x =-3
The equation is [tex]x^{2} + 7x +12 = 0[/tex]
To solve this equation using the table, we change the equation into a function
So, the function is F(x) = y = [tex]x^{2} + 7x + 12[/tex]
We have to find the value of x which we can substitute to the equation to get the y value to become 0. Since it is a quadratic equation, x will have 2 values, on [tex]x^{2}[/tex] & 7x
X [tex]x^{2} + 7x + 12[/tex] = Y
-4 = [tex]-4^{2}[/tex] +7(-4) + 12 = 0
-3 = [tex]-3^{2}[/tex] +7(-3) + 12 = 0
-2 = [tex]-2^{2}[/tex] +7(-2) + 12 = 2
-1 = [tex]-1^{2}[/tex] +7(-1) + 12 = 6
0 = [tex]0^{2}[/tex] +7(0) + 12 = 12
1 = [tex]1^{2}[/tex] +7(1) + 12 = 20
2 = [tex]2^{2}[/tex] +7(2) + 12 = 30
Based on table above we can see Y = 0, when X = -4 and X = -3
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The residents of a city voted on whether to raise property taxes. The ratio of yes votes to no votes was 7 to 5 If there were 4255 no votes, what was the total number of votes?
The total number of votes is given as 10212
How to solve for the number of votesThe total number of votes that would be in the city is given as 7400.
How to find the total number of votes
The ratio of votes is in terms of yes to no
This is 7 : 5
Given that the number of no votes is 4255, we can use this number in order to find the yes votes.
7 + 5 = 12
there were 4255 no votes.
we would have:
(5 / 12 )x= 4255
5x = 12 * 4255
5x = 51060
x = 10212
The total number of votes is given as 10212
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What is called prime factorization method?
Answer:
The process of writing all numbers as a product of prime.
Step-by-step explanation:
For example, if we have something like the number 20. We can break that down into two factors. We can say, “well, that's 4 times 5.” And notice, 5 is a prime number. Hope it helps!
25. Which correctly gives the center of dilation
and scale factor of the dilation below?
A. (-8,-1); k=3/4
B. (-8,-1); k=4/3
C. (0,-5): k = 3/4
D. (0,-5); k=4/3
The required center of dilation and scale factor of a dilation is given as (0, -5) and k = 4/3. Option D is correct.
What is the scale factor?The scale factor is defined as the ratio of the modified change in length to the original length.
here,
To determine the center of dilation,
Draw a straight line Joining BB' and AA', this line meets at a point (0, -5) so this point is the required center of dilation.
Now,
From the ceter of the dilation,
The scale factor of the dilation is given as,
SF = OB'/OB
SF = 4/3
Thus, the required center of dilation and scale factor of a dilation is given as (0, -5) and k = 4/3. Option D is correct.
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Potatoes cost $3.64/lb. How much would 0.47 lb of potatoes cost?
Answer:
Step-by-step explanation:
1.7108 is the answer
Lisa's father is 42 years old. His age is two more than five times Lisa's age. How old is Lisa?
Answer:
Lisa is 8 years old.
Step-by-step explanation:
If Lisa's father is 42, then we can easily find out what Lisa's age is from the problem. We simply just have to create an equation and then solve it.
The equation:
5x + 2 = 42
Solving said equation:
5x = 42 - 2
5x/5 = 40/5
x = 8
What you do to one side, must be done to the other.
Hope this helps!
Can the sides of a triangle have lengths 5'7 and 13?
The sum of the lengths of any two sides of a triangle must exceed the length of the third side; therefore, 5+7 > x, which cannot happen if x = 13.
Which Cannot be sides of a triangle?Assume that a triangle has sides that measure 5, 7, and 11. Finding the sum of the squares on the triangle's two shorter sides and comparing it to the square on the longest side will reveal the type of triangle. The triangle is therefore not acute. Thirteen plus seven equals twenty, and twenty is more than nine. The Triangle Inequality Theorem is satisfied by this set of side lengths. A triangle is formed by these lengths. No, not in the traditional sense of the word (Euclidean geometry). A "degenerate triangle" is possible, but it isn't exactly a triangle in the sense that we often think of triangles 8 + 7 = 15 the length of any two sides added together.To learn more about sides of a triangle refer to:
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Determine the type of dilation shown and the scale factor used.
The type of dilation that is used here as well as the factor used is given as: enlargement with scale of 2.5. Option 2
What is dilation in mathematics?Reducing the object's size by dilation is a change. The items are enlarged or shrunk by dilation. An picture that retains the original shape is created by this alteration. The size of the form does differ, though.
The smaller image is resized into a bigger image from what we can see in the images that we have before us.
We would have to divide all sides of the big square with the corresponding sides of the smaller square
20 / 8 = 2.5
15 / 6 = 2.5
Hence we would see that the scale factor that is used here is 2.5. Therefore the dilation that is used is the enlargement with scale factor 2.5. Option 2 is correct
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subtract 11 hrs from 6 hrs 51 min
Answer:
(A) -5 hr 51 mins (B) 7:51 am/pm
Step-by-step explanation:
(A):
6 hr 51 mins
-
11 hr 0 mins
___________
-5 hr 51 mins
(B):
6 hr 51 mins - 11 hr 0 mins = 18 hr 51 mins - 11 hr 0 mins = 7 hr 51 mins am/pm or 7:51 am/pm.
Hope this helps!
P.S. Feel free to suggest any edits.
maya wants to be able to take a series of four triangles and place them in a mobile design where all four triangles are attached at one point and are balanced so that they stay horizontal. Which statement best describes where she should attach each triangle?
A) Where the medians intersect
B) where the altitudes intersect
C) where perpendicular bisectors intersect
D) at any vortex
The correct statement is (D) at every vertex.
What is a vertex?The vertex of a triangle is the intersection of two sides. Vertices is the term used to describe objects with several vertices.
The ability to write an equation for a line given two points is a prerequisite. The intersection of two lines will then need to be solved, which will give you the intersection's coordinates.
One of the triangle's vertices is at this intersection of two lines.
So, we know that 4 triangles are joined to make them horizontal.
This can be done when the vertex is attached to all 4 triangles.
Then, statement (D) will be a correct statement.
Therefore, the correct statement is (D) at every vertex.
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Find the range, mean, median, and mode of the following data set. 8, 4, 12, 10, 6, 25, 6
The range of the set is 21
The mean of the set is 10.1
The median of the set is 8
The mode of the set is 6
What is Median?The median is the value that's exactly in the middle of a data set when it is ordered. It's a measure of central tendency that separates the lowest 50% from the highest 50% of values. The steps for finding the median differ depending on whether you have an odd or an even number of data points
Arrange the data points from smallest to largest.
If the number of data points is odd, the median is the middle data point in the list.
If the number of data points is even, the median is the average of the two middle data points in the list.
Given data ,
The range of the set is the difference of the largest and the smallest value in the set
So , the range R = 26 - 4 = 21
The mean of the set be represented as A
The value of A = sum of observations / total number of values in the set
The mean of A = ( 8 + 4 + 12 + 10 + 6 + 25 + 6 ) / 7
The mean of A = 71 / 7
The mean of A = 10.14
Now , the median of the set be represented as B
For calculating the median of the set
Arrange the data points from smallest to largest
The number of elements in the set is 7 which is odd
So , B = { 4 , 6 , 6 , 8 , 10 , 12 , 25 }
The median of B = the middle term of the set
The median B = 8
Now , the mode of the set be represented as C
The mode C = the mode is the value that appears the most often in a data set
So , the number 6 is the value that appear more in the set
The mode C = 6
Hence , the mean , median and mode is 10.14 , 8 and 6 respectively
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A rock climber descends 31 feet into a small canyon. Then she climbs up 5 feet. Which integer represents her final position in feet, relative to where she started?
Answer:
-15ft
Step-by-step explanation:
Answer: - 26
-31 + 5 = -26
After school Andy spends 15 minutes on science homework 20 minutes on math homework 15 minutes on literature homework what was the ratio of the number of minutes she spends on on math homework to the total she spends on homework each night
The ratio of the number of minutes she spends on on math homework to the total she spends on homework each night is 2:5.
What is ratio?
When b does not equal 0, an ordered pair of numbers a and b, represented as a / b, is said to be a ratio. Two ratios are set to be equal in an equation called a proportion. You might express the ratio as 1: 3 for instance, if there is 1 boy and 3 girls (for every one boy there are 3 girls)
Here Andy spends 15 minutes on science homework 20 minutes on math homework 15 minutes on literature homework then,
Total time is ,
Science home work time+ Math home work time+ Literature home work time
=> 15+20+15 = 50 mins.
Now the ratio to the number of minutes she spends on on math homework to the total she spends on homework each night then
=> 20 : 50
=> 2 : 5.
Hence the ratio of the number of minutes she spends on on math homework to the total she spends on homework each night is 2:5.
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What is the table of 4?
The multiplication table for 4's underlying pattern: To begin with, let your child know that 4 is an even number. Therefore, no matter what number you multiply by 4 .
what is sequence ?A list of items in a sequence is one that is ordered (or events).The number of elements in an ordered sequence determines its length (potentially infinite). Arithmetic sequences, geometric sequences, quadratic sequences, and special sequences are the four primary types of sequences that you need to be familiar with. Number sequences are collections of numbers connected by some kind of rule. If you understand the rule, you can determine the numbers that will come after that in the sequence. Each number differs from the other in this case by 6. In this case, the rule for this sequence is to add 6 each time.
given
The multiplication table for 4's underlying pattern: To begin with, let your child know that 4 is an even number.
Therefore, no matter what number you multiply by 4, the result will always have either 0, 2, 4, 6 or 8 at the end.
With the aid of this method, kids will be able to distinguish multiples of 4, which will make learning the table of four much simpler.
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i need the answer please
Answer:
(1.5 , -5.5)
Step-by-step explanation:
Finding the midpoint when endpoints are given:(9 , -8) ; x₁ = 9 & y₁ = -8
(-6 , -3) ; x₂= -6 & y₂ = -3
[tex]\sf \boxed{\bf Midpoint= \left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)}[/tex]
[tex]\sf = \left(\dfrac{9-6}{2},\dfrac{-8-3}{2}\right)\\\\\\=\left(\dfrac{3}{2},\dfrac{-11}{2}\right)\\\\\\=(1.5, -5.5)[/tex]
Sofia earned a score of 460 on Exam A that had a mean of 400 and a standard
deviation of 20. She is about to take Exam B that has a mean of 100 and a
standard deviation of 25. How well must Sofia score on Exam B in order to do
equivalently well as she did on Exam A? Assume that scores on each exam are
normally distributed.
X2 = 175 Sofia needed a certain grade on Exam B to perform at the same level as she did on Exam A.
What is standard deviation ?Data are said to be more closely grouped around the mean when the standard deviation is low and more dispersed when the standard deviation is high. The dispersion of the data around the mean value is measured by the standard deviation. It might be helpful when comparing data sets that may have the same mean but a different range. The means of the two numbers 15, 15, 15, 14, 16, and 2, 7, 14, 22, 30 are the same, for instance. The second, though, is unmistakably wider.
given
X1
u1 = 400 u2 = 100
20 25
X2
Z1 score = Z2 score
X1 - u1 / 20 = X2 - u2 / 25
460 - 400 / 20 = X2 -100 / 25
3 = X2 - 100 /25
X2 = 175 score on exam b
X2 = 175 Sofia needed a certain grade on Exam B to perform at the same level as she did on Exam A.
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Construct one table that includes relative frequencies based on the frequency distributions shown below, then compare the amounts of tar in non filtered and filtered cigarettes. Do the cigarette filters appear to be effective? ( Hint: the filters reduce the amount of tar ingested by the smoker)
The table including relative frequencies based on the frequency distributions is as follows:
Tar(mg) Relative frequency Relative frequency
(Nonfiltered) (Filtered)
3-8 0% 4%
9-14 0% 12%
15-20 4% 24%
21-26 0% 60%
27-32 56% 0%
33-38 28% 0%
39-44 12% 0%
Yes, the cigarette filters appear to be effective.
What are relative frequencies?
The ratio (fraction or proportion) of the number of times a data value appears in the set of all outcomes to the total number of outcomes is known as the relative frequency.
Tar(mg) Frequency Relative Frequency Relative
frequency frequency
(Nonfiltered) (Filtered)
3-8 0 0% 1 4%
9-14 0 0% 3 12%
15-20 1 4% 6 24%
21-26 0 0% 15 60%
27-32 14 56% 0 0%
33-38 7 28% 0 0%
39-44 3 12% 0 0%
Hence, the cigarette filters appear to be effective because the relative frequency of the higher tar classes is greater for non filtered cigarettes.
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What scale factor was used in the dilation?
The scale factor used in the dilation of the triangle ABC is 3.
What is the dilation of a triangle?Dilation is a method for generating analogous figures decreasing or increasing their dimensions by some scaling factors.
By dividing distances by the scale factor, each point is stretched outward from the central point D if the scaling factor is more than 1 or inwards if the scale factor is less than 1.
Considering the triangle ABC:
The length of the side AB is 2 units
Compared with triangle DEF, the length of the similar side DE is 6 units
The scaling factor = length of side DE /length of the side AB
The scaling factor = 6 / 2
The scaling factor = 3
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It is given that y is directly proportional to x.
1
2345
y
-4
-8
-12
-16
-20
What is the constant of variation?
When it is assumed that y is directly proportional to x, the variational constant is -4.
what is variation ?A variation is a relationship between a set of values for one variable and another set of values. direct change. If m is a nonzero constant and b = 0, the function y = mx (commonly written y = kx) that results from the equation y = mx + b is known as a direct variation. Variation Theory is a broad theory of learning that is mostly applied in mathematics teaching and pedagogy. It highlights a concept's key components by concentrating on what remains the same and what changes, which creates the possibility for meaningful connections.
given
If y and x are directly proportionate, we would write:
If we include the proportionality constant, or variation, k, we get: y = kx
Any ordered pair from the table can be used to determine the value of y when x=1. y = -4 = k (1) k = -4
When it is assumed that y is directly proportional to x, the variational constant is -4.
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Mia is rolling a pair of 666-sided dice. P(n)p(n)p, left parenthesis, n, right parenthesis models the probability of the event that the sum of the dice is nnn. Which number type is more appropriate for the domain of ppp?.
Mia is rolling a pair of 666-sided dice. P(n)p(n)p, left parenthesis, n, right parenthesis models the probability of the event that the sum of the dice is nnn. The number type is more appropriate for the domain of ppp is {n | n ∈ N, 2 ≤ n ≤ 12}
A perfect dice has six faces and is exactly cubical in shape. Each face has one of the numerals 1, 2, 3, 4, 5, or 6, and any two faces can have a different digit.
Now that two dice have been rolled, P(n) simulates the likelihood that the total of the two dice's faces will be n.
The domain of P is therefore given by,
{n | n ∈ N, 2 ≤ n ≤ 12}
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what is 270 divided by 4.25
Answer:
1080 / 17
Step-by-step explanation:
270 divided by 4.25 = 27000 ÷ 425
= [tex]\frac{27000}{425}[/tex] = [tex]\frac{1080}{17}[/tex]
The table represents a function.
Xx
-4
-1
3
5
f(x)
-2
5
4
-8
What is f(5)?
-8
-1
1
8
Therefore , the solution to the given problem of function comes out to be f(5) = -8.
What does the word function mean?Mathematics encompasses variables and their changes, equations and related structures, form locations and forms themselves. The relationship between a number of inputs, each of which has an associated output, is referred to as a "function." An association between inputs and outputs that yields a single, distinct outcome is known as a function. Each function has a realm and with a requested request, or scope, allocated to it. The letter f is frequently used to represent , equation functions (x). An x is entered. The four primary kinds of activities that are offered are on activities, each functions, numerous features, within characteristics, and on functions.
Here,
Given :
x -4 -1 3 5
=> f(x) -2 5 4 -8
The value of f(x) at x = 5 is f(5).
=> f(5) = -8
Therefore , the solution to the given problem of function comes out to be f(5) = -8.
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In what sense is the regression line the straight line that best fits the points in a scatterplot?
The straight line along which the squared vertical distances from the data points are reduced is known as the regression line. minimization of the squared vertical distance between the points and the line for a straight line.
What does regression line mean?
An estimated linear relationship between two variables, which is known as the real but unobserved linear relationship, is shown by a regression line. A given value for the explanatory variable is used to predict (or estimate) the value of the response variable using the equation of the regression line.
What does the term "regression line" mean?
Regress, which is derived from the Latin verb "regressus" (to go back), is the root of the word "regression" (to something). Regression, in this sense, is a method that enables "going back" from muddled, difficult-to-interpret data to a more understandable and valuable model.
The degree to which both variables can move in unison is defined by correlation. Regression, however, identifies how a change in the unit of a known variable (p) affects a variable that is being evaluated (q).
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The figure below is translated down 4 Units and then reflected over the x-axis.what are the coordinates of the images of point E after these transformations?
The coordinates of the image of point Y following these modifications are thus -2.
How do we find the Cartesian coordinate plane?There are no boundaries to the two dimensions of the Cartesian coordinate plane. Any two-dimensional figure can be drawn on an unending 2D plane. In a Cartesian plane, every point has a specific place.The provided number has been rotated 180 degrees anticlockwise and increased in scale by two units.
After these adjustments, the image of point Y has coordinates that are -2.
The coordinates of the image of point Y following these modifications are thus -2.
The x-axis and y-axis intersect to produce the two-dimensional coordinate plane known as the cartesian plane in mathematics. The origin is the point at where the x- and y-axes cross perpendicularly.The complete question is,
The figure below is translated up 2 units and then rotated 180° counterclockwise. What are the coordinates of the image of point Y after these transformations?
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work out 4 1/2×1 2/5 give your answer in its simplest form
a fitness coach needs to use 3 different exercises from a list of 25 in a workout. How many different groups of 3 exercises can she choose?
Answer:
This is a combination problem, where order doesn't matter.
We want to find the number of ways to choose 3 exercises from a list of 25, which is denoted as 25C3.
This is also equal to (252423) / (321) which is equal to 2300.
Therefore, there are 2300 different groups of 3 exercises the fitness coach can choose from the list of 25 exercises.
Step-by-step explanation:
For $2.75 you can buy 20 small bags of jelly beans. How much is 1 bag of jelly beans?
Answer:
The answer is $ 0.13
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The graph below shows the percentages of ingredients in a salad mix.
A pie chart titled Percentages in salad mix. 40 percent is iceberg, 27 percent is romaine, 15 percent is spinach, 12 percent is arugula, 6 percent is carrots.
Alyssa buys a 38-ounce container of salad mix. About how many ounces of iceberg lettuce does the container have?
8 ounces
16 ounces
27 ounces
38 ounces
Using percentage to determine the ounce of iceberg lettuce in the salad from the pie chart, we have 15.2 ounces
What is Pie ChartA pie chart is a circular chart that is divided into slices to illustrate numerical proportion. In a pie chart, the arc length of each slice (and consequently its central angle and area), is proportional to the quantity it represents. Pie charts are very useful for displaying data and are most commonly used to show percentages or proportional values of different categories relative to the whole.
We can use the concept of percentage to determine the number of ounces of iceberg lettuce present.
In the pie chart, the percentage of ice berg lettuce is 40% and the quantity of salad bought is 38 ounce
40 / 100 = x / 38
0.4 = x / 38
x = 38 * 0.4
x = 15.2 ounces
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PLS HELP!! 15 points!
The origin's coordinates are as follows: (0, 0). A pair of ordered numbers contains one point's coordinates in the coordinate system.
What do you meant by ordered pair ?To accomplish this, convert the total number of liters of fuel you purchased into gallons (multiply by 0.219). the initial mileage is subtracted from the total. Then, multiply the distance traveled by the number of gallons of fuel used. Miles per gallon are provided. The concept of proportionality refers to a graphed proportional relationship in which the independent variable is represented by x and the dependent variable by y.
Independent variables describe the input values in a relationship and are typically represented by the x coordinate in the ordered pairs (x, y). When two or more parameters are directly or indirectly proportional to one another, the relationship is denoted by the symbols a=kb or a = k/b, where k denotes the direction of the relationship between the two variables.
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