When y [(C1)+(C2)x]exp(Ax) is the general solution of the second order linear differential equation: (y'') + (-4y') + ( 4y) = 0 then the values of A that satisfy the given differential equation are A = ±2.
To determine the value of A in the second-order linear differential equation (y'') + (-4y') + (4y) = 0, we can use the general solution y = (C1) + (C2)[tex]x^Ae^{Ax}[/tex], where C1 and C2 are constants.
By comparing the general solution with the given differential equation, we can identify the value of A.
The given differential equation is (y'') + (-4y') + (4y) = 0.
We can substitute the general solution y = (C1) + (C2)[tex]x^Ae^{Ax}[/tex] into the differential equation to find the value of A.
First, let's calculate the first and second derivatives of y:
y' = C2([tex]Ax^{A-1}e^{Ax}[/tex]) + C1[tex]e^{Ax}[/tex]
y'' = C2(A(A-1)[tex]x^{A-2}e^{Ax}[/tex]) + C2([tex]A^2x^{A-1}e^{Ax}[/tex]) + C1([tex]Ae^{Ax}[/tex])
Now, substitute these derivatives into the differential equation:
C2(A(A-1)[tex]x^{A-2}e^{Ax}[/tex]) + C2([tex]A^2x^{A-1}e^{Ax}[/tex]) + C1([tex]Ae^{Ax}[/tex]) + (-4)(C2([tex]Ax^{A-1}e^{Ax}[/tex]) + C1[tex]e^{Ax}[/tex]) + 4(C1) + 4(C2)[tex]x^Ae^{Ax}[/tex] = 0
Simplifying the equation and collecting like terms:
C2[[tex](A^2 - 4) x^{A-1} + A x^{A-1}[/tex]][tex]e^{Ax}[/tex] + (C1A - 4C2A)[tex]e^{Ax}[/tex] + (4C1 + 4C2)[tex]x^Ae^{Ax}[/tex] + 4C1 = 0
For this equation to hold true for all x, the coefficient of each term must be zero.
Therefore, we can equate each coefficient to zero and solve for A.
Let's equate the coefficients:
For the term involving [tex]x^{A-1}e^{Ax}[/tex]:
C2[[tex](A^2 - 4) x^{A-1} + A x^{A-1}[/tex]] = 0
For the term involving x[tex]e^(Ax)[/tex]:
(4C1 + 4C2)[tex]x^A[/tex] = 0
For the constant term:
4C1 = 0
From the first equation, we have two possibilities:
([tex]A^2[/tex] - 4) = 0, which leads to A = ±2.
A = 0, which results in the trivial solution y = C1.
From the second equation, we have two possibilities:
[tex]x^A[/tex] = 0, which implies A < 0 (not valid for our general solution).
4C1 + 4C2 = 0, which means C1 = -C2.
Now, let's consider the value of A = ±2.
For A = 2:
The general solution becomes y = (C1 + C2[tex]x^2[/tex])[tex]e^{2x}[/tex].
For A = -2:
The general solution becomes y = (C1 + C2[tex]x^{-2}[/tex])[tex]e^{-2x}[/tex].
So, the values of A that satisfy the given differential equation are A = ±2.
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The Arnold's took out a loan for $195,000 to purchase a home. At a 4.3% interest rate compounded annually, how much total will they have paid after 30 years?
Answer:
just interest = 251,550. Plus loan = 446,550
Step-by-step explanation:
195,000 x 0.043(interest as a decimal) = 8385 per year.
8385 x 30(years) = 251,550
The total of $446550 amount of money have to pay after 30 years at the rate of 4.3% of the principal amount of $195000.
What is compound interest?Compound interest is applicable when there will be a change in principle amount after the given time period.
For example, if you give anyone $500 at the rate of 10% annually then $500 is your principle amount. After 1 year the interest will be $50 and hence principle amount will become $550 now for the next year the interest will be $550, not $500.
Given that
the principle amount of $195000
The rate of interest is 4.3%
Time period 30 years.
By compound interest formula
A = P × [tex](1 + 0.00r)^{n}[/tex]
where A is the total amount, P is the principal amount,r is the rate of interest and n is the total time period.
A = $195000[tex](1 + 0.043)^{30}[/tex]
A = $689546.985
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Use the distributive property to rewrite each algebraic expression 7(y+2) + (8+r) + 8(x + 9)
Answer:
7y+r+8x+94
Explanation
Given the algebraic expression 7(y+2) + (8+r) + 8(x + 9)
Given A, B and C. According to distributive law;
A(B+C) = AB + AC
A is distributed over C. In the same vein
On expanding the expression;
7(y+2) + (8+r) + 8(x + 9)
7y+7(2) + 8 + r + 8(x)+ 8(9)
7y+14+8+r+8x+72
Bringing the variables and constants together
7y+r+8x+14+8+72
7y+r+8x+94
3. Your friend shows you a scale drawing of her apart-
ment. The drawing of the apartment is a rectangle
4 inches by 6 inches. Your friend wants to know the
length of the shorter side of the apartment. If she
knows that the length of the longer side of the apart-
ment is 30 feet, how many feet long is the shorter side
of her apartment?
Answer:
The answer is 20 feet
Step-by-step explanation:
We know that her apartment sketch is 6 by 4 inches.
If the 6 inch side is actually 30 feet and 6 goes into 30 5 times, we should multiply 4 by 5 to get proportionate answers.
Therefore her apartment is 20*30 feet
Hope this helps
Please help!
I don't understand, so if you can explain, that would be amazing. If you just give me an answer, that's ok too!
Answer:
3 units
Step-by-step explanation:
-1, 0, 1 ,2
( 1, 2 ,3)
Give the other person Brainliest :)
A coin was tossed 50 times and landed on tails 10 times. What is the Probability that will land on tail on the next toss.
Answer:
4/5
Step-by-step explanation:
50-10 = 40
40/50 = 4/5
the answer is 40/50
Step-by-step explanation:
50-10=40
it's easy
can someone awnser the following screenshot
The answer is 23v
steps-
remove the parentheses
2x7v+9v
multiply the number
2x7=14+19v
add them
14v+19v=23v
Fiona is always looking for a great deal while shopping. Her favorite jeans regularly cost 40$. She found a sale rack where all of the jeans are marked 30$. What is the percentage of the discount on jeans
M∠ACB = 52° , find m∠CBE =________°
Answer:
(1) It is given that line AB is tangent to the circle at A.
∴ ∠CAB = 90º (Tangent at any point of a circle is perpendicular to the radius throught the point of contact)
Thus, the measure of ∠CAB is 90º.
(2) Distance of point C from AB = 6 cm (Radius of the circle)
(3) ∆ABC is a right triangle.
CA = 6 cm and AB = 6 cm
Using Pythagoras theorem, we have
BC2=AB2+CA2⇒BC=
√
62+62
⇒BC=6
√
2
cm
Thus, d(B, C) = 6
√
2
cm
(4) In right ∆ABC, AB = CA = 6 cm
∴ ∠ACB = ∠ABC (Equal sides have equal angles opposite to them)
Also, ∠ACB + ∠ABC = 90º (Using angle sum property of triangle)
∴ 2∠ABC = 90º
⇒ ∠ABC =
90°
2
= 45º
Thus, the measure of ∠ABC is 45º.
Step-by-step explanation:
Slove the system of linear equations by graphing y=-x+7 y=x-1
Answer: Point Form: ( 4 , 3 ) Equation Form: x = 4 , y = 3
Step-by-step explanation: Solve for the first variable in one of the equations, then substitute the result into the other equation.
Brainliest or a thank you please? :))
What value of n makes the
equation n x 3/4= 3/16 true
Answer:
1/4
Step-by-step explanation:
Try it if you don't believe me :)
The solution of the linear equation n x (3/4) = 3/16 will be 1/4.
What is the solution to the equation?In other words, the collection of all feasible values for the parameters that satisfy the specified mathematical equation is the convenient storage of the bunch of equations.
The equation is given below.
n x (3/4) = 3/16
The degree of the equation is one. Then the equation will be a linear equation.
Simplify the equation, then the value of the variable 'n' is calculated as,
n x (3/4) = 3/16
n = (3/16) x (4/3)
n = 4/16
n = 1/4
The solution of the linear equation n x (3/4) = 3/16 will be 1/4.
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Write an equation in point-slope form of the line that passes through the point (7, -4) and has a slope of m=-6.
Answer:
(-4-y) = -6(7 - x)
Step-by-step explanation:
find the shaded region of the figure below
Amber and her older brother Johnny were helping their father build a sand pit in their
backyard to go with the new swimming pool he just purchased. The sand pit is circular in
shape. The height of the p[it is to be 2 feet. The radius of the pit is to be 6 feet. If one 50
pound bag of sand covers.52ft of space, how many bags of sand will be needed to
completely fill the sand pit?
how do I graph this linear equation?
4x+6y=-12
Answer: Simplifying
Step-by-step explanation:
Simplifying
4x + 6y = 12
Solving
4x + 6y = 12
Solving for variable 'x'.
Move all terms containing x to the left, all other terms to the right.
Add '-6y' to each side of the equation.
4x + 6y + -6y = 12 + -6y
Combine like terms: 6y + -6y = 0
4x + 0 = 12 + -6y
4x = 12 + -6y
Divide each side by '4'.
x = 3 + -1.5y
Simplifying
x = 3 + -1.5y
Then you get your answer I think or hopeful get your answer. I hoped this helped!
Based on the two data sets represented below, complete the following sentences.
\text{DATA SET B}
DATA SET B
0
0
23
24
25
26
27
28
29
30
31
32
33
\text{DATA SET C}
DATA SET C
0
0
23
24
25
26
27
28
29
30
31
32
33
The median of Data Set B is
than the median of Data Set C. The minimum of Data Set B is
than the minimum of Data Set C.
Jenna works at a local sports bar and is interested in whether the percent tip she will receive is related to how many drinks people order. What test should she perform? O Correlation O ANOVA Independent samplest test O Single sample t test
She should perform a correlation test.
Jenna works at a local sports bar and is interested in whether the percent tip she will receive is related to how many drinks people order. What test should she perform?
She should perform a correlation test.
What is a correlation test?
A correlation test is a statistical method used to examine the relationship between two variables. It is used to determine the degree of association between two continuous variables.
How do you carry out a correlation test?
To perform a correlation test, follow these steps:
Step 1: Collect your data.
Step 2: Determine the level of measurement of each variable.
Step 3: Calculate the correlation coefficient.
Step 4: Determine the p-value.
Step 5: Interpret the results.
Therefore she should perform a correlation test
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the intercept in the multiple regression model: group of answer choices should be excluded if one explanatory variable has negative values. determines the height of the regression line. should be excluded because the population regression function does not go through the origin. is statistically significant if it is larger than 1.96.
The incorrect statement is "the intercept in the multiple regression model should be excluded if one explanatory variable has negative values".
The intercept, also known as the constant term, represents the value of the dependent variable when all independent variables are equal to zero.
It is a necessary component of the regression equation regardless of the values of the independent variables. Excluding the intercept when one explanatory variable has negative values may result in biased and unreliable estimates.
The intercept does not determine the height of the regression line, but rather its slope and position. The slope represents the change in the dependent variable for a unit change in an independent variable, while the intercept represents the starting point of the regression line on the y-axis.
The intercept should not be excluded because the population regression function does not go through the origin. While some models may have a theoretical justification for excluding the intercept, such as when all independent variables are proportions or percentages that sum to one, it is generally advisable to include it.
The statistical significance of the intercept depends on its estimated value and standard error, as well as the chosen level of significance. A common level of significance is 0.05, which corresponds to a critical value of 1.96 for a two-tailed test.
If the absolute value of the t-statistic for the intercept is larger than 1.96, we can reject the null hypothesis that it is equal to zero at a 5% level of significance.
In summary, excluding the intercept in a multiple regression model based on negative values of an explanatory variable is not recommended, and its significance should be assessed based on its estimated value and standard error.
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Find GCF (212, 96) and LCM (212, 96) using prime factoring.
Anyone know...
Answer:
8x12 and 2x2x53
Step-by-step explanation:
hey lol pls help <3
Ms. Philor is going on a trip to Hawaii. The function A(d)=0.80d+200 models the amount A, in dollars, that Ms.Philor's company pays her based on the round trip distance d, in miles that Ms. Philor travels to do a job.
Ms. Philor's pay increases by $
A)80.200
B) 0.80
C)200.80
Suppose your favorite coffee machine oder 14 ounce cup of coffee the actual amount of coffee pot in the cup by the machine vanes according to a normal distribution with mean equal to 15 ounces and standard deviation equal to 0.65 ounces. What percentage of cups will be filled with less than 14 ounces?
Approximately 6.3% of cups will be filled with less than 14 ounces is the answer.
Given, mean = 15 ounces and standard deviation = 0.65 ounces
The actual amount of coffee in the cup by the machine vanes according to a normal distribution.
For this, we need to calculate the z-score as z=(X-μ)/σ
We need to find the percentage of cups that will be filled with less than 14 ounces of coffee.
For this, we will calculate the probability that X < 14. So, we need to find P(X<14).
For this, we will first calculate the z-score as z = (X - μ) / σ= (14 - 15) / 0.65= -1.538
Now, we will find the area to the left of the z-score using the standard normal distribution table or calculator:
Using the standard normal distribution table, we get the area as 0.0630.
The percentage of cups that will be filled with less than 14 ounces is 6.3%.
Thus, approximately 6.3% of cups will be filled with less than 14 ounces.
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This year, the ACT score of a randomly selected student is normally distributed with a mean of 24.4 points and a standard deviation of 4.7 points. Let XX be the ACT score of a randomly selected student and let ¯¯¯XX¯ be the average ACT score of a random sample of size 16.
1. Describe the probability distribution of XX and state its parameters μμ and σσ:
The probability distribution of the sample mean is approximately normal, with mean of 24.4 points and standard deviation of 1.175 points.
How to obtain the distribution?By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation given by the equation presented as follows: [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
The mean is the same as the population mean, of 24.4 points, while the shape is approximately normal.
The standard error is given as follows:
[tex]s = \frac{4.7}{\sqrt{16}} = 1.175[/tex]
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Suppose NiceGirl is the set of all nice girls, Sailor is the set of all sailors, and loves is a relation between nice girls and sailors such that n loves s means that nice girls n loves sailor s. Make precise the sentence "All the nice girls love a sailor" to expose at least two distinct meanings of this ambiguous utterance.
The sentence "All the nice girls love a sailor" can be interpreted in at least two different ways when it comes to the given sets Nice Girl and Sailor and the relation loves between them.
They are: All the nice girls love the same sailor. This interpretation would mean that there exists a sailor s ∈ Sailor such that all the girls in the set Nice Girl love s, i.e., ∀n ∈ Nice Girl, n loves s. This interpretation assumes that there is only one sailor that is loved by all the nice girls.
2. Each of the nice girls loves a different sailor. This interpretation would mean that for every girl n ∈ Nice Girl, there exists a sailor s ∈ Sailor such that n loves s, but s may be different for different girls. This interpretation assumes that each nice girl loves a different sailor.
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Determine the value of variables a, b, and c that make each equation true.
What is the value of a in this equation?
1
(14496
30
a
What is the value of b in this equation?
b=
What is the value of c in this equation?
(x2)° = 222
Answer:
77
Step-by-step explanation:
I got it right on my question
x h(x)
-25 6
-13 0
-3 -5
0 -7
9 -11
11 -30
What is the average rate of change of h(x) over the interval -13 < x < 11 ?
Answer:
56
Step-by-step explanation:
Suppose f(x, y) = x + , where 0 <= x <=1 0 <=y <= 2
Find P(Y - X< 1)
To determine the area of a region, we need to know the specific shape and dimensions of the region in question. The process for finding the area varies depending on the shape. P(Y - X < 1) = Area of region below y - x = 1 within the given constraints= 1/6 sq. units.
Given, f(x, y) = x + y, where 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2 We are to find P(Y - X < 1)
Now, Y - X < 1 is the equation of the line in the XY plane. We have to find the area of the region below this line within the given constraints. To find this, we have to integrate over the region below the line,
i.e.,y - x = 1 implies x = y - 1∫∫R (x + y)dA = ∫₀² ∫₀^(y-1) (x + y) dx dy= ∫₀² [(y²/2) - y + (1/2)] dy= 1/6 sq. units.
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Answer:
Step-by-step explanation:
Given function f(x, y) = x + y, where 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2. The probability of P(Y - X < 1) is 1/4.
Now we have to find P(Y - X < 1).
The probability of (Y - X < 1) is given by:
P(Y - X < 1) = Area of the region {(x, y): y - x < 1} / Area of the region R.
Here R is the region bounded by 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2.
We know that the function f(x, y) = x + y is a plane with slope 1, and intersects the y-axis at (0, 0) and the x-axis at (0, 0).
Now the line y - x = 1 intersects the plane at x = 0 and y = 1, so the intersection point is (0, 1).
Now the area of the region {(x, y): y - x < 1} can be found by finding the area of the triangle ABC as shown below: [tex]A=\frac{1}{2}Base×Height[/tex], Where base BC is of length 1 and height AB is of length 2 - 1 = 1.
Therefore, the area of the triangle ABC is:
Area of the triangle ABC = 1/2 × 1 × 1
= 1/2
The area of the region R can be found by finding the area of the rectangle with sides of length 1 and 2.
Therefore, the area of R is:
Area of R = 1 × 2
= 2
P(Y - X < 1) = Area of the region {(x, y): y - x < 1} / Area of the region RP(Y - X < 1)
= (1/2) / 2P(Y - X < 1)
= 1/4
Therefore, P(Y - X < 1) = 1/4.
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Given x=e−t and y=te8t, find the following derivatives as functions of t .
dy/dx = ________
d²y/dx² = ________
Answer:
[tex]\frac{dy}{dx}=-(8t+1)e^{9t}\\\frac{d^{2}y}{dx^{2}}=(72t+17)e^{10t}[/tex]
Step-by-step explanation:
The explanation is attached below.
dy/dx = -te^(9t) (1 + 8t)
d²y/dx² = e^(8t) (16t + 1)
Firstly, we will find the value of dx/dt using the Chain rule as follows: dx/dt = - e^(-t)Then, the value of dy/dt can be calculated as
dy/dt = (d/dt) [t(e^(8t))] [Using product rule]= te^(8t) + t * 8e^(8t)
[Using the product rule again]= te^(8t) + 8te^(8t)
[Factoring out t]= te^(8t) (1 + 8t)So, we have obtained the first derivative of y w.r.t t.
Now, let's find the second derivative of y w.r.t t.d²y/dt² = (d/dt) [te^(8t) (1 + 8t)]
[Using the product rule]= e^(8t)(1 + 8t) + t * 8e^(8t)
[Using the product rule again]= e^(8t)(1 + 8t) + 8te^(8t)
[Factoring out 8e^(8t)]= e^(8t) (1 + 8t + 8t)
[Factorizing]= e^(8t) (16t + 1)
So, the value of the derivative dy/dx is:dy/dx = (dy/dt) / (dx/dt) = te^(8t) (1 + 8t) / - e^(-t) = -te^(9t) (1 + 8t)
Thus, the values of derivatives as functions of t are:
dy/dx = -te^(9t) (1 + 8t)
d²y/dx² = e^(8t) (16t + 1)
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Sequence: 13.9.5.1.-3. ...
5) is this Arithmetic, Geometric, or Neither?
6) Write and Equation for the Sequence.
7) What is the 15th Term in the Sequence?
Raghav, Rajit, Junsu, Mia, and Umema were finished with lunch and began playing with drink straws. Each one was making a line design using either 3 or 4 straws. Since they had just come from math class where they had been studying special angles, Raghav pulled his pencil out of his bookbag and labeled some of the angles and lines. He then challenged himself and the others to find the values of x and y in degrees.
Show how you would find the values of x and y for Raghav’s straw figure shown below using a system of linear equations. (You may use a decimal to the tenths place for your answers for the angle measures.)
Answer:
x = [tex]25.8^{o}[/tex] and y = [tex]51.4^{o}[/tex]
Step-by-step explanation:
Known that he sum of angles on a straight line is [tex]180^{o}[/tex]. From the diagram;
2y + x + y = [tex]180^{o}[/tex]
x + 3y = [tex]180^{o}[/tex] ................ 1
Also,
2x + x + 2y = [tex]180^{o}[/tex]
3x + 2y = [tex]180^{o}[/tex] .............. 2
Using the elimination method, multiply equation 1 by 3 and equation 2 by 1
3x + 9y = 540 ............ 3
3x + 2y = [tex]180^{o}[/tex] ............. 4
7y = 360
y = [tex]\frac{360}{7}[/tex]
y = [tex]51.4^{o}[/tex]
substitute the value of y in equation 1,
x + 3y = [tex]180^{o}[/tex]
x + 3(51.4) = [tex]180^{o}[/tex]
x + 154.2 = [tex]180^{o}[/tex]
x = [tex]180^{o}[/tex] - 154.2
x = [tex]25.8^{o}[/tex]
Thus, x = [tex]25.8^{o}[/tex] and y = [tex]51.4^{o}[/tex]
Let X denote the time to failure (in years) of a certain hydraulic component. Suppose the pdf of X is f(x) = 32/(x+4)³ for x < 0. a. Verify that f(x) is a legitimate pdf. b. Determine the cdf.C.
a. The function f(x) = 32/(x+4)³ for x < 0 is not a legitimate pdf
b. The function f(x) does not have a cumulative distribution function (cdf)
a. Verifying that f(x) is a legitimate pdf.From the question, we have the following parameters that can be used in our computation:
f(x) = 32/(x+4)³ for x < 0
The density function f(x) is a legitimate pdf if
∫ f(x) dx = 1
So, we have
[tex]\int\limits^{\infty}_{-\infty} {\frac{32}{(x + 4)^3} \, dx = 1[/tex]
Integrate the function
[tex]-\frac{16}{(x + 4)^2}|\limits^{\infty}_{-\infty} = 1[/tex]
Expand the equation
So, we have
[tex]-\frac{16}{(\infty + 4)^2} + \frac{16}{(-\infty + 4)^2} = 1[/tex]
Evaluate the exponents
-16/∝ + 16/∝ = 1
So, we have
0 + 0 = 1
Evaluate
0 = 1
The above equation is false
This means that f(x) is not a legitimate pdf.
b. Determine the cdf.In (a), we proved that
f(x) is not a legitimate pdf.
This means that it does not have a cumulative distribution function (cdf)
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find the area of a polygon with the vertices of (-4, 5), (-1, 5), (4, -3), and (-4, -3). suggestion: plot the points on graph paper and connect the vertices to form the polygon.
The area of the polygon with the vertices (-4, 5), (-1, 5), (4, -3), and (-4, -3) is 12 square units.
To calculate the area of the polygon, we can use the shoelace formula, also known as Gauss's area formula or the surveyor's formula. The formula involves writing the x-coordinates and y-coordinates of the vertices in a specific order and performing a series of calculations.
1. We write the x-coordinates of the vertices in one row, repeating the first coordinate at the end: -4, -1, 4, -4.
2. We write the y-coordinates of the vertices in the next row, in the same order: 5, 5, -3, -3.
3. Next, we multiply each pair of adjacent x and y coordinates and add them together in a counterclockwise direction.
4. Then, we subtract the sum of the products of the y-coordinates and the x-coordinates in a counterclockwise direction.
5. Taking the absolute value of this result, we divide it by 2 to obtain the area.
Applying the shoelace formula:
Area = |((-4*5) + (-1*-3) + (4*-3) + (-4*5)) - (5*-1 + 5*4 + -3*-4 + -3*-4)| / 2
= |-49 - (-25)| / 2
= |-24| / 2
= 12 / 2
= 12.
Therefore, the area of the polygon with the given vertices is 12 square units.
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