The exact area of the surface obtained by rotating the curve y = √(1 + eˣ) about the x-axis over the interval 0 ≤ x ≤ 7, we would need to use numerical methods to approximate the value of the integral since it does not have a simple closed-form solution.
To find the exact area of the surface obtained by rotating the curve y = √(1 + eˣ) about the x-axis, we can use the formula for the surface area of a solid of revolution.
The formula for the surface area of a curve y = f(x) rotated about the x-axis over the interval [a, b] is given by:
A = 2π∫[a, b] y * sqrt(1 + (dy/dx)²) dx
In this case, the given curve is y = √(1 + eˣ) and the interval of interest is 0 ≤ x ≤ 7. To calculate the area, we need to find the derivative dy/dx and substitute it into the formula.
Let's start by finding the derivative of y = √(1 + eˣ) with respect to x. Applying the chain rule, we have:
dy/dx = (1/2)(1 + eˣ)^(-1/2) * eˣ
Now, we can substitute y and dy/dx into the surface area formula:
A = 2π∫[0, 7] √(1 + eˣ) * sqrt(1 + [(1/2)(1 + eˣ)^(-1/2) * eˣ]²) dx
Simplifying the expression inside the integral, we have:
A = 2π∫[0, 7] √(1 + eˣ) * sqrt(1 + (eˣ/2)(1 + eˣ)^(-1)) dx
Now, we need to evaluate this integral over the interval [0, 7] to find the exact area of the surface.
Unfortunately, the integral for this particular curve does not have a simple closed-form solution. Therefore, to find the exact area, we would need to rely on numerical methods, such as numerical integration techniques or computer algorithms, to approximate the value of the integral.
Using these numerical methods, we can calculate an accurate estimate of the surface area by dividing the interval [0, 7] into smaller subintervals and applying techniques like the trapezoidal rule or Simpson's rule. The more subintervals we use, the more accurate the approximation will be.
In summary, to find the exact area of the surface obtained by rotating the curve y = √(1 + eˣ) about the x-axis over the interval 0 ≤ x ≤ 7, we would need to use numerical methods to approximate the value of the integral since it does not have a simple closed-form solution.
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9 Marty conducted a survey in his first period class to determine student preferences for music. Out of 25 students, 14 like hip-hop music best. There are 300 students in Marty's school. Based on the survey, how many students in the school like hip- hop music best? A. 50 students B. 132 students C. 168 students D. 261 students
Answer:
C
Step-by-step explanation:
14/25=0.56 0.56x300=168
Based on the survey,
168 students like hip-hop music.
What is ratio?The ratio is a numerical relationship between two values that demonstrates how frequently one value contains or is contained within another.
Given:
Marty conducted a survey in his first period class to determine student preferences for music.
Out of 25 students, 14 like hip-hop music best.
That means, the ratio is 14/25 = 0.56.
There are 300 students in Marty's school.
Based on the survey,
the number of students = 300 x 0.56 = 168 students like hip-hop music.
Therefore, 168 students like hip-hop music.
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(a-1)+(b+3)i = 5+8i
please answer me quickly i need it please
Answer:
a = 6, b = 5
Step-by-step explanation:
Assuming you require to find the values of a and b
Given
(a - 1) + (b + 3)i = 5 + 8i
Equate the real and imaginary parts on both sides , that is
a - 1 = 5 ( add 1 to both sides )
a = 6
and
b + 3 = 8 ( subtract 3 from both sides )
b = 5
Let Z= max (X, Y) and W = min (X, Y) are two new random variables as functions of old random variables X and Y. (a). Determine fz (z) and fw (w) in terms of marginal CDFs of X and Y random variables, by first drawing the region of interest on X and Y plane. (b). Let x and y be independent exponential random variables with common parameter A. Define W = min (X, Y). Find fw (w).
(a) fz (z) and fw (w) in terms of cumulative distribution functions (CDFs) are:
fz(z) = Fx(z) * (1 - Fy(z)) + Fy(z) * (1 - Fx(z))
fw(w) = 1 - fz(w)
(b) If X and Y are independent exponential random variables with parameter λ, then fw(w) = [tex]1 - e^{-2\lambda w}[/tex] for w ≥ 0.
To determine fz(z) and fw(w) in terms of the marginal cumulative distribution functions (CDFs) of X and Y random variables, we need to consider the region of interest on the X-Y plane.
(a) Drawing the region of interest on the X-Y plane:
The region of interest can be visualized as the area where Z = max(X, Y) and W = min(X, Y) take specific values. This region is bounded by the line y = x (diagonal line) and the lines x = z (vertical line) and y = w (horizontal line).
Determining fz(z):
To find fz(z), we need to consider the cumulative probability that Z takes a value less than or equal to z. This can be expressed as:
fz(z) = P(Z ≤ z) = P(max(X, Y) ≤ z)
Since X and Y are independent random variables, the probability can be calculated using the joint CDF of X and Y:
fz(z) = P(max(X, Y) ≤ z) = P(X ≤ z, Y ≤ z)
Using the marginal CDFs of X and Y, denoted as FX(x) and FY(y), respectively, we can express fz(z) as:
fz(z) = P(X ≤ z, Y ≤ z) = P(X ≤ z) * P(Y ≤ z) = FX(z) * FY(z)
Determining fw(w):
To find fw(w), we need to consider the cumulative probability that W takes a value less than or equal to w. This can be expressed as:
fw(w) = P(W ≤ w) = P(min(X, Y) ≤ w)
Since X and Y are independent random variables, the probability can be calculated using the joint CDF of X and Y:
fw(w) = P(min(X, Y) ≤ w) = 1 - P(X > w, Y > w)
Using the marginal CDFs of X and Y, denoted as FX(x) and FY(y), respectively, we can express fw(w) as:
fw(w) = 1 - P(X > w, Y > w) = 1 - [1 - FX(w)][1 - FY(w)]
Special case when X and Y are independent exponential random variables with parameter A:
If X and Y are independent exponential random variables with a common parameter A, their marginal CDFs can be expressed as:
[tex]FX(x) = 1 - e^{-Ax}\\FY(y) = 1 - e^{-Ay}[/tex]
Using these marginal CDFs, we can substitute them into the formulas for fz(z) and fw(w) to obtain the specific expressions for the random variables Z and W.
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Andy has $ 200 to buy a new TV . One- forth of that money came from his grandmother and he saved the rest . How much money did Andy save?
Answer:
$150
Step-by-step explanation:
200/4=50
200-50=150
Please help me!! No files allowed. I need the answer and an explanation!
Answer:
1/324
Step-by-step explanation:
encontre as raízes quadradas dos números:
a)²√625
b)²√100
c)²√81
Answer:
a.) 25, b.)10, c.)9
Step-by-step explanation:
a.) 25x25=625
b.)10x10=100
c.) 9x9=81
Find the value of the variables in the simplest form
Answer:
Step-by-step explanation:
Answer:
x = 15[tex]\sqrt{3}[/tex] , y = 15
Step-by-step explanation:
Using the sine and cosine ratios in the right triangle and the exact values
sin60° = [tex]\frac{\sqrt{3} }{2}[/tex] , cos60° = [tex]\frac{1}{2}[/tex] , then
sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{x}{30}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )
2x = 30[tex]\sqrt{3}[/tex] ( divide both sides by 2 )
x = 15[tex]\sqrt{3}[/tex]
---------------------------------------------------------
cos60° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{y}{30}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )
2y = 30 ( divide both sides by 2 )
y = 15
A random variable X has density function fx(x) e*, x<0, 0, otherwise. The moment generating function My(t)= Use My(t) to compute E(X)= and Var(x)= Use My(t) to compute the compute the mgf for 3 Y= X-2. That is My(t)= = 2
To compute the moment generating function (MGF) for the random variable X, we need to use the formula:
[tex]My(t) = E(e^(tx))[/tex]
Given that the density function for X is fx(x) = e^(-x), x < 0, and 0 otherwise, we can write the MGF as follows:
[tex]My(t) = ∫[from -∞ to ∞] e^(tx) * fx(x) dx[/tex]
Since the density function fx(x) is non-zero only for x < 0, we can rewrite the integral accordingly:
[tex]My(t) = ∫[from -∞ to 0] e^(tx) * e^x dx + ∫[from 0 to ∞] e^(tx) * 0 dx[/tex]
The second integral is zero because the density function is zero for x ≥ 0. We can simplify the expression:
[tex]My(t) = ∫[from -∞ to 0] e^(x(1+t)) dx[/tex]
Using the properties of exponents, we can simplify further:
[tex]My(t) = ∫[from -∞ to 0] e^((1+t)x) dx[/tex]
Now we can evaluate this integral:
[tex]My(t) = [1 / (1+t)] * e^((1+t)x) | [from -∞ to 0)[/tex]
= [tex][1 / (1+t)] * (e^((1+t)(0)) - e^((1+t)(-∞)))[/tex]
= [tex][1 / (1+t)] * (1 - 0)[/tex]
= [tex]1 / (1+t)[/tex]
The moment generating function My(t) simplifies to 1 / (1+t).
To compute the expected value (E(X)) and variance (Var(X)), we can differentiate the MGF with respect to t:
E(X) = My'(t) evaluated at t=0
Var(X) = My''(t) evaluated at t=0
Taking the derivative of My(t) = 1 / (1+t) with respect to t, we get:
[tex]My'(t) = -1 / (1+t)^2[/tex]
Evaluating My'(t) at t=0:
E(X) = [tex]My'(0) = -1 / (1+0)^2 = -1[/tex]
Thus, the expected value of X is -1.
To compute the second derivative, we differentiate My'(t) =[tex]-1 / (1+t)^2[/tex]again:
[tex]My''(t) = 2 / (1+t)^3[/tex]
Evaluating My''(t) at t=0:
Var(X) =[tex]My''(0) = 2 / (1+0)^3 = 2[/tex]
Thus, the variance of X is 2.
Now, let's compute the MGF for the random variable Y = X - 2:
[tex]My_Y(t) = E(e^(t(Y)))= E(e^(t(X - 2)))= E(e^(tX - 2t))[/tex]
Using the properties of the MGF, we know that if X is a random variable with MGF My(t), then e^(cX) has MGF My(ct), where c is a constant. Therefore, we can rewrite the MGF for Y as:
[tex]My_Y(t) = e^(-2t) * My(t)[/tex]
Substituting My(t) = 1 / (1+t) from the previous calculation, we get:
[tex]My_Y(t) = e^(-2t) * (1 / (1+t))[/tex]
Simplifying further:
[tex]My_Y(t) = e^(-2t) / (1+t)[/tex]
Thus, the MGF for Y = X
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Colby made a scale model of the Washington Monument. The monument has an actual height of 554 feet. Colby’s model used a scale in which 1 inch represents 100 feet. What is the height in inches of Colby’s model?
Answer:
500043004030405.3
Step-by-step explanation:
a - 2/3 = 3/5 how much is a?
Answer:
19/15
Step-by-step explanation: In order to solve for A add 2/3 to both sides of the equation to get A alone and 2/3 + 3/5 is equal to 10/15 + 9/15 which means the answer is 19/15.
Are the following true or false? Justify your answers briefly. a) Let f, g (0, [infinity]) → R. If limx→[infinity] (fg)(x) exists and is finite then so are both limx→[infinity] f(x) and limx→[infinity] g(x). b) Let {n} and {n} be sequences such that n < yn for all n € N. If → x and Yny, then x
False. The limit of f(x) as x approaches infinity does not exist (it approaches zero), and the limit of g(x) as x approaches infinity is infinite. Therefore, the statement is false.
False. The statement is not necessarily true. The existence of the limit of the product (fg)(x) as x approaches infinity does not guarantee the existence of the limits of f(x) and g(x) individually.
Counterexamples can be found by considering functions that approach zero at different rates. For instance, let f(x) = 1/x and g(x) = x. As x approaches infinity, the product (fg)(x) = x/x = 1 approaches 1, which is finite. However, the limit of f(x) as x approaches infinity does not exist (it approaches zero), and the limit of g(x) as x approaches infinity is infinite. Therefore, the statement is false.
For instance, let f(x) = 1/x and g(x) = x. As x approaches infinity, the product (fg)(x) = x/x = 1 approaches 1, which is finite.
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calculate the double integral ∫∫r(10x 10y 100)da where r is the region: 0≤x≤5,0≤y≤5
The solution of the double integral ∫∫r(10x+10y+100)dA is found to be 5937.5.
To calculate the double integral ∫∫r(10x+10y+100)dA over the region r: 0 ≤ x ≤ 5, 0 ≤ y ≤ 5, we can integrate with respect to x first and then with respect to y. Let's start by integrating with respect to x,
∫∫r(10x+10y+100) dA = ∫[0,5] ∫[0,5] (10x+10y+100)dxdy
Integrating with respect to x, we treat y as a constant,
= ∫[0,5] [(10x²/2) + 10xy + 100x] dx dy
Next, we integrate the expression [(10x²/2) + 10xy + 100x] with respect to x over the range [0,5],
= ∫[0,5] [(10x²/2) + 10xy + 100x] dx dy
= [5x³/3 + 5xy²/2 + 50x²] evaluated from x=0 to x=5 dy
= [(5(5)³/3 + 5(5)y²/2 + 50(5)²) - (5(0)³/3 + 5(0)y²/2 + 50(0)²)] dy
= [(125/3 + 125y²/2 + 250) - 0] dy
= (125/3 + 125y²/2 + 250) dy
Now, we integrate the expression (125/3 + 125y/2 + 250) with respect to y over the range [0,5],
= ∫[0,5] (125/3 + 125y²/2 + 250) dy
= [(125/3)y + (125/6)y³ + 250y] evaluated from y=0 to y=5
= [(125/3)(5) + (125/6)(5³) + 250(5)] - [(125/3)(0) + (125/6)(0³) + 250(0)]
= [625/3 + (125/6)(125) + 1250] - [0 + 0 + 0]
= 625/3 + 125/6 * 125 + 1250
= 625/3 + 15625/6 + 1250
= 2083.33 + 2604.17 + 1250
= 5937.5
Therefore, the double integral ∫∫r(10x+10y+100)dA over the region r: 0 ≤ x ≤ 5, 0 ≤ y ≤ 5 is equal to 5937.5.
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4(8x - 3) - 6 = 5 + 2x
WHATS THE SOLUTION???
Answer:
x = 23/30
Step-by-step explanation:
4(8x - 3) - 6 = 5 + 2x
32x - 12 - 6 = 5 + 2x
32x - 18 = 5 + 2x
32x - 2x = 5 + 18
30x = 23
x = 23/30
Answer:
30x=14
Step by Step Explanation:
32x-9=5+2x
32x-2x=30x
30x-9=5
5+9 is 14
30x=14
Find the area of the figure.
HELP PLZZ
Answer:
159.25 ft²
I hope this helps! :)
Step-by-step explanation:
Formulas:
For the Rectangle... bh = a
For the Semicircle... 1/2 × πr²
Step 1:
Solve the area for the rectangle:
bh = a
10 × 12 = 120
a = 120 ft²
Step 2:
Solve the Area for the Semicircle:
1/2 × πr²
1/2 × 3.14 = 1.57
Radius = Diameter ÷ 2
10 ÷ 2 = 5
Radius = 5
1.57 × 5²
1.57 × 5 × 5
= 39.25 ft²
Step 3:
Add the two areas together:
120 + 39.25 = 159.25 ft²
11 - x when x= -4 how do you solve this
Answer:
15 is the answer
Step-by-step explanation:
We know that x = -4, so substitute x for -4 in the problem
11 - (-4)
2 negative signs make a positive sign
11 + 4
=15
Answer:
Hi! The answer to your question is [tex]15[/tex]
How to solve is whenever there is an x, replace it with a -4 so the problem would be set up like this 11-(-4) and at that point you can just solve it in a calculator
Step-by-step explanation:
☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆
☁Brainliest is greatly appreciated!!☁
Hope this helps!!
- Brooklynn Deka
Find the area of the shape shown below.
3
3
units?
Answer:
find the answer of the rectangle (7×3=21)
than find the area if one triangle and do 7/2 to get base then multiply 1/2base×height. because there are two triangles add the area to itself then add it to the area of the rectangle. the two triangles shoukd equal 21 together and 21 plus 21 equals 42.
Step-by-step explanation:
im sorry if this is incorrect but it should be right
Linear programming can be used to find the optimal solution for profit, but cannot be used for nonprofit organizations. False True
The statement "Linear programming can be used to find the optimal solution for profit, but cannot be used for nonprofit organizations" is False.
Linear programming can be used to find the optimal solution for profit as well as for non-profit organizations. Linear programming is a method of optimization that aids in determining the best outcome in a mathematical model where the model's requirements can be expressed as linear relationships. Linear programming can be used to solve optimization problems that require maximizing or minimizing a linear objective function, subject to a set of linear constraints.
Linear programming can be used in a variety of applications, including finance, engineering, manufacturing, transportation, and resource allocation. Linear programming is concerned with determining the values of decision variables that will maximize or minimize the objective function while meeting all of the constraints. It is used to find the optimal solution that maximizes profits for for-profit organizations or minimizes costs for non-profit organizations.
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How many turns must an ideal solenoid 10 cm long have if it is to generate a magnetic field of 1.5 mT when a current of 1.0 A passes through it?
a) 3.5
b) 1.8
c) 2.2
d) 0.50
e) 2.8
1.8 turns must an ideal solenoid should have if it is to generate a magnetic field of 1.5 mT when a current of 1.0 A passes through it
To calculate the number of turns required for an ideal solenoid, we can use the formula for the magnetic field inside a solenoid: B = μ₀ * n * I, where B is the magnetic field, μ₀ is the permeability of free space (constant), n is the number of turns per unit length, and I is the current.
Rearranging the formula, we have n = B / (μ₀ * I).
Given B = 1.5 mT (or 1.5 x 10⁻³ T) and I = 1.0 A, and knowing that μ₀ is a constant, we can substitute these values into the formula to find n.
n = (1.5 x 10⁻³) / (4π x 10⁻⁷ * 1.0) ≈ 1.19 x 10⁴ turns/m.
Since the solenoid is 10 cm (0.1 m) long, we can multiply n by the length to find the total number of turns:
Total turns = (1.19 x 10⁴ turns/m) * 0.1 m ≈ 1.19 x 10³ turns.
Rounding to the nearest whole number, the closest option is (b) 1.8.
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can someone please help me out its important please.
PLISSSS HELP 20 POINTS
Answer:
x=8
Step-by-step explanation:
Because you are solving for x, you want to cancel out the y terms. You can do this by multiplying the entire equations by numbers that will make the y terms have equal numbers but opposite signs.
2(2x-5y=1)
5(-3x+2y=-18)
This turns into
4x-10y=2
-15x+10y=-90
The y terms cancel out, and the other terms can be added together.
-11x=-88
x=8
ok so i thought i knew what i was doing but then i didn't know what i was doin-
class 9 help who are clever will get a brainlist
will give 20 brainly PLEASE NEED HELP NOW
plz put the answer as simple as a b c or d
Answer:
1. A
2. C
Step-by-step explanation:
The mean score of a competency test is 64, with a standard deviation of 4. Between what two values do about 99.7% of the values lie? (Assume the data set has a bell-shaped distribution.) Between 56 and 72 Between 60 and 68 O Between 52 and 76 Between 48 and 80
In a dataset with a bell-shaped distribution, approximately 99.7% of the values lie within three standard deviations of the mean. Given a mean score of 64 and a standard deviation of 4 on a competency test, we can determine the range within which about 99.7% of the values will fall. The correct range is between 56 and 72.
To calculate the range, we need to consider three standard deviations above and below the mean. Three standard deviations from the mean account for approximately 99.7% of the data in a bell-shaped distribution.
Lower limit: Mean - (3 * Standard Deviation)
= 64 - (3 * 4)
= 64 - 12
= 52
Upper limit: Mean + (3 * Standard Deviation)
= 64 + (3 * 4)
= 64 + 12
= 76
Therefore, about 99.7% of the values lie between 52 and 76.
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My friend Yoy purchased some rews for $3 each and some jooghs for
$5 each. The total cost was about $60. Altogether, he purchased 18
items.
Write a system of equations, in standard form, to model the
relationship between Yoy's rews (x) and jooghs (y).
Answer:
x+Y =x68 i thinkStep-by-step explanation:
Answer:
86
Step-by-step explanation:
Example 1
Make a graph for the table in the Opening Exercise.
Example 2
Use the graph to determine which variable is the independent variable and which is the dependent variable. Then state the relationship between the quantities represented by the variables
Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint: Use an energy argument-multiply the ODE by y and integrate from p=0 to r=R; use integration by parts and use the boundedness at r = 0 to get the boundary term to vanish.
The eigenvalue problem (4.75-4.77) has no negative eigenvalues.
In the eigenvalue problem (4.75-4.77), we aim to show that there are no negative eigenvalues. To do this, we employ an energy argument.
First, we multiply the ordinary differential equation (ODE) by the eigenfunction y and integrate from p=0 to r=R. By applying integration by parts, we manipulate the resulting equation to obtain a boundary term. Utilizing the boundedness at r=0, we can show that this boundary term vanishes.
Consequently, this implies that there are no negative eigenvalues in the given eigenvalue problem.
By employing this energy argument and carefully considering the properties of the ODE, we can confidently conclude the absence of negative eigenvalues.
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8 ft
Find the area of the figure.
Answer:
Area of a rectangle is length multiplied by the width. In this case, length is equal to width. So, Area is 8 ft * 8 ft which is 64 ft2.
Use the decimal grid to write the percent and fraction equivalents.
0.53
Answer:
53%
53/100
Step-by-step explanation:
The figures are similar. Give the ratio of the perimeters and the ratio of the areas of the first figure to the second.
a. 7:8 and 49:64
b. 8:9 and 49:64
c. 8:9 and 64:81
d. 7:8 and 64:81
The correct answer is: c. 8:9 and 64:81. The ratio of the areas of the first figure to the second figure is 64:81. This means that the area of the second figure is larger by a factor of 81/64 compared to the first figure.
When two figures are similar, their corresponding sides are proportional. This means that the ratio of the perimeters is equal to the ratio of the corresponding side lengths. Additionally, the ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding side lengths.
In this case, the ratio of the perimeters of the first figure to the second figure is 8:9. This means that the perimeter of the second figure is larger by a factor of 9/8 compared to the first figure.
The ratio of the areas of the first figure to the second figure is 64:81. This means that the area of the second figure is larger by a factor of 81/64 compared to the first figure.
Therefore, the correct answer is c. 8:9 and 64:81.
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Solve system of equations given below using both inverse matrix (if possible) and reduced row echelon forms. (20 Points each)
a) xy + 2x_2 + 2x_3 = 1
x_1 - 2x_2 + 2x_3 = - 3
3x_1 - x_2 + 5x_3 = 7
b) x_1 + 2x_2 + 2x_3 + 5x_4 = 0
x_1 - 2x_2 + 2x_3 - 4x_4 = 0
3x_1 - x_2 + 5x_3 + 2x_4 = 0
3x_1, -2x_2 + 6x_3 - 3x_4 = 0.
The solution to the system of equations is: x1 = 1/2, x2 = 9/4, x3 = 1, x4 = 0
a) Solving the system of equations using inverse matrix:
Let's write the system of equations in matrix form: AX = B
The coefficient matrix A is:
A = [[y, 2, 2], [1, -2, 2], [3, -1, 5]]
The variable matrix X is:
X = [[x], [y], [z]]
The constant matrix B is:
B = [[1], [-3], [7]]
To solve for X, we need to find the inverse of matrix A (if it exists):
Calculate the determinant of matrix A: |A|
|A| = y((-2)(5) - (-1)(2)) - 2((1)(5) - (3)(2)) + 2((1)(-1) - (3)(-2))
= -9y + 4
Check if |A| is non-zero. If |A| ≠ 0, then the inverse of A exists.
Since |A| = -9y + 4, it can only be zero if y = 4/9.
If y ≠ 4/9, then |A| ≠ 0, and we can proceed to find the inverse of A.
Calculate the matrix of minors of A: Minors(A)
Minors(A) = [[(-2)(5) - (-1)(2), (1)(5) - (3)(2), (1)(-1) - (3)(-2)],
[(2)(5) - (2)(2), (3)(5) - (3)(2), (3)(-1) - (3)(-2)],
[(2)(-1) - (2)(-2), (3)(-1) - (1)(2), (3)(-2) - (1)(-1)]]
= [[-8, -1, -1],
[6, 9, -3],
[2, -1, -5]]
Calculate the matrix of cofactors of A: Cofactors(A)
Cofactors(A) = [[(-1)^1(-8), (-1)^2(-1), (-1)^3(-1)],
[(-1)^2(6), (-1)^3(9), (-1)^4(-3)],
[(-1)^3(2), (-1)^4(-1), (-1)^5(-5)]]
= [[-8, 1, -1],
[6, -9, 3],
[-2, 1, -5]]
Calculate the adjugate of A: Adj(A) = Transpose(Cofactors(A))
Adj(A) = [[-8, 6, -2],
[1, -9, 1],
[-1, 3, -5]]
Calculate the inverse of A: A^(-1) = Adj(A)/|A|
A^(-1) = [[(-8)/(9y - 4), 6/(9y - 4), (-2)/(9y - 4)],
[1/(9y - 4), (-9)/(9y - 4), 1/(9y - 4)],
[(-1)/(9y - 4), 3/(9y - 4), (-5)/(9y - 4)]]
Multiply A^(-1) by B to find X:
X = A^(-1) * B
= [[(-8)/(9y - 4), 6/(9y - 4), (-2)/(9y - 4)],
[1/(9y - 4), (-9)/(9y - 4), 1/(9y - 4)],
[(-1)/(9y - 4), 3/(9y - 4), (-5)/(9y - 4)]] * [[1], [-3], [7]]
Simplifying the multiplication will give the solution for X in terms of y.
b) Solving the system of equations using reduced row echelon form:
Let's write the system of equations in augmented matrix form [A | B]:
The augmented matrix [A | B] is:
[1, 2, 2, 5 | 0]
[1, -2, 2, -4 | 0]
[3, -1, 5, 2 | 0]
[3, -2, 6, -3 | 0]
Using Gaussian elimination and row operations, we can transform the augmented matrix to reduced row echelon form.
Performing row operations:
R2 = R2 - R1
[1, 2, 2, 5 | 0]
[0, -4, 0, -9 | 0]
[3, -1, 5, 2 | 0]
[3, -2, 6, -3 | 0]
R3 = R3 - 3R1
[1, 2, 2, 5 | 0]
[0, -4, 0, -9 | 0]
[0, -7, -1, -13 | 0]
[3, -2, 6, -3 | 0]
R4 = R4 - 3R1
[1, 2, 2, 5 | 0]
[0, -4, 0, -9 | 0]
[0, -7, -1, -13 | 0]
[0, -8, 0, -18 | 0]
R2 = (-1/4)R2
[1, 2, 2, 5 | 0]
[0, 1, 0, 9/4 | 0]
[0, -7, -1, -13 | 0]
[0, -8, 0, -18 | 0]
R3 = R3 + 7R2
[1, 2, 2, 5 | 0]
[0, 1, 0, 9/4 | 0]
[0, 0, -1, -1 | 0]
[0, -8, 0, -18 | 0]
R4 = R4 + 8R2
[1, 2, 2, 5 | 0]
[0, 1, 0, 9/4 | 0]
[0, 0, -1, -1 | 0]
[0, 0, 0, -6 | 0]
R4 = (-1/6)R4
[1, 2, 2, 5 | 0]
[0, 1, 0, 9/4 | 0]
[0, 0, -1, -1 | 0]
[0, 0, 0, 1 | 0]
R1 = R1 - 2R2 - 2R3
[1, 0, 0, 1/2 | 0]
[0, 1, 0, 9/4 | 0]
[0, 0, -1, -1 | 0]
[0, 0, 0, 1 | 0]
R3 = -R3
[1, 0, 0, 1/2 | 0]
[0, 1, 0, 9/4 | 0]
[0, 0, 1, 1 | 0]
[0, 0, 0, 1 | 0]
The reduced row echelon form of the augmented matrix is obtained.
From the reduced row echelon form, we can write the system of equations:
x1 = 1/2
x2 = 9/4
x3 = 1
x4 = 0
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