Answer:
The general solution to the given linear system is x = 3z - 1, y = -z + 1, where z is a free variable. This means that the solution consists of infinitely many points that lie on a straight line in three-dimensional space.
To solve the linear system, we can use the method of elimination or Gaussian elimination. Here, we'll use Gaussian elimination to find the general solution.
We start by writing the augmented matrix of the system:
[1 1 -2 | 0]
[2 2 3 | 1]
[3 3 1 | 7]
To simplify the matrix, we perform row operations to create zeros in the first column below the first entry. We subtract twice the first row from the second row and subtract three times the first row from the third row:
[1 1 -2 | 0]
[0 0 7 | 1]
[0 0 7 | 7]
Next, we divide the second and third rows by 7 to create leading ones:
[1 1 -2 | 0]
[0 0 1 | 1/7]
[0 0 1 | 1]
Now, we perform row operations to create zeros in the second column below the second entry. We subtract the third row from the second row:
[1 1 -2 | 0]
[0 0 1 | 1/7]
[0 0 0 | 0]
From the last row, we can see that 0z = 0, which means that z is a free variable. We can assign a parameter to z, say t, and solve for x and y in terms of t. From the first row, we have x + y - 2z = 0. Plugging in the values for x and y, we get x = 3z - 1 and y = -z + 1. Therefore, the general solution to the linear system is x = 3z - 1, y = -z + 1, where z is a free variable.
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3. The angle between two equal sides of an isosceles triangle is 52º.
Each of the equal sides is 18 cm long.
a) Determine the measures of the two equal angles in the triangle.
b) Determine the length of the third side.
c) Determine the perimeter of the triangle.
Answer:
a) 64° each
b) 15.78 cm
c) 51.78 cm
Step-by-step explanation:
a) (180 - 52)/2 = 64°
b) cos 64 = x/18, x = 7.89, 2x = 15.78 cm
c) 18 + 18 + 15.78 = 51.78 cm
Solve the following system of IVP: [3 -1 01 7 x' = Ax where A = 4 -2 0 and x(0) = 10x -4 21 Hint: The eigenvalues are λ₁ = -1,A₂ = 2, A3 = 2.
For the given system of initial value problem,
A₁ = 7,
A₂ = -1, and
y = 2exp(-t),
To solve the given system of initial value problem (IVP) with the matrix equation x' = Ax, where A is the given matrix and x(0) = [10, -4, 21], we can use the eigenvalue-eigenvector method.
The matrix A is given as:
A = | 4 -2 0 |
| 3 -1 0 |
| 1 7 2 |
Let's find the eigenvalues and eigenvectors of matrix A.
To find the eigenvalues, we solve the characteristic equation:
det(A - λI) = 0
Substituting the values, we get:
| 4-λ -2 0 |
| 3 -1-λ 0 |
| 1 7 2-λ |
Expanding the determinant, we have:
(4-λ)[(-1-λ)(2-λ)] + 2[3(2-λ) - (-1-λ)7] = 0
Simplifying and solving the equation, we find the eigenvalues:
λ₁ = -1
λ₂ = 2
λ₃ = 2
Next, let's find the corresponding eigenvectors.
For λ₁ = -1:
(A + I)v₁ = 0
| 5 -2 0 | | v₁₁ | | 0 |
| 3 -1 0 | * | v₁₂ | = | 0 |
| 1 7 1 | | v₁₃ | | 0 |
Solving the system of equations, we find the eigenvector corresponding to λ₁:
v₁ = | 2 |
| 5 |
|-1 |
For λ₂ = 2:
(A - 2I)v₂ = 0
| 2 -2 0 | | v₂₁ | | 0 |
| 3 -3 0 | * | v₂₂ | = | 0 |
| 1 7 0 | | v₂₃ | | 0 |
Solving the system of equations, we find the eigenvector corresponding to λ₂:
v₂ = | 1 |
| 1 |
|-1 |
For λ₃ = 2:
(A - 2I)v₃ = 0
| 2 -2 0 | | v₃₁ | | 0 |
| 3 -3 0 | * | v₃₂ | = | 0 |
| 1 7 0 | | v₃₃ | | 0 |
Solving the system of equations, we find the eigenvector corresponding to λ₃:
v₃ = | 2 |
| 3 |
| 1 |
Now that we have the eigenvalues and eigenvectors, we can write the general solution to the system of differential equations as:
x(t) = c₁ * exp(λ₁ * t) * v₁ + c₂ * exp(λ₂ * t) * v₂ + c₃ * exp(λ₃ * t) * v₃
Substituting the given initial condition x(0) = [10, -4, 21], we can find the specific solution by solving the following system of equations:
c₁ * v₁ + c₂ * v₂ + c₃ * v₃ = x(0)
Substituting the values of the eigenvectors and the initial condition, we get:
2c₁ + c₂ + 2c₃ = 10 (Equation 1)
5c₁ + c₂ + 3c₃ = -4 (Equation 2)
-c₁ - c₂ + c₃ = 21 (Equation 3)
Solving this system of equations will give us the values of c₁, c₂, and c₃, which will determine the specific solution.
By solving Equations 1, 2, and 3, we find:
c₁ = 1
c₂ = -5
c₃ = -3
Therefore, the specific solution to the initial value problem is:
x(t) = exp(-t) * | 2 | + exp(2t) * | 1 | + exp(2t) * |-3 |
| 5 | | 1 |
|-1 | | 1 |
Simplifying this expression, we get:
x(t) = | 2exp(-t) + 5exp(2t) - 3exp(2t) |
| 5exp(-t) + exp(2t) + exp(2t) |
|-exp(-t) + exp(2t) + exp(2t) |
Finally, we can rewrite the solution in the given form (7) - ₁ (1) ¹²+₂ (1) ¹:
x(t) = 7 * | 2exp(-t) | + (-1) * | 5exp(-t) |
| 5exp(-t) | | -exp(-t) |
Therefore, A₁ = 7, A₂ = -1, and y = 2exp(-t).
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Please help me guys!
Answer:
11+13+16 or 14x15x18
Step-by-step explanation:
HOPE IT HELPS! :)
a rectangular field 50m wide and ym long requires 260m fencing. find y
Answer:
5.2m
Step-by-step explanation:
Factor as the product of two binomials.
x^2-3x+2=
Answer: (x - 2) (x - 1）
Step-by-step explanation:
x²-3x+2
= (x - 2) (x - 1）
someone help me ill mark you brainliest x
Answer:
x= 64° y= 18° z= 98°
Step-by-step explanation:
x= 64°
y = 18°
z = 180° - x - y
z = 180° - 64° - 18°
z= 98°
A doctor is measuring body temperature for patients visiting the office. The doctor believes the average body temperature is less than 98.6 degrees Fahrenheit and would like to test this claim. During the process of hypothesis testing, the doctor computes a value from the sample data, which will be used to compare the sample data to the population parameter. What value did the doctor compute?
Answer:
Test statistic
Step-by-step explanation:
let a be a countably infinite set. which of the following must be true?
a. there is a one-to-one function from set A to the set of rational numbers
b. the set of integers have the same cardinality as the set A
c. there can be no surjective function from the set of rational numbers
d. the open interval (0, 1) has the same cardinality as the set A
The set of integers has the same cardinality as the countably infinite set A. The correct option is b.
A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers (integers starting from 1). This means that we can establish a one-to-one function (bijection) between the countably infinite set A and the set of integers.
Options a, c, and d are not necessarily true.
Option a: There is no guarantee that there is a one-to-one function from the set A to the set of rational numbers. The set of rational numbers, also known as Q, is uncountably infinite, while A is only countably infinite.
Option c: It is possible to have a surjective function (onto) from the set of rational numbers to a countably infinite set. For example, a function that maps every rational number to its numerator can be surjective onto the set of integers.
Option d: The open interval (0, 1) has a higher cardinality (uncountable) than a countably infinite set. The interval (0, 1) contains an uncountably infinite number of real numbers between 0 and 1, whereas A is countably infinite.
The correct option is b.
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The data set below is a random sample of the heights (in meters) of women belonging to a certain ethnic subgroup. Assume the population is normally distributed 1.63 1.62 1.61 162 1.39 1.55 a) Find the mean and standard deviation of the data.
The mean of the data set is approximately 1.57 meters. The standard deviation of the data set is approximately 0.0968 meters, indicating the average deviation of data points from the mean.
To compute the mean and standard deviation of the data set, we'll use the following formulas:
Mean (μ) = (sum of all data values) / (total number of data values)
Standard Deviation (σ) = sqrt((sum of squared differences from the mean) / (total number of data values))
Let's calculate the mean and standard deviation for the given data set:
Data set: 1.63, 1.62, 1.61, 1.62, 1.39, 1.55
Mean (μ) = (1.63 + 1.62 + 1.61 + 1.62 + 1.39 + 1.55) / 6 = 9.42 / 6 ≈ 1.57
Next, we calculate the sum of squared differences from the mean:
(1.63 - 1.57)^2 + (1.62 - 1.57)^2 + (1.61 - 1.57)^2 + (1.62 - 1.57)^2 + (1.39 - 1.57)^2 + (1.55 - 1.57)^2 ≈ 0.0666
Finally, we calculate the standard deviation:
Standard Deviation (σ) = sqrt(0.0666 / 6) ≈ 0.0968
Therefore, the mean of the data set is approximately 1.57 meters and the standard deviation is approximately 0.0968 meters.
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please solve for x!!!
Answer:
2x+156=4x+192
192-156=-4x+2x
36=-2x
x=-18
Step-by-step explanation:
Assume that the U.S. Mint manufactures dollar coins so that the standard deviation is 0.0390 g. The accompanying list contains weights (grams) of dollar coins manufactured with a new process designed to decrease the standard deviation so that it is less than 0.0390 g. This sample has these summary statistics: n=16, x = 8,068 9. s=0.02 g. A significance level is used to test the claim that the sample is from a population with a standard deviation less than 0.0390 g. 11 we want to use a 0.05 significance level and a parametric method to test the claim that the sample is from a population with a standard deviation less than 0.0390 g, what requirements must be satisfied? How does the normality requirement for a hypothesis test of a claim about a standard deviation differ from the normality requirement for a hypothesis test of a claim about a mean? Click the icon to view the weights of dollar coins manufactured with the new process, What requirements must be satisfied? Select all that apply. A. Either or both of these conditions are satisfied:
A. the population is normally distributed or the sample size is greater than 30. B. The population has a chi-square distribution C. The sample is a simple random sample. D. The sample size is greater than 30. E. The population has a normal distribution F. No requirements must be satisfied.
To test the claim that the sample is from a population with a standard deviation less than 0.0390 g, and using a 0.05 significance level and a parametric method, certain requirements must be satisfied. The requirements include:
A. Either or both of these conditions are satisfied:
A. the population is normally distributed or the sample size is greater than 30.
C. The sample is a simple random sample.
The normality requirement for a hypothesis test of a claim about a standard deviation differs from the normality requirement for a hypothesis test of a claim about a mean. For a hypothesis test of a claim about a standard deviation, it is not necessary for the population to be normally distributed. Instead, either the population should be normally distributed or the sample size should be sufficiently large (typically greater than 30) for the Central Limit Theorem to apply. In this case, the requirement is satisfied if either the population is normally distributed or the sample size is greater than 30. However, the sample should still be a simple random sample, which ensures that the observations are independent and representative of the population.
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1/2 x 240 im just too lazy to do it lol
Answer: 120
Step-by-step explanation:
Factor the expression using the GCF.
42 – 12 =
Answer:
30
Step-by-step explanation:
Well 42
-
12
30
Solve the equation 4x-12=(11-3x)
Answer:
Step-by-step explanation:
4x - 12 = 11 - 3x
Bringing like terms on one side
4x + 3x = 11 + 12
7x = 23
x = 23/7
this ladybird is rotated. choose the correct word to complete eachsentence. this ladybird made a __ turn clockwise this ladybird made a -__ anticlockwise
We can see here that when the ladybird is rotated:
This ladybird made a 90° turn clockwise.
This ladybird made a 180° anticlockwise.
What is rotation?In mathematics, rotation refers to a transformation that turns or rotates an object around a fixed point called the center of rotation. It is a fundamental concept in geometry and is used to describe the movement of points, shapes, or figures in the plane or in three-dimensional space.
A rotation involves specifying the center of rotation, the angle of rotation, and the direction of rotation (clockwise or counterclockwise).
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suzy randomly picks marbles from a bag containing 12 identical marbles. how many possible outcomes are there if she selects 9 marbles?
There are 12 identical marbles in a bag, and Suzy is going to select 9 marbles from the bag at random.
The problem asks how many possible outcomes there are.
To begin with, we need to understand the concept of combinations. A combination is a way to select a subset of objects from a larger set, without regard to the order of the objects. For example, if we have four marbles (A, B, C, and D), there are six possible combinations of two marbles: AB, AC, AD, BC, BD, and CD.
In this problem, we have 12 marbles and we are choosing 9 of them. To find the total number of combinations, we can use the formula for combinations:
nCr = n! / r!(n-r)!
where n is the total number of objects, r is the number of objects we are choosing, and ! represents factorial (i.e. multiplying a number by all the positive integers less than it).
So, plugging in our numbers:
12C9 = 12! / 9!(12-9)! = 12! / 9!3! = (121110) / (321) = 220
Therefore, there are 220 possible outcomes if Suzy selects 9 marbles at random from a bag containing 12 identical marbles.
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Can y’all help me?
The teacher wants me to do work but ion understand this bc Ian done school for 3 months so I’m a bit behind...
Answer:
402.1 in^2
Step-by-step explanation:
Find the area of the circle with the 18 in. radius first.
Formula for a circle is πr^2
So the area of the circle with the 8 in. radius is 1017.9 in^2
Now do the same for the 14 in. radius circle.
The area for the 14 in. radius circle is 615.8
Now subtract:
1017.9 - 615.8 =
402.1 in^2
hope this helped!
What is the probability of selecting an orange candy
Answer:
how many colors are there and how many of each color?
Step-by-step explanation:
Probability of an event is the measurement of its chance of occurrence.
The probability of selecting an orange candy depends on how many candies are there. If its the only candy, and at least one candy must be chosen, then its probability is 1 or say in percent it is 100%How to calculate the probability of an event?Suppose that there are finite elementary events in the sample space of the considered random experiment, and all are equally likely.
Then, suppose we want to find the probability of an event E.
Then, its probability is given as
[tex]P(E) = \dfrac{\text{Number of favorable cases}}{\text{Number of total cases}} = \dfrac{n(E)}{n(S)}[/tex]
where favorable cases are those elementary events who belong to E, and total cases are the size of the sample space.
How to convert percent to probability?Percent counts the number compared to 100 whereas probability counts it compare to 1.
So, if we have a%, that means for each 100, there are 'a' parts. If we divide each of them with 100, we get:
For each 1, there are a/100 parts.
Thus, 50% = 50/100 = 0.50 (in probability)
The probability of selecting an orange candy depends on how many candies are there.
If its the only candy, and at least one candy must be chosen, then its probability is 1 or say in percent it is 100% because then n(E) = 1 and n(S) = 1 where
E is event of selecting one orange candy from only 1 candy availablen(E) = count of ways E can be done = 1n(S) = number of ways 1 candy can be selected from only 1 candy availableThus, the probability of selecting an orange candy depends on how many candies are there. If its the only candy, and at least one candy must be chosen, then its probability is 1 or say in percent it is 100%
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can someone please explain how to do this??
Notice that the weird figure looks like a rectangle that is missing a piece.
In this case, find the area of the rectangle... then subtract the area of the missing piece from that area of the rectangle.
-----------------------------------------------------------------------
Area of rectangle: 4 x 5 = 20 sq. inches
-----------------------------------------------------------------------
How to find the area of the missing piece:
Notice that the other side is 2 inches while the left side is 4 inches, which means that if I subtract 2 from 4... then I will know the width of the missing piece. [4 - 2 = 2 inches is the width of the missing piece.]Notice that the top side is 4 inches while the bottom side is 5 inches, which means that if I subtract 4 from 5... then I will know the length of the missing piece. [5 - 4 = 1 inch is the length of the missing piece].So, the area of the missing piece: A = lw = (1)(2) = 2 sq. inches
-----------------------------------------------------------------------
Now, subtract the area of the missing piece from the area of the rectangle:
20 - 2 = 18 sq. inches
-----------------------------------------------------------------------
ANSWER: 18 sq. inchesA cup and saucer together cost N$ 20.50 . A cup and two saucer cost N$ 27.00 . Find the cost of the cup and the saucer.
Answer: Cup-$14, Saucer-$6.5
Step-by-step explanation:
Given
Cup and saucer costs $20.5
A cup and two saucer costs $27
Assume the price of cup and saucer are x and y
So, we can write
[tex]\Rightarrow x+y=20.5\quad \ldots(i)\\\\\Rightarrow x+2y=27\quad \ldots(ii)[/tex]
Solving [tex](i)\ \text{and}\ (ii)[/tex] we get
[tex]x=\$\ 14, y=\$\ 6.5[/tex]
Thus, the cost of the cup is $14 and that of the saucer is $6.5
Simplify the following and leave your answer in smallest form
1 + 2 + 1
_ _ _
2 3 6
Answer:
1/59
Step-by-step explanation:
Answer:
1/59
think of it dude
NO LINKS PLEASE
SOMEONE HELP IM DESPERATE
FIND THE EXACT VALUE OF EACH TRIGONOMETRIC FUNCTION USING THE UNIT CIRCLE
Answer: 45
Step-by-step explanation: 360-315=45
Antonio has six white socks, two black socks, and four gray socks in a drawer. If he randomly chooses a sock from the drawer, what is the probability he will choose a gray sock?
A.1/12
B.1/4
C.1/3
D.1/2
Answer:
C
Step-by-step explanation:
The probability is 1/3
If the two triangles are similar,what is the length of the missing side?
Answer:
8 m
Step-by-step explanation:
7.5 - 1.5 = 6
2 + 6 = 8
If y varies inversely as x, and y = 33 when x = 7, what is the constant and what is value of y when x=11?
Answer:
We have that y = 21 when x = 11.
Step-by-step explanation:
We solve this question by proportions, using a rule of three.
Since they vary inversely, we apply the inverse rule of three, that is, with lateral multiplication instead of diagonal.
33 - 7
y - 11
So
[tex]11y = 33*7[/tex]
Dividing both sides by 11
[tex]y = 3*7 = 21[/tex]
We have that y = 21 when x = 11.
A quantitatively savvy, young couple is interested in purchasing a home in southern Sydney. They collected data on 48 houses that had recently sold in the area. They want to predict the selling price of homes (in thousands of dollars) based on the size of the home (in square feet).
The regression equation is House Price (in thousands) = 17.1 + 0.0643 Size (sq. ft.)
Predictor Coef SE Coef T P
Constant 17.06 24.59 0.69 0.491
Size (sq. ft.) 0.06427 0.01224 5.25 0.000
S = 48.5733 R-Sq = 37.5% R-Sq(adj) = 36.1%
Use the computer output to test the slope, at the 5% level, to determine whether size (in square feet) is an effective predictor of the selling price of recently sold homes. Include all details of the test.
a) What are the null (H) and alternative hypotheses (Ha) ?
b) What is the t-test statistics?
c) degree of freedom for this t-test statistics?
d) final conclusion of this test is ?
Since the p-value is less than the significance level of 0.05, we can reject the null hypothesis, the size of the home is a significant predictor of the selling price of homes sold recently in the southern Sydney area.
(a) Null hypothesis: H0: β1 = 0 (The size of the home is not a significant predictor of the selling price of homes sold recently).
Alternative hypothesis: Ha: β1 ≠ 0 (The size of the home is a significant predictor of the selling price of homes sold recently).
(b) The t-test statistic: We can use the t-test statistic to test whether the slope (β1) of the regression line is significant at the 5% level.
t = (β1 - 0) / SE(β1)
= 0.0643 / 0.01224
= 5.25
(c) The degrees of freedom (df) for this t-test statistic will be n - 2, where n is the sample size.
Since the sample size is 48, the degrees of freedom will be 48 - 2 = 46.
(d) Final conclusion of this test: Since the p-value is less than the significance level of 0.05, we can reject the null hypothesis.
Thus, we conclude that the size of the home is a significant predictor of the selling price of homes sold recently in the southern Sydney area.
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Carmen y Catalina comparan la nota que obtuvieron en su examen bimestral de Aritmética y mencionan lo siguiente: Nuestras notas juntas es igual a 34 puntos, pero se sabe que Carmen obtuvo 4 puntos más que Catalina. ¿Cuál es la nota de Catalina?
Answer:
La nota de Carolina es 15 puntos.
Step-by-step explanation:
Un sistema de ecuaciones lineales es un conjunto de ecuaciones lineales que tienen más de una incógnita, relacionadas mediante las ecuaciones.
En los sistemas de ecuaciones, se debe buscar los valores de las incógnitas. Al reemplazar en las ecuaciones, deben dar la solución planteada.
En este caso se debe plantear un sistema de ecuaciones, donde las incógnitas son:
x: nota de Carolinay: nota de CarmenAmbas notas juntas es igual a 34 puntos. Expresado matemáticamente es:
x + y= 34
Se sabe que Carmen obtuvo 4 puntos más que Catalina. Esto es:
y= x + 4
Entonces el sistema de ecuaciones a resolver es:
[tex]\left \{ {{x+y=34} \atop {y=x+4}} \right.[/tex]
A través del método de sustitución debes despejar una de las incógnitas en una de las ecuaciones y sustituir su valor en la siguiente. En este caso, ya tienes despejada la incógnita y en la segunda ecuación. Por lo que sustituyes esta expresión en la primer ecuación:
x+ x+4=34
Resolviendo obtienes:
x + x= 34 - 4
2*x=30
x= 30÷ 2
x=15
Sustituyendo este valor en la segunda ecuación obtienes:
y= x+4
y= 15 +4
y= 19
Recordando que "x" representa la nota de Carolina e "y" la nota de Carmen, entonces la nota de Carolina es 15 puntos y la nota de Carmen es 19 puntos.
The most efficient first step in the process to factor the trinomial 3x³-24x² + 36x is :
O A. factor out 3
O B. factor out 3x
O C. factor out (x - 2)
O D. factor out-1
Answer:
B
Step-by-step explanation:
If the angles
are represented in degrees, find both angles:
sin(x + 7) = cos(4x + 8)
Answer:
x = -1/3
Step-by-step explanation:
We have,
sin(x + 7) = cos(4x + 8)
We need to find the value of x.
We know that,
sin(90-x) = cosx
So,
sin(90-(x + 7)) = cos(4x + 8)
cos(x+7) = cos(4x + 8)
i.e.
(X+7) = (4x+8)
7-8 = 4x-x
-1 = 3x
x = -1/3
Answer:
sin(22 ∘ ) = cos(68∘ )
Step-by-step explanation:
Sin(x+7)=cos(4x+8)
(x+7) + (4x+8) = 90
x + 7 + 4x + 8 = 90
5x + 15 = 90
−15 = −15
5x/5 = 75/5
x = 15
Answer: sin(22 ∘ ) = cos(68∘ )
PLEASE HELP ME :(
The questions and answer choices are in the picture.
It would also reallyyyyy help me if you explain how you solved it. At least just show the steps.
Thank you!
Hi army! I found the answer!
Answer:
F. An 18-month loan with an annual simple interest rate of 4.75%
Step-by-step explanation:
I just used a credit card payoff calculator to solve this problem. You can use that calculator to solve other problems like this too!
Also, stream BTS Film Out for clear skin and good grades ;)
Have a great day!