Answer: i dont know
Step-by-step explanation:
1) The Pet Company has recently discovered a type of rock which, when crushed, is extremely absorbent. It is expected that the firm will experience (beginning now) an unusually high growth rate (20%) during the period (3 years) when it has exclusive rights to the property where this rock can be found. However, beginning with the fourth year the firm's competition will have access to the material, and from that time on the firm will assume a normal growth rate of 8% annually. During the rapid growth period, the firm's dividend payout ratio will be relatively low (20%), to conserve funds for reinvestment. However, the decrease in growth will be accompanied by an increase in dividend payout to 50%. Last year's earnings were $2.00 per share (E0) and the firm's cost of equity is 10%. What should be the current price of the common stock?
Answer:
$ 71.83
Step-by-step explanation:
This year's earnings=last year's earnings*(1+growth rate)=$2*(1+20%)=$2.4
This year's dividend=2.4 *payout ratio=2.4 *20%=$0.48
Next year's earnings=$2.4*1.2=$2.88
Next year's dividend=2.88 *0.2=$ 0.58
Year 3 earnings=$2.88*1.2=$ 3.46
Year three dividend= 3.46*0.2=$0.69
year 4 earnings =$3.46*1.08=$ 3.74
Dividend from year 4 onward= 3.74 *0.5=1.87
From year 4 terminal value=1.87 /(cost of equity-growth rate)=1.87/(10%-8%)=$93.5
Share price is present value of the above dividends amnd terminal value=0.48/(1+0.1)^0+0.58/(1+0.1)^1+0.69/(1+0.1)^2+93.5/(1+0.1)^3=$ 71.83
(50 POINTS!) The figure is transformed as shown in the diagram. Describe the transformation.
A) dilation, then reflection
B) reflection, then rotation
C) rotation, then translation
D) translation, then reflection
Answer:
The solution is rotation, then translation. The figure has been rotated about the origin by 90° and then translated 6 units to the right.
Lucy places five cards that are labeled 1 to 6 face down on the table and mixes them up. What is the likelihood that her friend Harry will draw an even numbered card?
Answer:
1/2 or 0.5
Step-by-step explanation:
Within 1 to 6,
Even numbers are 2, 4, 6
While odd numbers are 1, 3, 5
Assuming cards are mixed up properly and the probabilities of drawing a card with any number from 1 to 6 is constant,
Probability of drawing even number cards = number of units of even number cards ÷ total number of units of cards from 1 to 6 aka total number of cards
= 3 / 6
= 1 / 2
The probability of drawing even number cards is 1/2.
What is probability?Probability is "possibility that deals with occurrence of random events".
According to the question,
Lucy places five cards that are labeled 1 to 6 face down on the table and mixes them up.
Numbers of space = 1,2,3,4,5,6
Even numbers are 2, 4, 6 and odd numbers are 1, 3, 5.
To find Probability of drawing even number cards is equal to number of units of even number cards is divided by total number of units of cards.
Probability of even numbers of cards = 3/6
= 1/2
Hence, Probability of even numbers of cards is 1/2.
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I need help! I don't understand!
The process of completing the square is done for f(x) = x^2 +10x + 21 to be changed into a vertex form..which step shows Three steps toward this process? why did you pick your answer ?
Answer:
And for this case we can begin completing the square like this:
[tex] f(x) = x^2 +10x + (10/2)^2 +21 -(10/2)^2[/tex]
And after aggrupate the terms we got:
[tex] f(x) = (x+5)^2 +21 -25 [/tex]
And finally we have:
[tex] f(X)= (x+5)^2 -4[/tex]
And for this case our vertex would be:
[tex] (h,k) = (-5,-4) [/tex]
Step-by-step explanation:
For this case we have the following equation given:
[tex] f(x) = x^2 +10x +21[/tex]
And we want to find a formula in terms in the vertex form given by:
[tex] f(x) = a(x-h)^2 +k[/tex]
And for this case we can begin completing the square like this:
[tex] f(x) = x^2 +10x + (10/2)^2 +21 -(10/2)^2[/tex]
And after aggrupate the terms we got:
[tex] f(x) = (x+5)^2 +21 -25 [/tex]
And finally we have:
[tex] f(X)= (x+5)^2 -4[/tex]
And for this case our vertex would be:
[tex] (h,k) = (-5,-4) [/tex]
Maribel surveyed 55 people to find out their favorite types of music. The results are shown in the bar graph. Based on the information in the graph, which types of music were chosen by 40% of the people surveyed
Answer:
B. Jazz and opra
Step-by-step explanation:
40 percent of 55 is 22. Find whatever is equal to 22
B) Jazz & Opera music combo were chosen by 40% of people surveyed
Calculation of percentage respondentsGiven : Total respondents = 55
So, 40% of total respondents = 40% of 55 = 22
County & Opera are chosen by 15 + 10 = total 25 respondents, ie not equal to 40%
Jazz & Opera are chosen by 12 + 10 = total 22 respondents, ie equal to 40%
Jazz, Opera, Rock & Country, Jazz, Rock are totally more respondents.
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dane used a ruler and a protractor to draw a rhombus with a side of 4 cm inlength and an angle that measures 110 degrees choose the angle measure that is missing
Answer:
70 °
Step-by-step explanation:
We have that one side of the rhombus is equal to 4 cm and an angle of the rhombus is 110 degrees. We know that the adjacent sides of a rhombus are supplementary, which means that the sum of the angles must be 180 degrees.
We can assume that the other angle is x.
110 ° + x = 180 °
x = 180 ° -110 °
x = 70 °
The measure of the missing angle on the diagram will be 70 °.
Find g(x) if it is known that g(2t)=8t−1. (IWILLMARKBRAINLIEST)
Answer:
g(x) = 4x - 1
Step-by-step explanation:
We have that:
[tex]g(x) = ax + b[/tex]
So
[tex]g(2t) = a*(2t) + b[/tex]
Equaling both sides:
[tex]g(2t) = 8t - 1[/tex]
[tex]a*(2t) + b = 8t - 1[/tex]
b = -1.
And
[tex]2a = 8[/tex]
[tex]a = \frac{8}{2}[/tex]
[tex]a = 4[/tex]
Then
g(x) = 4x - 1
Rivet holes are punched in steel beams. To ensure that the rivets will fit and that the joint will have adequate strength, it is necessary to control the standard deviation o the diameter, and measurements are made periodically. Ten measurements are made of nominally 1-inch-diameter holes, and the standard deviation is found to be 0.002 inches. What is the 95% confidence interval on the standard deviation? Required – confidence interval in the standard deviation → Chi-squared distribution. S = 0,002 in., n = 10, = n – 1 = 9. 95% confidence level: = 1-0.95 = 0.05.
Answer:
The 95% confidence interval for the standard deviation of the diameter is (0.0014; 0.0036).
Step-by-step explanation:
We have to calculate a confidence interval for the standard deviation.
The confidence level is 95%.
The size of the sample is n=10.
The sample standard deviation is s=0.002.
The lower limit is calculated as:
[tex]LL=s\sqrt{\dfrac{n-1}{\chi_{(1-\alpha)/2;n-1}}}\\\\\\LL=0.002\sqrt{\dfrac{10-1}{19.02}}=0.002\sqrt{0.473}=0.002*0.688=0.0014[/tex]
[tex]UL=s\sqrt{\dfrac{n-1}{\chi_{\alpha/2;n-1}}}\\\\\\UL=0.002\sqrt{\dfrac{9}{2.7}}=0.002\sqrt{3.33}=0.002*1.826=0.0036[/tex]
The 95% confidence interval for the standard deviation of the diameter is (0.0014; 0.0036).
The table of values below shows data for a five-word spelling test. Number of Letters vs. Percentage who Spelled Correctly Word Number of letters in the word Percentage of students who spelled the word correctly 1 7 48 2 9 12 3 5 73 4 6 59 5 8 27 Which describes how the data for word 2 should be plotted on a coordinate grid? From the origin, go 2 units to the right and 9 units up. From the origin, go 2 units to the right and 12 units up. From the origin, go 9 units to the right and 12 units up. From the origin, go 12 units to the right and 9 units up.
Answer:
I'm pretty sure it would be B; go 2 units to the right and 12 units up.
Step-by-step explanation:
the question is asking for the percentage that got it right not the number of letters in the word, and 9 is the number of letters, and 12 is the number that got it right.
The correct data for 2 in coordinate grid will be from the origin, go 9units to the right and 12units up.
How to locate a point on graph?
To locate a point having coordinate (x,y) in graph, we have to go x direction right or left depending on the sign of x-coordinate(for right +,for left -), then we have to go y direction up or down depending on the sign of y-coordinate(for up +,for down -).
The graph is done by taking Number of letters in the word in x axis and Percentage of students who spelled the word correctly in y axis.
According to the table, it is clear that for data 2 Number of letters in the word is 9. so x=9
And Percentage of students who spelled the word correctly is 12
so y=12
So the coordinate of the point for data 2 is (9,12)
To locate the point (9,12) in coordinate grid we have to go 9units to the right in x axis and then we have to go 12units up in y axis direction.
Therefore The correct data for 2 in coordinate grid will be from the origin, go 9units to the right and 12units up.
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Find the value of x.
72 + 24x = 288
Answer:
X = 9
Step-by-step explanation:
1. Subtract 72 from both sides.
24x = 216
2. Divide 24 on both sides
x = 9
Answer:
X = 9
Step-by-step explanation:
We have to separate the like terms
72 + 24x = 288
24x = 288 - 72
24x = 216
We then divide both sides by 24
X = 9
ANYONE PLZZ THIS DUE IN AN HOUR!!!! Will mark the brainliest
Answer:
Option C is correct.
Step-by-step explanation:
The area of 1 side of the divider is approximately 8 x 3 = 24 < 25
Hope this helps!
:)
circle is centered at (-4,-7)The circle passes through the point (-5,-9) . What is its radius?
Answer:
r = sqrt of 5 or 2.2
Step-by-step explanation:
(1)²+(2)²=r²
5 = r²
r = sqrt of 5 or 2.2
According to a 2017 survey conducted by the technology market research firm The Radicati Group, U.S. office workers receive an average of 121 emails per day. Suppose for a particular office the number of emails received per hour follows a Poisson distribution and that the average number of emails received per hour is nine. (Round your answers to four decimal places.)
A. What is the probability of receiving no emails during an hour?B. What is the probability of receiving at least three emails during an hour?C. What is the expected number of emails received during 15 minutes?D. What is the probability that no emails are received during 15 minutes?
Answer:
a) 0.0065 = 0.65% probability of receiving no emails during an hour.
b) 0.1212 = 12.12% probability of receiving at least three emails during an hour
c) The expected number of emails received during 15 minutes is 1.2604.
d) 0.2835 = 28.35% probability that no emails are received during 15 minutes
Step-by-step explanation:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
e = 2.71828 is the Euler number
[tex]\mu[/tex] is the mean in the given time interval.
In this question.
121 emails per day.
A day has 24 hours, so per hour, 121/24 = 5.0417, which means that [tex]\mu = 5.0417n[/tex], in which n is the number of hours.
A. What is the probability of receiving no emails during an hour?
n = 1, so [tex]\mu = 5.0417[/tex]
This is P(X = 0).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-5.0417}*(5.0417)^{0}}{(0)!} = 0.0065[/tex]
0.0065 = 0.65% probability of receiving no emails during an hour
B. What is the probability of receiving at least three emails during an hour?
Either you receive less than three emails during an hour, or you receive at least 3. The sum of the probabilities of these events is decimal 1. So
[tex]P(X < 3) + P(X \geq 3) = 1[/tex]
We want [tex]P(X \geq 3)[/tex]
Then
[tex]P(X \geq 3) = 1 - P(X < 3)[/tex]
In which
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-5.0417}*(5.0417)^{0}}{(0)!} = 0.0065[/tex]
[tex]P(X = 1) = \frac{e^{-5.0417}*(5.0417)^{1}}{(1)!} = 0.0326[/tex]
[tex]P(X = 2) = \frac{e^{-5.0417}*(5.0417)^{2}}{(2)!} = 0.0821[/tex]
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0065 + 0.0326 + 0.0821 = 0.1212[/tex]
0.1212 = 12.12% probability of receiving at least three emails during an hour
C. What is the expected number of emails received during 15 minutes?
15 minutes is a fourth of an hour, so n = 1/4 and [tex]\mu = \frac{5.0417}{4} = 1.2604[/tex]
The expected number of emails received during 15 minutes is 1.2604.
D. What is the probability that no emails are received during 15 minutes?
[tex]P(X = 0) = \frac{e^{-1.2604}*(1.2604)^{0}}{(0)!} = 0.2835[/tex]
0.2835 = 28.35% probability that no emails are received during 15 minutes
ASAP! GIVING BRAINLIEST! Please read the question THEN answer CORRECTLY! NO guessing. Show your work OR give an explaination.
Answer:
d.
Step-by-step explanation:
x^2 because their is no other x^2
2x because their is no more x.
-7+1 equals -6
Answer:
D
Step-by-step explanation:
(f + g)(x) means that you add the two functions together, which results in (f + g)(x) = 2x + 1 + x² - 7 = D) x² + 2x - 6. Hope this helps!
When it is operating properly, a chemical plant has a daily production rate that is normally distributed with a mean of 885 tons/day and a standard deviation of 42 tons/day. During an analysis of period, the output is measured with random sampling on 60 consecutive days, and the mean output is found to be x=875 tons/day. The manager claims that at least 95 % probability that the plant is operating properly. Is he right? Justify your answer!
Answer:
The test statistic Z = 1.844 < 1.96 at 0.05 level of significance
Null hypothesis is accepted
Yes he is right
The manager claims that at least 95 % probability that the plant is operating properly
Step-by-step explanation:
Explanation:-
Given data Population mean
μ = 885 tons /day
Given random sample size
n = 60
mean of the sample
x⁻ = 875 tons/day
The standard deviation of the Population
σ = 42 tons/day
Null hypothesis:- H₀: The manager claims that at least 95 % probability that the plant is operating properly
Alternative Hypothesis :H₁: The manager do not claims that at least 95 % probability that the plant is operating properly
Level of significance = 0.05
The test statistic
[tex]Z = \frac{x^{-} -mean}{\frac{S.D}{\sqrt{n} } }[/tex]
[tex]Z = \frac{875 -885}{\frac{42}{\sqrt{60} } }[/tex]
[tex]Z = \frac{-10}{5.422} = -1.844[/tex]
|Z| = |-1.844| = 1.844
The tabulated value
[tex]Z_{\frac{0.05}{2} } = Z_{0.025} = 1.96[/tex]
The calculated value Z = 1.844 < 1.96 at 0.05 level of significance
Null hypothesis is accepted
Conclusion:-
The manager claims that at least 95 % probability that the plant is operating properly
A device that continuously measures and records seismic activity is placed in a remote region. The time, T, to failure of this device is exponentially distributed with mean 3 years. Since the device will not be monitored during its first two years of service, the time to discovery of its failure is X = max(T, 2).
Answer:
E[X]= [tex]2 + 3 e^{(-2/3)[/tex]
Step-by-step explanation:
The objective of this question is to determine E[X].
T is defined (0,infinity)
X=max(c,T)
where; c=constant
E[X]=c+function (c,infinity) Sf(t)dt
E[X] =[tex]e^{-t/3[/tex]
E[X]=2+function (2,infinity)[tex]e^{-t/3[/tex]dt
E[X] =[tex](2+e^{-t/3})/(1/3)[/tex] function (2,infinity)
E[X]= [tex]2 + 3 e^{(-2/3)[/tex]
If X = T if T ≥ 2 and X = 2 if 0 ≤ T < 2,
So Since T is exponentially distributed with mean 3, the density function of T is [tex]f(t) = (1/3)e^{(-t/3)[/tex]
B =
Round your answer to the nearest hundredth.
please help
Answer:
26.39
Step-by-step explanation:
sin B = 4/9
B = inverse sin 4/9 = 26.387
2. Kate's family saved 2,573 pennies last
year. Zoe's family saved 3 times as
many. How many pennies did Zoe's
family save last year?
A 8,519
B 7.729
C 7.719
D 7519
Answer:
7719
Step-by-step explanation:
2,573*3=7719
Answer:
[tex] \boxed{C. \: 7,719} [/tex]
Step-by-step explanation:
Amount saved by Kate's family last year = 2,573 pennies
Amount saved by Zoe's family last year = 3 × Amount saved by Kate's family last year
= 3 × 2,573
= 7,719 pennies
21=b/-19 solve for b
Answer:
-399
Step-by-step explanation:
21=b/-19
b=21 x -19
b=-399
Answer:
-399
explanation
b=21×-19
b=-399
f(x)=-9x^2-2x and g(x)=3x^2+6x-9, find (f-g)(x) and (f-g)(-4)
Answer:
[tex](f-g)(x) = -12x^{2} - 8x + 9[/tex]
(f-g)(-4) = -151
Step-by-step explanation:
To find f-g(x) we subtract the common terms. So
[tex]f(x) = -9x^{2} - 2x[/tex]
[tex]g(x) = 3x^{2} + 6x - 9[/tex]
(f-g)(x)
[tex](f-g)(x) = f(x) - g(x) = -9x^{2} - 2x - (3x^{2} + 6x - 9) = -9x^{2} - 2x - 3x^{2} - 6x + 9 = (-9 - 3)x^{2} + (-2 - 6)x + 9 = -12x^{2} - 8x + 9[/tex]
(f-g)(-4)
[tex](f-g)(x) = -12x^{2} - 8x + 9[/tex]
We replace x by -4
[tex](f-g)(-4) = -12(-4)^{2} - 8(4) + 9= -151[/tex]
(f-g)(-4) = -151
Match each equation to a story. (Two of the stories match the same equation.)
Answer:
Step-by-step explanation:
Considering story 1,
The weight of each book is x pounds. The weight of 5 books would be 5x pounds. Since the weight of the empty bag is 3 pounds and the weight of the bag and the books is 17 pounds, then the equation is
5x + 3 = 17
Considering story 2
x represents the increase in length of each rectangle. The new length of each rectangle would be x + 5. Since the three identical triangles has a total length of 17 feet, the equation would be
3(x + 5) = 17
Considering story 3,
x represents the cost of each bookmark. The cost of a $3 book and 5 booksmarks is 5x + 3. Since the amount spent is $17, the equation is
5x + 3 = 17
Considering story 4,
x represents the weight of each packet of paper. The weight of each bag is x + 3. Since the weight of 5 bags is 17 pounds, the equation is
5(x + 3) = 17
Considering story 5,
x represents the number of pencils that Noah has. The number of pencils that Andre has is 3x. Since he has 17 pens and pencils altogether, the equation would be
3x + 5 = 17
Answer:
Step-by-step explanation: first you take x and multiply it by z and u start taking the guys p and start blowing. if u dont know how 2 pls look it up<33 then ur answer should end up at 69
<3333
A ladder, leaning against a wall, makes an angle of 20° with the ground. The foot of the ladder is 3 m from the wall. How long is the ladder?
(please show work)
Answer:
0.136m
This is because if you draw a diagram, and label all the sides of the triangle (opp, hyp, adj), the adjacent angle is 3m. You can now use the sine rule to find the hypotenuse (length of ladder) by doing: cos 20= 3/h. You divide cos(20) by 3 and you get the answer of 0.13602m
Cómo se resuelven las ecuaciones químicas?
Answer:
La ecuación química necesita ser equilibrada para que siga la ley de conservación de la masa.
Step-by-step explanation:
Which angles are alternate exterior angles with angle 11?
5 and 13
7 and 15
6 and 16
8 and 14
The above questions answer is 8 and 14
HELP PLEASE ASAP 55 POINTS I BEG YOU!!!!!!
A store had a three-day sale. On the first day the store sold 1 bicycle, 3 tricycles, and 1 unicycle for a total of $561. On the second day, 7 bicycles and 1 tricycle were sold for a total of $906. And at the third day, 2 bicycles, 7 tricycles, and 5 unicycles were sold for a total of $1758.
Set up a system and use row reduction to find the price of each item
Answer:
Bicycle : $117
Tricycle : $87
Unicycle : $183
Step-by-step explanation:
Write a system of equations:
x + 3y + z = 561
7x + y = 906
2x + 7y + 5z = 1758
Where:
x = price of bicycle
y = price of tricycle
z = price of unicycle
Solve for x, y, and z.
How to you write 5,678,209 in expand form
if 6a/2=12, then a =
Answer:
a = 4
Step-by-step explanation:
Multiply by 2 on each side
6a = 24
a = 4
Which statement is always false?
A. All squares are rhombuses.
B. All trapezoids are quadrilaterals.
C. All rectangles are parallelograms.
D. All parallelograms are squares
PLEASE HELP QUICKLY! THANK YOU
Answer:
All parallelograms are squares
Step-by-step explanation:
Reasoning:
You can have a parallelogram with side lengths, i dont know, 5 and 3, right? That isn't a square :)
Given the following system of equations:
6X1 - 6x2 - 4x3 = 0
X1 - 7x2 - 6x3 = 2
X1 +5x2 + nx3 = -2
Rewrite the system in Ax = b format and determine the following:
a. By reduction of the augmented matrix [A|b] to ref, find a value for n such that the system is consistent with an infinite number of solutions.
b. Based on your solution in part A, identify the rank of matrix A and rank of the augmented matrix [A|b].
c. Based on the value of the rank, how many equations (the row vectors of the augmented matrix [Ab]) are linearly independent?
d. Using your solution in part A, solve the system of equations using Gauss-jordan elimination.
Answer:
Step-by-step explanation:
Given:-
- The following system of equations is given:
[tex]6x_1 - 6x_2 -4x_3 = 0\\\\x_1 - 7x_2 -6x_3 = 0\\\\x_1 - 5x_2 -nx_3 = 0\\[/tex]
Solution:-
- The matrix equation consists of coefficient matrix "A" and a variable matrix " x ". These two matrices undergo multiplication to yield a solution column vector "b".
- The matrix A, is a symmetrical square matrix with its elements representing the coefficients of each variable as follows:
[tex]A = \left[\begin{array}{ccc}a_1_1&a_1_2&a_1_3\\a_2_1&a_2_2&a_2_3\\a_3_1&a_3_2&a_3_3\end{array}\right][/tex]
- Where the elements first subscript denotes the equation number and second subscript denotes the variable number.
[tex]A = \left[\begin{array}{ccc}6&-6&-4\\1&-7&-6\\1&5&n\end{array}\right][/tex]
- Similarly, the variable matrix " X " is a column vector that lists all the variables in the the system of equations in a ascending order.
[tex]X = \left[\begin{array}{c}x_1&x_2&x_3\end{array}\right][/tex]
- The solution vector " b " is the corresponding solution or any number written on the right hand side of the equals to sign " = " :
[tex]b = \left[\begin{array}{c}0&2&-2\end{array}\right][/tex]
- Now, we can express the given system in the asked format:
[tex]A*X = b\\\\\left[\begin{array}{ccc}6&-6&-4\\1&-7&-6\\1&5&n\end{array}\right]*\left[\begin{array}{c}x_1&x_2&x_3\end{array}\right] = \left[\begin{array}{c}0&2&-2\end{array}\right][/tex]
- The augmented matrix is a matrix that combines the coefficient matrix " A " and the solution vector " b ". A solution vector "b" as an extra column to the coefficient matrix:
[tex][ A | b ]\\\\ \left[\begin{array}{ccccc}6&-6&-4&|&0\\1&-7&-6&|&2\\1&5&n&|&-2\end{array}\right][/tex]
- Now we will perform row reduction operation such that the system is consistent and has infinite number of solution.
- Row operation: R3 - R2 & R1/6
[tex]\left[\begin{array}{ccccc}1&-1&-\frac{2}{3} &|&0\\1&-7&-6&|&2\\0&12&n+6&|&-4\end{array}\right][/tex]
- Row operation: R2 - R1 & R3 / 12
[tex]\left[\begin{array}{ccccc}1&-1&-\frac{2}{3} &|&0\\0&-6&-\frac{16}{3} &|&2\\0&1&\frac{n+6}{12} &|&-\frac{1}{3}\end{array}\right][/tex]
- Row operation: R2 / 6
[tex]\left[\begin{array}{ccccc}1&-1&-\frac{2}{3} &|&0\\0&-1&-\frac{8}{9} &|&\frac{1}{3} \\0&1&\frac{n+6}{12} &|&-\frac{1}{3}\end{array}\right][/tex]
For the above system to be consistent and have infinite many solution then the coefficient of " x3 " for the 2nd and 3rd row must be equal:
[tex]-x_2 - ( \frac{n+6}{12})*x_3 = \frac{1}{3}[/tex]
[tex]-x_2 - ( \frac{8}{9})*x_3 = \frac{1}{3}[/tex]
The coefficient of " x_3 " must be equal:
[tex]( \frac{n+6}{12}) = \frac{8}{9} \\\\\\\n = \frac{14}{3}[/tex]
- The augmented matrix in reduced form becomes:
[tex]\left[\begin{array}{ccccc}1&-1&-\frac{2}{3} &|&0\\0&1&\frac{8}{9} &|&-\frac{1}{3} \\0&0&0 &|&0\end{array}\right][/tex]
Answer: Rank = Number of non-zero rows = 2
- The number of linearly independent rows are equal to the rank of the augmented matrix.
Hence,
Answer: Number of linearly independent rows = 2
Row operation: R1 + R2
[tex]\left[\begin{array}{ccccc}1&0&\frac{2}{9} &|&-\frac{1}{3} \\0&1&\frac{8}{9} &|&-\frac{1}{3} \\0&0&0 &|&0\end{array}\right][/tex]
- The variable "x_3" will take any arbitrary value for which the solution holds infinitely many solutions.
[tex]x_2 + \frac{8}{9}*x_3 = -\frac{1}{3} \\\\x_2 = - ( \frac{8}{9}*x_3 + \frac{1}{3} )\\\\x_1 + \frac{2}{9}*x_3 = -\frac{1}{3} \\\\x_1 = - ( \frac{2}{9}*x_3 + \frac{1}{3} )\\[/tex]
- Taking x_3 = α:
Answers:
[tex]x_1 = -\frac{1}{3} + \frac{2}{9} \alpha \\\\x_2 = -\frac{1}{3} + \frac{8}{9} \alpha[/tex]