a. If A is the coefficient matrix, solve det(A - λI) = 0 for the eigenvalues λ :
[tex]\det\begin{bmatrix}-6-\lambda & -4 \\ 12 & 8-\lambda\end{bmatrix} = (-6-\lambda)(8-\lambda)+48 = 0 \implies \lambda(\lambda-2)=0[/tex]
[tex]\implies \lambda = 0, \lambda = 2[/tex]
Let v = [v₁, v₂]ᵀ be the eigenvector corresponding to λ. Solve Av = λv for v :
[tex]\lambda=0 \implies \begin{bmatrix}-6&-4\\12&8\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix} \implies 3v_1 + 2v_2 = 0[/tex]
If we pick v₂ = -3, then v₁ = 2, so [2, -3]ᵀ is the eigenvector for λ = 0.
[tex]\lambda = 2 \implies \begin{bmatrix}-8&-4\\12&6\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix} \implies 2v_1 + v_2 = 0[/tex]
Let v₁ = 1, so v₂ = -2.
b. λ = 0 and v = [2, -3]ᵀ contributes a constant solution,
[tex]\vec y_1 = e^{\lambda t} v = \begin{bmatrix}2\\-3\end{bmatrix}[/tex]
while λ = 2 and v = [1, -2]ᵀ contribute a solution of the form
[tex]\vec y_2 = e^{\lambda t} v = e^{2t} \begin{bmatrix}1\\-2\end{bmatrix}[/tex]
c. Yes; compute the Wronskian of the two fundamental solutions:
[tex]W(1, e^{2t}) = \det\begin{bmatrix}1 & e^{2t} \\ 0 & 2e^{2t}\end{bmatrix} = 2e^{2t} \neq 0[/tex]
The Wronskian is non-zero, so the solutions are independent.
The line passes through point (0,6) and has a slope of 7. The equation of the line is?
Answer:
y = 7x + 6
Step-by-step explanation:
The general equation for a line in slope-intercept form is:
y = mx + b
In this form, "m" is the slope and "b" is the y-intercept. Remember, the y-intercept is the y-value when x = 0. As such, the point (0,6) gives you the y-intercept.
You have been given the slope (7) and the y-intercept (6). Therefore, you can just substitute these values in for their variables to determine the equation of the line.
slope = m = 7
y-intercept = b = 6
y = mx + b
y = 7x + 6
CAN SOMEONE HELP ME PLEASE? I’LL GIVE YOU FREE EASY POINTS IF YOU GIVE ME THE RIGHT ANSWER.. NO WRONG ANSWERS PLS!!
see the attachment..........
A fighter tank targets an enemy helicopter flying at an altitude of 1250 ft. If the ground
distance between the tank & the helicopter is 875 ft, what angle of elevation must the
cannon be raised to in order to hit the target?
By using trigonometric relations, we will see that the angle of elevation must be 55°.
How to find the angle of elevation?
We can see this as a right triangle, where one cathetus measures 1250ft (altitude) and the other cathetus measures 875ft.
The angle of elevation is the angle such that the adjacent cathetus is the one measuring 875ft.
Then we can use the relation:
tan(a) = (opposite cathetus)/(adjacent cathetus)
tan(a) = 1250ft/875ft
To find the angle of elevation, we can use the inverse tangent function:
a = Atan(1250ft/875ft) = 55°
The angle of elevation must be 55 degrees.
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HELP WITH THE RIGHT ASNWER FOR BRAINLIEST!!!!
Graph the line with slope 3 passing through the point (2,1).
need help would be apperacited .
A function on a graph to figure it out you can do the line test where you draw a line through the graph and if the line touches two points its not a function. With this in mind bottom right is for sure one and the top left looks like one but i cant really tell...
Step-by-step explanation:
What is the value of x? 2 3 6 7
Answer:
6
Step-by-step explanation:
Answer:
2 or A
Step-by-step explanation:
Correct on edge22
Which expression is equivalent to – 2 log2 x + 4 log2 y + 4 log2 z? log base 2 of the quantity x squared y to the fourth all divided by z to the fourth log base 2 of the quantity y to the fourth z to the fourth all divided by x squared log base 2 of the quantity y times z to the fourth all divided by x squared log base 2 of the quantity yz divided by x end quantity raised to the sixth
The expression equivalent to -2log₂x + 4log₂y + 4log₂z is log₂(yz)⁴/x².
To answer the question, we need to know the laws of logarithm
Laws of logarithmpower law - nlogₐx = logₐxⁿaddition law - logₐx + logₐy = logₐxySince we want to find the expression equivalent to -2log₂x + 4log₂y + 4log₂z, we simplify using these laws, we have
-2log₂x + 4log₂y + 4log₂z = log₂x⁻² + log₂y⁴ + log₂z⁴ (appying the power law)
= log₂x⁻²y⁴z⁴ (applying the multiplication law)
= log₂y⁴z⁴/x²
= log₂(yz)⁴/x²
So, the expression equivalent to -2log₂x + 4log₂y + 4log₂z is log₂(yz)⁴/x².
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Answer:
C.log_2(yz^4/x^2)
Step-by-step explanation:
I took the Algebra exam
6E and 7 with full explanation
Answer:
See below ~
Step-by-step explanation:
Question 6(e) :
⇒ x² ≤ 16
Take square root on each side :
⇒ √x² ≤ √16
⇒ x ≤ 4 and x ≥ -4
⇒ -4 ≤ x ≤ 4
⇒ x² ≤ 16
Apply the absolute rule if x² < a then -√a < x < √a
⇒ -√16 ≤ x ≤ +√16
⇒ -4 ≤ x ≤ 4
Part 7First of all, The student should have subtracted both sides by 4
What he should have done:
[tex]\rightarrow \sf \dfrac{3}{x-2} > 4[/tex]
[tex]\rightarrow \sf \dfrac{3}{x-2} -4 > 4 -4[/tex]
[tex]\rightarrow \sf \dfrac{3}{x-2} -\dfrac{4(x-2)}{x-2} > 0[/tex]
[tex]\rightarrow \sf \dfrac{3-4(x-2)}{x-2}} > 0[/tex]
[tex]\rightarrow \sf \dfrac{3-4x+8}{x-2}} > 0[/tex]
[tex]\rightarrow \sf \dfrac{-4x+11}{x-2}} > 0[/tex]
[tex]\sf \large \boxed{\sf {\left \{ {{-4x+11 = 0} \atop {x-2 = 0}} \right. }}[/tex]
⇒ -4x + 11 = 0, x - 2 = 0
⇒ -4x = -11, x = 2
⇒ x = 11/4 , x = 2
Solution satisfying the inequality:
⇒ 2 < x < 11/4
The table lists the test scores William and Andre received on five math assessments.
William’s Scores Andre’s Scores
89
90
97
74
78
73
81
87
91
82
Which statement best describes the difference of the mean of the two data sets?
statement best describes the difference of the mean of the two data sets 2.8 is about 1.5 times the mean absolute deviation of either set.
What is mean?Mean is the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers.
The mean of William's set.
To find the mean, add all of the data points together and divide by the number of data points:
=89+90+97+74+78/5
=85.6
Now, mean absolute deviation,
=|89-85.6|+|90-85.6|+|97-85.6|+|74-85.6|+|78-85.6|/5
= 3.4+4.4+11.4+11.6+7.6/5
=7.68.
The mean of Andre's set.
To find the mean, add all of the data points together and divide by the number of data points:
=73+81+87+91+82/5
=82.8
Now, mean absolute deviation,
=|73-82.8|+|81-82.8|+|87-82.8|+|91-82.8|+|82-82.8|/5
=9.8+1.8+4.2+8.2+0.8/5
= 4.96.
Hence, the difference between the means of the two sets is
85.6-82.8=2.8.
2.8 is about 1.5 times the mean absolute deviation of either set.
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Answer:
Step-by-step explanation:
nothing im just as lost as everyone else
From the set {1, 2, 3, 4}, use substitution to determine which value of x makes the equation true. 4x = 8
50pts PLEASE HURRY UP I WILL GIVE BRAINLEST SHOW ALL OF YOUR WORK ANSWER ALL THE QUESTIONS
Use the spinner to answer the practice problems below:
Alice spins the spinner 2 times
What is the probability that the spinner stops on A and then B. or on B and then A?
Maria will spin the arrow on the spinner 2 times.
What is the probability that the arrow will stop on the same letter twice?
Answer:
In two spins, there are 9 possible outcomes: AA, AB, AC, BA, BB, BC, CA, CB, CCIf we assume each of these is equally likely, then the probability of AB or BA is ... 2/9
Step-by-step explanation:
If f(n) = 2n^2 + 2 and g(n) = n + 1, what is (f -g)(n) when n = -1?
What percent of the first 20 natural numbers are prime one-digit numbers?
Answer:
5%
Step-by-step explanation:
The first 20 single digit first numbers are 2,3,5 and 7. Thus they are four.
So next we calculate the percentage:
⇒ 4 × 100% = 1 × 100% = 5%
20 20
Activity 2 :Collect the information on any one of the famous Mathematician :a) b) Bhaskara c) Shri Niwasan Ramanujan Aryabhatta please
Answer:
Bhāskara was a 7th-century mathematician and astronomer, who was the first to write numbers in the Hindu decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work.
Aryabhatta is the father of Indian mathematics. He was a great mathematician and astronomer of ancient India. His major work is known as Aryabhatiya. It consists of spherical trigonometry, quadratic equations, algebra, plane trigonometry, sums of power series, arithmetic.
Step-by-step explanation:
hope it helps you and give brainliest
ch of these is the absolute value parent function?
A. f(x) = x
B. f(x) = x^2
C. f(x) = |x|
D. f(x) = 2^x
Answer:
[tex] \text{C.} \: f(x) \: = \: |x| [/tex]
Step-by-step explanation:
To identify the absolute value of a number, it will always be represented between two sticks.
MissSpanishWork out the value of y in the equation
below.
бу
6y = 44 - 5у
Answer: Y = 6 1/2
Step-by-step explanation: Let's begin,
step 1) Isolate the variable by dividing each side by factors that don't contain the variable.
Then, Exact Form:
y = 13/2
Decimal Form:
y = 6.5
Mixed Number Form:
y = 6 1/2
this is probability can someone please give me an answer
Answer:
The experimental probability is 4%
Step-by-step explanation:
A cylinder has a base radius of 9 feet and a height of 4 feet. What is its volume in
cubic feet, to the nearest tenths place?
The volume of the cylinder will be equal to 1017.87 cubic feet.
What is volume?Volume is defined as the space occupied by any object in the three-Dimensions. For the cylinder, the volume will be calculated by radius and the height of the cylinder.
Given that:-
The volume of the cylinder will be calculated by using the formulas below:-
V = πr²h
V = π ( 9 )² 4
V = 1017.87 cubic feet.
Therefore the volume of the cylinder will be equal to 1017.87 cubic feet.
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Decagon, a=22 m, s=14.3 m
The area of the decagon is 1573.39 square meters
How to determine the area of the decagon?The given parameters are:
Side length, s = 14.3 mApothem, a = 22 mThe area of a decagon is calculated using:
[tex]A = \frac 52s^2\sqrt{5 + 2\sqrt 5[/tex]
Substitute 14.3 for s
[tex]A = \frac 52 * 14.3^2\sqrt{5 + 2\sqrt 5[/tex]
Evaluate the exponent
[tex]A = \frac 52 * 204.49\sqrt{5 + 2\sqrt 5[/tex]
Evaluate the product and the roots
A = 1573.39
Hence, the area of the decagon is 1573.39 square meters
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You bought 3 pounds of strawberries and 1 pound of blueberries for $8.40 to make pies for the summer festival. Your friend helped and bought 2 pounds of strawberries and 2 pounds of blueberries for $10.08. How much did a pound of blueberries cost? Let the cost of each pound of strawberries =x and the cost of each pound of blueberries = y
Step-by-step explanation:
3x + y = 8.4
2x + 2y = 10.08
x + y = 5.04
y = 5.04 - x
we use this then in the first equation
3x + 5.04 - x = 8.4
2x = 3.36
x = 3.36/2 = $1.68
y = 5.04 - x = 5.04 - 1.68 = $3.36
blueberries cost $3.36 per pound.
Sheila counted 9 blue jays and 21 sparrows at the bird feeder. What was the ratio of blue jays to sparrows?
Answer:
9/21
= 1/7
Step-by-step explanation:
ratio means the comparison of two things
The radius of the larger circle is 5m. The radius of the smaller circle is 2.5m. What’s the approximate area of the shaded region of the larger circle?
Answer: C
Step-by-step explanation:
By using a problem-solution structure, the author is able to show readers how people are hurting the planet. The structure also shows how we can change and what we can do to help solve the problems we have caused. It inspires people to take action
Answer:
One of the problems humans have made is a large amount of pollution.
And so some of the ways we can fix this is;
1- Using electric cars.
2- Using reusable packaging.
3- For the final one you could use stuff that easily and quickly to break down naturally materials.
Step-by-step explanation:
What is the probability of drawing an ace, a 2, and a 3, without replacement?
The probability of drawing an ace will be 8 / 16575.
What is probability?Probability is defined as the ratio of the number of favourable outcomes to the total number of outcomes in other words the probability is the number that shows the happening of the event.
The probability will be calculated as follows:-
P = [tex]\dfrac{4}{52}\times \dfrac{4}{51}\times\dfrac{4}{50}[/tex]
P = [tex]\dfrac{8 }{ 16575}[/tex]
Therefore the probability of drawing an ace will be 8 / 16575.
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In a class of 28 students, 21 have a cat and 9 have a dog. There are 7 students who
have a cat and a dog. What is the probability that a student has a dog given that they
do not have a cat?
The probability that a student has a dog given that they do not have a cat is 1/14.
What is the probability?Probability refers to a possibility that deals with the occurrence of random events.
The probability of all the events occurring need to be 1.
P(E) = Number of favorable outcomes / total number of outcomes
In a class of 28 students, 21 have a cat and 9 have a dog.
There are 7 students who have a cat and a dog.
The probability that a student has a dog given that they do not have a cat
= 2/ 28
= 1/14
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-6 3/5 - (-7 4/15) + 2 1/5
The answer to your question would be 43/15 or 2.866 and ongoing.
Nayeli lights a 4\,\text{m}^24m 2 4, start text, m, end text, squared space with 121212 candles. Citlali lights a 10\,\text{m}^210m 2 10, start text, m, end text, squared space with 303030 of the same type of candles. Whose space is lit brighter? Choose 1 answer: help me im graded at this
Using proportions, it is found that both spaces have the same brightness, as both have 3 candles per m².
The correct question is as given below:-
Nayeli lights a 4 square meter space with 12 candles. Citlali lights a 10 square meter space with 30 of the same type of candles. Whose space is lit brighter? Choose 1 answer: help me im graded at this
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
For Nayeli and Citlali, the rate of candles per m² is given by:
N = 12/4 = 3 candles per m².
C = 30/10 = 3 candles per m².
Hence both spaces have the same brightness, as both have 3 candles per m².
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Use the functions a(x) = 7x + 10 and b(x) = 12x - 18 to complete the function operations listed below.
Part A: Find (a + b)(x). Show your work.
Part B: Find (a - b)(x). Show your work.
Part C: Find (a - b)(x). Show your work.
Part D: Find a[b(x)]. Show your work.
The value of the sum, difference and composite function a[b(x)] are 19x - 8, -5x + 28 and 84x - 136 respectively
Sum and difference of functionsGiven the following functions a(x) = 7x + 10 and b(x) = 12x - 18
Take the sum
(a+b)(x) = a(x) + b(x)
(a+b)(x) = 7x + 10 + 12x - 18
(a+b)(x) = 19x- 8
Take the difference
(a-b)(x) = a(x) - b(x)
(a-b)(x) = 7x + 10 - 12x + 18
(a-b)(x) = -5x + 28
For the composite function a[b(x)].
a(12x- 18) = 7(12x-18) - 10
a[b(x)] = 84x - 126 - 10
a[b(x)] = 84x - 136
Hence the value of the sum, difference and composite function a[b(x)] are 19x - 8, -5x + 28 and 84x - 136 respectively
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Find the slope of a line parallel to the line that passes through (-4, -1) and (-1, 3)
Answer:
[tex] \boxed{\rm \: Slope(m)= \cfrac{4}{ 3} }[/tex]
OR
[tex] \boxed{\rm \: Slope(m) \approx \: 1.30} \rm \: (nearest \: tenth)[/tex]
Step-by-step explanation:
Given two coordinates:
(-4, -1) and (-1, 3)To Find:
The slope of given two coordinatesSolution:
In this case,we'll need to use the slope's formula,to get the slope.
Here's the formula:
[tex] \rm \: Slope(m)= \: \cfrac{ y_2 -y_1 }{x_2 - x_1} [/tex]
Slope is usually denoted as m.According to the question:
[tex] \rm (y_2 , y _1) = (3,-1)[/tex][tex]\rm (x_2 , x_1) = (-1,-4)[/tex]Substitute them accordingly:
[tex] \rm \: Slope(m)= \cfrac{3 - ( - 1)}{ - 1 - ( - 4)} [/tex]
Simplify then, using a rule called PEMDAS.
[tex] \rm \: Slope(m)= \cfrac{3 + 1}{ - 1 +4 } [/tex]
[tex] \rm \: Slope(m)= \cfrac{4}{ 3} [/tex]
[tex] \rm \: Slope(m) \approx \: 1.30[/tex]
Hence, we can conclude that:
The slope of a line parallel to the line that passes through (-4, -1) and (-1, 3) is 4/3 in exact form and 1.30 in approximate decimal form.Hi student, let me help you out! :)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
We are asked to find the slope of a line parallel to the line that passes through
[tex]\star~\mathrm{(-4,-1)}\star[/tex]
[tex]\star~\mathrm{(-1,3)}[/tex]
[tex]\triangle~\fbox{\bf{KEY:}}[/tex]
The slope formula will help us out! :)Here's the formula:
[tex]\star\boxed{\mathrm{\cfrac{y2-y1}{x2-x1}}}[/tex]
y2 and y1 are y-coordinates
x2 and x1 are x-coordinates
Substitute the given values:
[tex]\star\mathrm{\cfrac{3-(-1)}{-1-(-4)}}[/tex]
Simplify!
[tex]\star\mathrm{\cfrac{3+1}{-1+4}}[/tex]
Simplify more!
[tex]\star\mathrm{\cfrac{4}{3}}[/tex]
Now, what is the slope of the line parallel to this one?
Well, parallel lines have the same slope.
[tex]\star \ \mathrm{The \ slope \ of \ the \ line \ parallel \ to \ this \ one \ is \ \cfrac{4}{3}}.[/tex]
Hope it helps you out! :D
Ask in comments if any queries arise.
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~Just a smiley person helping fellow students :)
Solve the proportion: 35=x20
Answer:
either x = 1.75 or x = 700 , question wasn't clear
Step-by-step explanation:
Because you put the x in front of 20 , you might be meaning x/20 (second answer listed)? Or, you put the variable on the wrong side of 20. (first answer listed)
If you meant to type 20x
35 = 20x
÷20 ÷20 *
1.75 = x
*divide both sides by 20 to isolate x
x = 1.75
(x = [tex]\frac{35}{20}[/tex])
--------------------------------------------------------------------------
If you meant to type x/20
35 = [tex]\frac{x}{20}[/tex]
·20 ·20 *
700 = x
*multiply both sides by 20 to isolate x
700 = x