Answer:
Mike's sisters age is 5 years old
Step-by-step explanation:
Since Mike is 17 years old, he is six years older than 2 times her age, so we subtract 6 from his age; we get 11 since he is 2 times her age divide 11 by 2 = 5.5. That means that his sister is 5 years old
Divide $80 among three people so that the second will have twice as much as
the first, and the third will have $5 less than the second. Using an algebraic
equation, find the amount that each will get.
Answer: 19,38, and 33.
Step-by-step explanation:
80=2x+x+2x-5=5x=95=x=19
6. You roll a 6 sided die and flip a coin. What is the
probability of rolling a number less than 5 one the die
and flipping tails on the coin?
Answer:
for the die it's a 4 to 6 chance and the coin is a 50% chance
Select the correct answer.
Which statement best describes the solution to this system of equations?
3x + y= 17
x+2y= 49
OA. It has no solution.
OB.
It has infinite solutions.
OC. It has a single solution: x= 15, y= 17.
OD. It has a single solution: x= -3, y = 26.
Reset
Next
Answer:
D. (-3, 26)
Step-by-step explanation:
The ratios of x- and y-coefficients are different in the two equations, so there will be a single solution. (Different ratios mean the slopes are different. Lines with different slopes must intersect in exactly one point.)
We can rearrange the first equation to give an expression for y:
y = 17 -3x
This can be substituted into the second equation to give ...
x +2(17 -3x) = 49
-5x +34 = 49 . . . . . . simplify
-5x = 15 . . . . . . . subtract 34
x = -3 . . . . . . divide by -5
Then the value of y is ...
y = 17 -3(-3) = 26
The single solution is (x, y) = (-3, 26).
Let A be a given matrix below. First, find the eigenvalues and their corresponding eigenspaces for the following matrices. Then, find an invertible matrix P and a diagonal matrix such that A = PDPâ’1.
(a) [ 3 2 2 3 ]
(b) [ 1 â 1 2 â 1 ]
(c) [1 2 3 0 2 3 0 0 3]
(d) [3 1 1 1 3 1 1 1 3]
It looks like given matrices are supposed to be
[tex]\begin{array}{ccccccc}\begin{bmatrix}3&2\\2&3\end{bmatrix} & & \begin{bmatrix}1&-1\\2&-1\end{bmatrix} & & \begin{bmatrix}1&2&3\\0&2&3\\0&0&3\end{bmatrix} & & \begin{bmatrix}3&1&1\\1&3&1\\1&1&3\end{bmatrix}\end{array}[/tex]
You can find the eigenvalues of matrix A by solving for λ in the equation det(A - λI) = 0, where I is the identity matrix. We also have the following facts about eigenvalues:
• tr(A) = trace of A = sum of diagonal entries = sum of eigenvalues
• det(A) = determinant of A = product of eigenvalues
(a) The eigenvalues are λ₁ = 1 and λ₂ = 5, since
[tex]\mathrm{tr}\begin{bmatrix}3&2\\2&3\end{bmatrix} = 3 + 3 = 6[/tex]
[tex]\det\begin{bmatrix}3&2\\2&3\end{bmatrix} = 3^2-2^2 = 5[/tex]
and
λ₁ + λ₂ = 6 ⇒ λ₁ λ₂ = λ₁ (6 - λ₁) = 5
⇒ 6 λ₁ - λ₁² = 5
⇒ λ₁² - 6 λ₁ + 5 = 0
⇒ (λ₁ - 5) (λ₁ - 1) = 0
⇒ λ₁ = 5 or λ₁ = 1
To find the corresponding eigenvectors, we solve for the vector v in Av = λv, or equivalently (A - λI) v = 0.
• For λ = 1, we have
[tex]\begin{bmatrix}3-1&2\\2&3-1\end{bmatrix}v = \begin{bmatrix}2&2\\2&2\end{bmatrix}v = 0[/tex]
With v = (v₁, v₂)ᵀ, this equation tells us that
2 v₁ + 2 v₂ = 0
so that if we choose v₁ = -1, then v₂ = 1. So Av = v for the eigenvector v = (-1, 1)ᵀ.
• For λ = 5, we would end up with
[tex]\begin{bmatrix}-2&2\\2&-2\end{bmatrix}v = 0[/tex]
and this tells us
-2 v₁ + 2 v₂ = 0
and it follows that v = (1, 1)ᵀ.
Then the decomposition of A into PDP⁻¹ is obtained with
[tex]P = \begin{bmatrix}-1 & 1 \\ 1 & 1\end{bmatrix}[/tex]
[tex]D = \begin{bmatrix}1 & 0 \\ 0 & 5\end{bmatrix}[/tex]
where the n-th column of P is the eigenvector associated with the eigenvalue in the n-th row/column of D.
(b) Consult part (a) for specific details. You would find that the eigenvalues are i and -i, as in i = √(-1). The corresponding eigenvectors are (1 + i, 2)ᵀ and (1 - i, 2)ᵀ, so that A = PDP⁻¹ if
[tex]P = \begin{bmatrix}1+i & 1-i\\2&2\end{bmatrix}[/tex]
[tex]D = \begin{bmatrix}i&0\\0&i\end{bmatrix}[/tex]
(c) For a 3×3 matrix, I'm not aware of any shortcuts like above, so we proceed as usual:
[tex]\det(A-\lambda I) = \det\begin{bmatrix}1-\lambda & 2 & 3 \\ 0 & 2-\lambda & 3 \\ 0 & 0 & 3-\lambda\end{bmatrix} = 0[/tex]
Since A - λI is upper-triangular, the determinant is exactly the product the entries on the diagonal:
det(A - λI) = (1 - λ) (2 - λ) (3 - λ) = 0
and it follows that the eigenvalues are λ₁ = 1, λ₂ = 2, and λ₃ = 3. Now solve for v = (v₁, v₂, v₃)ᵀ such that (A - λI) v = 0.
• For λ = 1,
[tex]\begin{bmatrix}0&2&3\\0&1&3\\0&0&2\end{bmatrix}v = 0[/tex]
tells us we can freely choose v₁ = 1, while the other components must be v₂ = v₃ = 0. Then v = (1, 0, 0)ᵀ.
• For λ = 2,
[tex]\begin{bmatrix}-1&2&3\\0&0&3\\0&0&1\end{bmatrix}v = 0[/tex]
tells us we need to fix v₃ = 0. Then -v₁ + 2 v₂ = 0, so we can choose, say, v₂ = 1 and v₁ = 2. Then v = (2, 1, 0)ᵀ.
• For λ = 3,
[tex]\begin{bmatrix}-2&2&3\\0&-1&3\\0&0&0\end{bmatrix}v = 0[/tex]
tells us if we choose v₃ = 1, then it follows that v₂ = 3 and v₁ = 9/2. To make things neater, let's scale these components by a factor of 2, so that v = (9, 6, 2)ᵀ.
Then we have A = PDP⁻¹ for
[tex]P = \begin{bmatrix}1&2&9\\0&1&6\\0&0&2\end{bmatrix}[/tex]
[tex]D = \begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}[/tex]
(d) Consult part (c) for all the details. Or, we can observe that λ₁ = 2 is an eigenvalue, since subtracting 2I from A gives a matrix of only 1s and det(A - 2I) = 0. Then using the eigen-facts,
• tr(A) = 3 + 3 + 3 = 9 = 2 + λ₂ + λ₃ ⇒ λ₂ + λ₃ = 7
• det(A) = 20 = 2 λ₂ λ₃ ⇒ λ₂ λ₃ = 10
and we find λ₂ = 2 and λ₃ = 5.
I'll omit the details for finding the eigenvector associated with λ = 5; I ended up with v = (1, 1, 1)ᵀ.
• For λ = 2,
[tex]\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}v = 0[/tex]
tells us that if we fix v₃ = 0, then v₁ + v₂ = 0, so that we can pick v₁ = 1 and v₂ = -1. So v = (1, -1, 0)ᵀ.
• For the repeated eigenvalue λ = 2, we find the generalized eigenvector such that (A - 2I)² v = 0.
[tex]\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}^2 v = \begin{bmatrix}3&3&3\\3&3&3\\3&3&3\end{bmatrix}v = 0[/tex]
This time we fix v₂ = 0, so that 3 v₁ + 3 v₃ = 0, and we can pick v₁ = 1 and v₃ = -1. So v = (1, 0, -1)ᵀ.
Then A = PDP⁻¹ if
[tex]P = \begin{bmatrix}1 & 1 & 1 \\ 1 & -1 & 0 \\ 1 & 0 & -1\end{bmatrix}[/tex]
[tex]D = \begin{bmatrix}5&0&0\\0&2&0\\0&2&2\end{bmatrix}[/tex]
The sizes of the angles in degrees of e triangles are.
2x+7
2x
X+18
Use this information to write down an equation in terms of x
ordered pairs on a vertical line have the same y coordinate. true, sometimes true or not true?
Write an equation in vertex form for each graph or given information: 31. Vertex (-5, 12) and through the point (-2, 15)
Answer:
(-3,16)
Step-by-step explanation:
(-5,12)
f (x) = ( (x+5) (x+5) ) + 12
f (x) = x^2 + 10x + 37
(-2,15)
f (x) = ( (x+2) (x+2) ) + 15
f (x) = x^2 + 4x + 19
i need help
[tex]\bf 3x-5=16[/tex]
Answer: x = 7
Step-by-step explanation:
do opposites to both sides. what is the opposite of -5? it would be plus 5.
add 5 to both sides
3x = 21
now what is the opposite of 3 times x? 3 divided by.
divide 3 to both sides
x = 7
Answer:
x = 7
Step-by-step explanation:
First add 5 on both sides to get 3x = 21, then divide 3 on both sides to get x = 7.
Hope this helps you :)
find the value of x if possible
Answer:
Step-by-step explanation:
6x - 17 = 3x + 7
3x = 24
x = 8
please help this is homework and I don’t know how to do it!!!
Step-by-step explanation:
you don't understand what the multiplication or division with 10 (or a higher power of 10) does ?
or you don't know. what 10² means ?
this means 10 to the power of 2 or 10 squared and is simply 10 multiplied with itself 2 times : 10×10.
and 10×10 is ... 100.
as you can imagine, this is not restricted to the number 2 (and 10). "power of" can be any number and just indicates how often the base number is multiplied with itself.
a surcharge case is "power of 0", which is simply 1.
now that we understand that, let's have a quick look at how the numbers we are dealing with (in daily life as well as in class and in science) are actually built :
when I write e.g the number
1234
then this actually means
4×10⁰ + 3×10¹ + 2×10² + 1×10³
when I multiply this now by e.g. 10 this becomes
4×10⁰×10 + 3×10¹×10 + 2×10²×10 + 1×10³×10 =
4×10¹ + 3×10² + 2×10³ + 1×10⁴
as you can see, the previous "single" position 4 now becomes a 40.
and the result looks in short form :
12340
so, a multiplication by 10 adds a 0 at the low end of the number.
a multiplication by 10² or 100 (= 10×10) adds then 2 0s at the low end. and so forth.
a division by 10 does the opposite and removes a 0 at the low end of the number.
but what if we don't have any 0 at this position of the number ? then we enter the system of the decimal point.
because on the right side of the decimal point we continue to countdown the exponent of 10 below 0.
the first position right of the decimal point stands for 10^‐1, the second position for 10^‐2, and so forth. and that is nothing else than tenths, hundredths, thousandths, ...
so, in short, a multiplication by 10 moves the decimal point one position to the right. and if we run out of written positions, then we can simply assume an infinite sequence of 0s continuing further to the right.
and a division by 10 moves the decimal point one position to the left. and if we run out of written positions, then we can simply assume an infinite sequence of 0s continuing further to the left too that are then showing up on the right side of the decimal point.
now, I think we covered all the basics and we can look at the question here :
10² × 18.72
10² = 100 = 10×10
so, we are multiplying twice by 10 and move the decimal point therefore 2 positions to the right.
the result is then
1872
as a little guide : when it is clear that the number has to get bigger by the operation (like a multiplication by 10 or 100), you need to move the decimal point in the direction that makes the number bigger (which is to the right).
and if the operation is clearly trying to make the number smaller, then you have to move the decimal point in the direction to make the number smaller (to the left).
Which function could be used to represent the sequence 8, 20, 50,
125, 312.5...., given that a, = 8?
(1) a, = 4,- 1 ta
(3) 4, = 4, + 1.5(, -1)
(2) a. = 2.5(
4-1) (4) 4 = (a), -1)
Answer: a = 2.5 (an-1)
Step-by-step explanation: trust me;)
The function which is used to represent the geometric progression 8, 20, 50, 125, 312.5, ... is [tex]a_{n} = 8(2.5)^{n-1}[/tex].
What is geometric progression?sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. This progression is as a geometric sequence of numbers.
Formula for nth term of geometric progression[tex]a_{n} =ar^{n-1}[/tex]
Where,
[tex]a_{n}[/tex] is the nth term of the sequence or geometric progression
n is the total number of terms
r is the common ratio
and a is the first term
According to the given question
We have
A geometric progression
8, 20, 50, 125, 312.5
Now the common ratio for the above progression is given by
[tex]r = \frac{20}{8} = 2.5[/tex]
And the first term is
a = 8
Therefore, the function which is used to represent the above sequence is given by
[tex]a_{n} = 8(2.5)^{n-1}[/tex]
Hence, the function which is used to represent the geometric progression 8, 20, 50, 125, 312.5, ... is [tex]a_{n} = 8(2.5)^{n-1}[/tex].
Learn more about geometric progression here:
https://brainly.com/question/4853032
#SPJ2
а Ava and her children went into a movie theater and she bought $48 worth of drinks and candies. Each drink costs $4 and each candy costs $3. She bought 2 more candies than drinks. Graphically solve a system of equations in order to determine the number of drinks, x, and the number of candies, y, that Ava bought.
Answer:
x = 6
y = 8
Explanation and graph is given in picture
I hope it helps.
What does the degree of the polynomial tell you.
A degree in a polynomial function is the greatest exponent of that equation, which determines the most number of solutions that a function could have and the most number of times a function will cross the x-axis when graphed.
Aroha is 3 years older than her brother. The sum of Aroha's age and her brother's age is 31. Write and solve and equation to find their ages
Answer:
Therefore, Aroha is 17 yrs old and her brother is 14 yrs old.
Step-by-step explanation:
Let's represent Aroha's brother's Agee with the letter "b" yrs old.
If Aroha is 3yrs older than her brother, she's : (b+3) yrs old.
The sum of their ages is 31,
b+(b+3)=31
2b+3=31
Collect like terms,
2b=31-3
2b=28
Divide both sides by 2,
b=14
b+3=17
A cheerful teen hoping this helps,
stay techy, brilliant and positive!
A package of breakfast bars has 12 breakfast bars in it. The total weight of the breakfast bars in the package is
552 grams. The mass of each breakfast bar is the same. What is the mass in grams of each breakfast bar?
Solve.
315−14x=−45
Enter your answer in the box.
x =
plzzzzz helpp me
Answer:
180/7
Step-by-step explanation:
\sf 45-2x=75-3x
Free
[tex]\sf \longmapsto45 - 2x = 75 - 3x[/tex]
[tex]\sf \longmapsto45+−2x=75+−3x[/tex]
[tex]\sf \longmapsto−2x+45=−3x+75[/tex]
[tex]\sf \longmapsto \: −2x+45+3x=−3x+75+3x[/tex]
[tex]\sf \longmapsto \: x+45=75[/tex]
[tex]\sf \longmapsto \: x+45−45=75−45[/tex]
[tex]\sf \longmapsto \: x=30[/tex]
[tex] \boxed{\sf 30}[/tex]
HELP!!
Formula
Y=___x +___
Answer:
look at the slope rise of 2 run of -1 or other way around so slope is
[tex] y = - \frac{2}{1}x + 5[/tex]
which shows 2^-2 * 2^6 in exponential form
Answer:
16
Step-by-step explanation:
2^-2=0.25
2^6=64
times
=16
Help me pleaseee asap !!!
Step-by-step explanation:
5x+3x-14=180
8x-14=180
8x=180+14
8x=174
x=174/8
x= 21.75°
if x is 21.75°, then 5x is 108.75°
which means 3x-14 is 51.25°
therefore, f +108.75+51.25=180
f = 180-160
f=20°
What is the value of the function y= 2x – 3 when x = -1?
O --5
O-1
o 2
03
Answer:
y= 2x-3
when X= -1
y= 2(-1)-3
y= -2-3
y= -5
The value of the given function y=2x-3 is -5 when x=-1
What is a function?A function is a correspondence from the two set A to set B in which each element of set A has a unique image in B. Generally we denote a function by f or y.
Here the given function is y=2x-3
Putting x= -1 in the given function y=2x-3 , we get
y= 2(-1) -3
y= -2-3
y= -5
Hence the value of the function y= 2x-3 is -5 when x=-1 .
To learn more about function click here:
https://brainly.com/question/25649287
#SPJ7
Which number sentence is true?
Answer:
B
Step-by-step explanation:
Edge 2021.
f(x)=-2x^2-2x+10 f(x)=−2x 2 −2x+10 \text{Find }f(2) Find f(2)
Given that:
f(x) = -2x²-2x+10
f(2) = -2(2)²-2(2)+10
= -2(2*2) - 4 + 10
= -2(4) - 4 + 10
= -8 - 4 + 10
= -12 + 10
= -2 Ans.
Read more:
Similar Question
Given f(x) = 1/2x - 5, find f^-1(x) a. f^-1 (x) = -2x+10 b. f^-1(x) = 2x+5 c. f^-1(x) = 2x-10 d. f^-1(x) = 2x+10...
https://brainly.com/question/11148895?referrer
The humerus is the bone in a person's upper arm.With this bone as a clue,an anthropologist can tell about how tall a person was.If the bone belongs to that of a female,then height of the person is about (2.75 × humerus length)+71.48 cm .Supposed that the humerus of the female.Supposed that the humerus of a female was found to be 31cm long,about how tall was she? Show your solution
Nonsense=acc in brainly will be deleted
Answer:
156.73 cm
Step-by-step explanation:
height of the person = (2.75 × humerus length)+71.48 cm
length of the found humerus = 31cm
∴ height of the person = (2.75 × 31)+71.48 cm
= 85.25 + 71.48 cm
= 156.73 cm
How to calculate the surface area of a football.
Answer:
A football is a sphere, so the formula for calculating its surface area is 4πr²
Hope it helps:)
(3 questions for 50 points)PLZ HELP
Answer:
see explanation
Step-by-step explanation:
(10)
Since the triangles are congruent then corresponding sides are congruent, so
EF = BC , that is
4x - 1 = 19 ( add 1 to both sides )
4x = 20 ( divide both sides by 4 )\
x = 5
and
DE = AB , that is
y - 6 = 8 ( add 6 to both sides )
y = 14
-------------------------------------------------------
(11)
Since the triangles are congruent the corresponding angles are congruent, so
∠ K = ∠ Y , that is
3x - 37 = 41 ( add 37 to both sides )
3x = 78 ( divide both sides by 3 )
x = 26 , then
∠ K = 3x - 37 = 3(26) - 37 = 78 - 37 = 41°
The sum of the 3 angles in Δ ZMK = 180° then
∠ Z = 180° - (41 + 112)° = 180° - 153° = 27°
So
∠ A = ∠ Z
2y + 7 = 27 ( subtract 7 from both sides )
2y = 20 ( divide both sides by 2 )
y = 10
------------------------------------------------------------------------
(12)
Since the triangles are congruent the corresponding sides and angles are congruent
DG = BS
4x - 11 = 25 ( add 11 to both sides )
4x = 36 ( divide both sides by 4 )
x = 9
∠ T = 180° - (56 + 21)° = 180° - 77° = 103°
Then
∠ H = ∠ T
7y + 5 = 103 ( subtract 5 from both sides )
7y = 98 ( divide both sides by 7 )
y = 14
After a baby was born, she began to gain weight at a rate of 1.5 pounds per month. The weight of the baby at birth was 12 pounds. Write an equation for W,W, in terms of t,t, representing weight, in pounds, of the newborn baby tt months after birth.
What we know:
The baby gains 1.5 pounds per month.
Its weight at birth is 12 pounds.
Therefore, when looking to find the weight after t time, we have to take the constant and the rate of change and place them into the equation. This means our equation will look something like this:
W(t) = Rate of Change x number of months + weight at birth.
Or
W(t) = 1.5(t) + 12
So if in 4 months we look at the weight, the baby will be:
1.5 (4) + 12 = W(t)
6 + 12 = W(t)
18 = W(t)
After 4 months the baby is 18 pounds.
Answer: W(t) = 1.5(t) + 12
I hope this helps! :)
A 12-in. steel cable weighs 0.428 lb. How much does 12.8 ft. weigh?
Answer:
5.48 lb
Step-by-step explanation:
12 inches is 1 foot.
12.8 times that length will have 12.8 times that weight:
12.8 · 0.428 lb = 5.4784 lb ≈ 5.48 lb
__
The given values have 3 significant figures, so the answer needs to be rounded to 3 significant figures.
what times what equals 48 and adds to -16
Answer:
-4 and -12
Step-by-step explanation:
For what times what equals 48:
-4(-12) = 48
-4 - 12 = -16
Answer:
x= -4 and y= -12
Step-by-step explanation:
x*y=48
x+y=-16
solve one equation for x
y=-16-x
then plug that y into the other equation
x(-16-x)=48
-16x-x^2=48
multiply by -1
x^2+16x=-48
complete the square
x^2+16x+64=-48+64
(x+8)^2=16
solve for x
x+8=sqrt(16)
x+8=4
x=4-8
x=-4
plug x back into original equation
x+y=-16
-4+y=-16
y=-16+4
y=-12
so your answers are x=-4 and y=-12
I just need answers ;-;
Answer:
1. 5⁴
2. d¹⁰
3. w⁶