how many distinct ordered triples $(x,y,z)$ satisfy the following equations? \begin{align*} x 2y 4z
Answers
There are 55 distinct ordered triples (x, y, z) that satisfy the given equations.
To find the number of distinct ordered triples (x, y, z) that satisfy the given equations, we need to analyze each equation separately and then find the common solutions.
First equation: x + 2y + 4z = 20
To solve this equation, we can fix the value of z and express x in terms of y and z:
x = 20 - 2y - 4z.
Let's consider the possible values of z:
When z = 0, the equation becomes x + 2y = 20.
When z = 1, the equation becomes x + 2y = 16.
When z = 2, the equation becomes x + 2y = 12.
When z = 3, the equation becomes x + 2y = 8.
When z = 4, the equation becomes x + 2y = 4.
Now, for each value of z, we can find the corresponding values of (x, y) that satisfy the equation.
When z = 0:
x + 2y = 20
x = 20 - 2y
For y = 0, x = 20.
For y = 1, x = 18.
For y = 2, x = 16.
For y = 10, x = 0.
When z = 1:
x + 2y = 16
x = 16 - 2y
For y = 0, x = 16.
For y = 1, x = 14.
For y = 2, x = 12.
For y = 8, x = 0.
When z = 2:
x + 2y = 12
x = 12 - 2y
For y = 0, x = 12.
For y = 1, x = 10.
For y = 2, x = 8.
For y = 6, x = 0.
When z = 3:
x + 2y = 8
x = 8 - 2y
For y = 0, x = 8.
For y = 1, x = 6.
For y = 2, x = 4.
For y = 4, x = 0.
When z = 4:
x + 2y = 4
x = 4 - 2y
For y = 0, x = 4.
For y = 1, x = 2.
For y = 2, x = 0.
From the above calculations, we can see that for each value of z, there are 11 possible solutions (including the solution where x = 0). Therefore, the total number of distinct ordered triples (x, y, z) that satisfy the equation x + 2y + 4z = 20 is 5 (the number of values for z) multiplied by 11 (the number of solutions for each z), which equals 55.
So, there are 55 distinct ordered triples (x, y, z) that satisfy the given equations.
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Related Questions
Let f be a function that has derivatives of all orders for allreal numbers.
Assume f(1) = 3, f '(1) = -2. f ''(1) = 2, and f '''(1) =4.
a) Write the second-degree Taylor polynomial for f about x = 1and use it to approximate f(0.7).
b) Write the third-degree Taylor polynomial for f about x = 1and use it to approximate f(1.2).
c) Write the second-degree Taylor polynomial for f ', thederivative of f , about x = 1 and use it to approximate f'(1.2).
Answers
a) The second-degree Taylor polynomial for f about x=1 is 1.69
b) The third-degree Taylor polynomial for f about x=1 is 0.867
c) The second-degree Taylor polynomial for f' about x=1 is -0.4
a) The second-degree Taylor polynomial for f about x = 1 is:
f(x) ≈ f(1) + f'(1)(x-1) + (f''(1)/2)(x-1)^2
Plugging in the given values, we get:
f(x) ≈ 3 - 2(x-1) + (2/2)(x-1)^2
f(x) ≈ 1 - 2(x-1) + (x-1)^2
To approximate f(0.7), we substitute x = 0.7 in the polynomial:
f(0.7) ≈ 1 - 2(0.7-1) + (0.7-1)^2
f(0.7) ≈ 1 + 2(0.3) + 0.09
f(0.7) ≈ 1.69
Therefore, f(0.7) ≈ 1.69 using the second-degree Taylor polynomial.
b) The third-degree Taylor polynomial for f about x = 1 is:
f(x) ≈ f(1) + f'(1)(x-1) + (f''(1)/2)(x-1)^2 + (f'''(1)/6)(x-1)^3
Plugging in the given values, we get:
f(x) ≈ 3 - 2(x-1) + (2/2)(x-1)^2 + (4/6)(x-1)^3
f(x) ≈ 1 - 2(x-1) + (2/3)(x-1)^3
To approximate f(1.2), we substitute x = 1.2 in the polynomial:
f(1.2) ≈ 1 - 2(0.2) + (2/3)(0.2)^3
f(1.2) ≈ 0.867
Therefore, f(1.2) ≈ 0.867 using the third-degree Taylor polynomial.
c) The second-degree Taylor polynomial for f', the derivative of f, about x = 1 is:
f'(x) ≈ f'(1) + f''(1)(x-1)
Plugging in the given values, we get:
f'(x) ≈ -2 + 2(x-1)
f'(x) ≈ -2 + 2x - 2
f'(x) ≈ 2x - 4
To approximate f'(1.2), we substitute x = 1.2 in the polynomial:
f'(1.2) ≈ 2(1.2) - 4
f'(1.2) ≈ -0.4
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The probability in your area that a voter is a republican is 0.35. What is the probability that the first republican you find is the 4th voter in line ?
Answers
The probability that the first republican you find is the 4th voter in line is 12.25%.
In the gievn question, the probability in your area that a voter is a republican is 0.35.
We have to find the probability that the first republican you find is the 4th voter in line.
The probability of a voter is a republican is 0.35.
The first republican we find is the 4th voter in line is
=0.35*0.35
Now multiplying;
=0.1225
To convert in percentage multiply and divide by 100.
We get
=12.25%
Hence, the probability that the first republican you find is the 4th voter in line is 12.25%.
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Given f(x) and g(x) = f(x) + k, use the graph to determine the value of k.
Graph of f of x and g of x. f of x equals 1 over 3 x minus 2 and g of x equals 1 over 3 x plus 3.
2
3
4
5
Answers
Answer:
k = 5
Step-by-step explanation:
Find the arclength of the curve r(t) = (6 sint, -10t, 6 cost), -9
Answers
the arclength of the curve is 10 units for the given curve r(t) = (6 sint, -10t, 6 cost).
The given curve is r(t) = (6sint,-10t,6cost) with a range of t from 0 to 1, so t ∈ [0,1].
To find the arclength of the curve, use the following formula: s = ∫√(dx/dt)² + (dy/dt)² + (dz/dt)² dt
Here, dx/dt = 6 cost, dy/dt = -10, dz/dt = -6sint.
Substitute the above values in the formula to obtain:
s = ∫(√(6 cost)² + (-10)² + (-6sint)²) dt = ∫√(36 cos²t + 100 + 36 sin²t) dt = ∫√(100) dt = ∫10 dt = 10t
The range of t is from 0 to 1.
Hence, substitute t = 1 and t = 0 in the above expression.
Then, subtract the values: s = 10(1) - 10(0) = 10 units.
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Solve for x: 5x+2-10x=22
Answers
Answer:
X=-4
Step-by-step explanation:
5x-10x=22-2
-5x=20
x=-4
Type the missing numbers in this sequence:-70,
-46, -38, -30
Answers
Answer:
70, 62, 54, 46, 38, 30...
Step-by-step explanation:
The sequence is going DOWN by 8.
Answer:
The missing numbers in the sequence are -54 and -42. The sequence is decreasing by 16, then 8, then 16, then 8 and so on.
Step-by-step explanation:
The first number is -70 and the second number is -46. The difference between these two numbers is 24. The third number is -38, which is 8 less than -46. So, the sequence is decreasing by 16, then 8, then 16, then 8 and so on.
what is the meaning of permutations and combinations?
Answers
Answer:
Step-by-step explanation:
Permutations and combinations are mathematical concepts that deal with counting the number of different arrangements or selections of objects from a set.
Permutations:
A permutation is an arrangement of objects in a specific order. For example, consider a set of three letters: A, B, and C. There are 3! = 6 possible permutations of these letters, which are ABC, ACB, BAC, BCA, CAB, and CBA. The notation "3!" is the factorial notation, which means 3 × 2 × 1.
Combinations:
A combination is a selection of objects, where the order does not matter. For example, consider a set of three letters: A, B, and C. There are 3 C 2 = 3 possible combinations of two letters from this set, which are AB, AC, and BC. The notation "3 C 2" is the combination notation, which means the number of ways to choose k objects from a set of n objects.
In summary, permutations deal with the arrangement of objects in a specific order, while combinations deal with the selection of objects without regard to the order.
Xaviera went out to play soccer at 5:31 and returned home at 7:00 how long was she playing soccer
Answers
Step-by-step explanation:
she played for about 1 hour 29 minutes
Answer:
1 hour and 29 minutes or 89 minutes
PLEASE HELP ASAP! NEED REAL ANSWERS!
Answers
The vertex of y=|2x+4|+5 is (-2, 5) and Range of the y=|2x+4|+5 is 5≤y<∞
What is a function?A relation is a function if it has only one y-value for each x-value.
We need to find the vertex and range of y=|2x+4|+5
f(x)=2x+4
f of x is equal to zero
2x+4=0
Subtract 4 from both sides
2x=-4
Divide both sides by 2
x=-2
y=|2x+4|+5
=|2(-2)+4|+5=|-4+4|+5=5
Hence vertex is (-2, 5)
The range of the y=|2x+4|+5 is 5≤y<∞
Hence vertex is (-2, 5) and Range of the y=|2x+4|+5 is 5≤y<∞
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Write the coordinates of the vertices after a rotation *
270° counterclockwise around the origin. P(-8,3) Q(-8,10) R(-4,2)
Answers
The new coordinates for P(-8,3) Q(-8,10) R(-4,2) are (3,-3) (10,-10) and (2,-2) .
What are coordinates ?
Coordinates can be defined as pair of numbers which are used to determine the position of a given point.
Given
to Write the coordinates of the vertices after a rotation 270° counterclockwise around the origin.
Given points are P(-8,3) Q(-8,10) R(-4,2) .
we know that if (x,y) are rotated at an θ, the new coordinates could be ,
x′=xcos(θ)−ysin(θ) y′=xsin(θ)+ycos(θ).
So,
for p (-8,3)
x' = -8*cos270 - 3 * sin270
x' = 0 + 3
x' = 3
y' = -8*cos270 + 3 * sin270
y' = 0 - 3
y' = -3
(x' , y' ) = (3,-3)
similarly,
for q (-8,10)
x' = -8*cos270 - 10 * sin270
x' = 0 + 10
x' = 10
y' = -8*cos270 + 10* sin270
y' = 0 -10
y' = -10
(x' , y' ) = (10,-10)
similarly,
for q (-4,2)
x' = -4*cos270 - 2 * sin270
x' = 0 + 2
x' = 2
y' = -4*cos270 + 2 * sin270
y' = 0 -2
y' = -2
(x' , y' ) = (2,-2)
Therefore, The new coordinates for P(-8,3) Q(-8,10) R(-4,2) are (3,-3) (10,-10) and (2,-2) .
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the individual most closely associated with innovations in photographic equipment was
Answers
The individual most closely associated with innovations in photographic equipment was George Eastman
Who is George Eastman?American businessman George Eastman helped popularize the use of roll film in photography by founding the Eastman Kodak Company.
George Eastman's entrepreneurial passion, fearless leadership, and amazing vision revolutionized the globe. He revolutionized the photography, film, and motion picture industries and is credited with establishing the Eastman Kodak Company, which will live on throughout history.
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complete question;
The individual most closely associated with innovations in photographic equipment was ________
Write down the percentage multiplier to find 53% of a price
Answers
The percentage multiplier to find the 53% of a price is 0.53
The given percentage value = 53%
Percentage of a quantity is defined as the number or the ratio, that can be expressed as the fraction of 100. To find the percentage of a number, divide the number by the whole quantity and multiply by 100. Percentage of a number is usually expressed as by using percentage sign "%"
Here we have to find the percentage multiplier
To find the percentage multiplier divide the given percentage by 100
The percentage value = 53%
The percentage multipler = 53 / 100
Divide the numbers
53 / 100 = 0.53
Therefore, the percentage multiplier is 0.53
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John spent 80% of his money and saved the rest. Peter spent 75% of his money and saved the rest. If they saved the same amount of money, what is the ratio of John’s money to Peter’s money? Express your answer in its simplest form.
Answers
The ratio of John's money to Peter's money is 5/4. This means if John has a total amount of 5 then Peter will have a total of 4 as his amount.
Let's assume John has 'x' amount of money, Peter has 'y' amount of money, The money John saved is 'p' and the money Peter saved is 'q'
So,
p = x - 80x/100 (equation 1)
q = y - 75y/100 (equation 2)
According to the given question, the amount John saved is equal to the amount Peter saved. Hence, we can equate equations 1 and 2.
p = q
x- 80x/100 = y - 75y/100
x - 0.8x = y - 0.75y
0.2x = 0.25y
x = 0.25y/0.2
x/y = 0.25/0.2
x/y = 25/20
x/y = 5/4
Hence, the ratio of John's money to Peter's money is 5/4.
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Suppose that you have $3000 to invest. Which investment yields the greater return over a 10 year period: 7.23% compounded daily or 7.3% compounded quarterly?
Answers
Answer:
you would have less than $3000 or more that $3000 since the value could have gone up or down, $3000/10=300/7.23=41.493.Hope this helps :)
La figura MNPQ es un rombo. Si 2θ – Φ =60°. Calcula el valor de “x”. Dime los fundamentos y principios que utilizas.
Answers
Answer:
x = 30°
Step-by-step explanation:
Donde 2θ - Φ = 60 °, tenemos;
∠QMN = ∠QPM
Ф + ∠P / 2 + (180 - θ) = 180 (Suma de ángulos en un triángulo ΔQLP)
Por lo tanto, Ф + ∠P / 2 - θ = 0
Por lo tanto, ∠P / 2 = θ - Ф, también
270 ° + θ + x + ∠P / 2 = 360 (Suma de ángulos en un NOLP cuadrilátero) da
270 ° + θ + x + θ - Ф = 270 ° + 2 · θ - Ф + x = 360
Como 2 · θ - Ф = 60 °, tenemos;
270 ° + 60 + x = 360
x = 360 - 270 - 60 = 30 °
What are the solutions of the equation cscxtanx=2sqrt3/3 on the interval [0,2pi)?
im having trouble understanding this, can someone help?
Answers
The solutions of the trigonometric equation are x₁ = π / 6 and x₂ = 5π / 3.
How to solve a trigonometric equation on a given interval
In this problem we find a trigonometric equation which has to be simplified and solved for x to find the family of solutions within the given interval:
csc x · tan x = 2√3 / 3
(1 / sin x) · (sin x / cos x) = 2√3 / 3
1 / cos x = 2√3 / 3
cos x = 3 / 2√3
cos x = √3 / 2
Then, by inverse trigonometric functions:
x₁ = π / 6, x₂ = 5π / 3
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dr carl is a mathematics and statistics lecturer. he is interested in studying a new approach at teaching mathematics and statistics in high school. recently this new approach has been adopted. it was previously the case that the mean score for all high school students doing their end of year exam was 71, however, the population standard deviation is unknown. the scores of all students are approximately normally distributed. he thinks that, with this new teaching approach, the mean score may have increased. a random sample of 39 scores is gathered and the sample mean and standard deviation calculated. dr carl will conduct a hypothesis test using a level of significance of 0.01. select the correct null and alternative hypotheses for this test:
Answers
The test is one-tailed to the right due to the Alternate hypothesis.
In a hypothesis test, a Null hypothesis (H0) is a statement of "no effect" or "no difference," whereas the Alternate hypothesis (HA) is a statement of "some effect" or "some difference." Here are the correct null and alternative hypotheses for the given scenario: Null hypothesis: H0: μ = 71, Alternative hypothesis: HA: μ > 71 (One-tailed test) we have a one-tailed hypothesis test since the research hypothesis is directed. In a one-tailed test, we're looking for a difference in one direction only. In this case, Dr Carl thinks that there has been an improvement, so the alternative hypothesis reflects this. Therefore, the test is one-tailed to the right.
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Fractions that have the same value , such as 1/4 and 2/8 are?
Answers
Answer: 4/16, 5/20, 6/24
Step-by-step explanation:
The fraction of 1/4 will multiply by the same number n on both numerator and denominator.
1/4 and 2/8 is equivalent fractions
1/4 and 2/8 has the following equivalent fractions:
3/12, 4/16, and so forth.
After simplifying the numerator and denominators, the equivalent fractions have the same amount or value. If both the numerator and denominator are cancelled by a common factor, the fractions will produce the same value.
To put it another way,
a/b = c/d = e/f
A fraction of 1/4, has the equal of 2/8 because 2/8 is simplified to the same value as 1/4.
What are Equivalent FractionsEquivalent fractions are two or more fractions that, after simplification, have the same value. If a/b and c/d are two fractions, the fractions are comparable only if each fraction's simplification yields e/f.HELP PLEASE FAST THANK U <3
Answers
Answer:
x would be 44
Step-by-step explanation:
i looked at the equation i did -1+45 and it is 44
Answer:
x=2
Step-by-step explanation:
alright so here just make those two equations equal to 135
-1+45x+23x=135 add like terms
-1+68x=135
68x=136 divide
x=2
MARKING BRAINLIEST FIND X PLEASE PLEASE HELP PLEEZ
Answers
there are 2 radius given and the radius extends to form a right angle triangle
6^2+ x^2 = (4+6)^2
36 + x^2 = 100
x^2=100-36
x=√44
x=6.6
hope it helps
The height of a helicopter above the ground is given by h = 3.05t3, where h is in meters and t is in seconds. At t = 2.35 s, the helicopter releases a small mailbag. How long after its release does the mailbag reach the ground?
Answers
the height of the mailbag decreases as time increases, and the mailbag reaches the ground at t = 5.33 seconds.
The height of the mailbag after its release is given by the equation h = 3.05t3. At t = 2.35 seconds, the height of the mailbag is h = 3.05 * 2.35 * 2.35 = 39.58 meters. This means that the mailbag is still 39.58 meters above the ground. The equation for the height of the mailbag tells us that the height of the mailbag is decreasing at a rate of 9.15t2 meters per second. This means that the mailbag will take 39.58 / 9.15 = 4.33 seconds to reach the ground.
Therefore, the mailbag will reach the ground after 1 + 4.33 = 5.33 seconds
Here is a table of the height of the mailbag over time:
Time (seconds) | Height (meters)
------- | --------
2.35 | 39.58
2.36 | 35.43
2.37 | 31.28
... | ...
5.33 | 0
As you can see, the height of the mailbag decreases as time increases, and the mailbag reaches the ground at t = 5.33 seconds.
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Find the midpoint of the segment with the following endpoints. (10, 10) and (2, 6)
Answers
Answer:
(6, 8 )
Step-by-step explanation:
given endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint is
( \(\frac{x_{1}+x_{2} }{2}\) , \(\frac{y_{1}+y_{2} }{2}\) )
here (x₁, y₁ ) = (10, 10 ) and (x₂, y₂ ) = (2, 6 ) , then
midpoint = ( \(\frac{10+2}{2}\) , \(\frac{10+6}{2}\) ) = ( \(\frac{12}{2}\) , \(\frac{16}{2}\) ) = (6, 8 )
The curve through the ordered pairs (0, 10), (1, 5), and (2, 2. 5) can be represented by the function f(x) = 10(0. 5)x. What is the multiplicative rate of change of the function? 0. 5 2 2002. 5 5.
Answers
The multiplicative rate of change of the function f(x) = 10(0.5)^x is 0.5, indicating that the function decreases by multiplying by 0.5 with each increase in x.
The function f(x) = 10(0.5)^x represents an exponential decay function. The base of the exponent is 0.5, indicating that the function decreases by multiplying by 0.5 with each increase in x. In other words, the function value decreases to half its previous value for every increase of 1 in x.
When we examine the given ordered pairs, we can see this pattern. As x increases from 0 to 1, the function value decreases from 10 to 5, representing a multiplicative decrease of 0.5. Similarly, as x increases from 1 to 2, the function value decreases from 5 to 2.5, again demonstrating a multiplicative decrease of 0.5.
Therefore, the multiplicative rate of change of the function is 0.5.
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5 litres into grams.
Answers
Step-by-step explanation:
Do like this only with no. mark me as brainliest answer
Answer:
Weight in Grams of:
Water Cooking Oil
5,000 g 4,400 g
Step-by-step explanation:
Hope this helps
please help me! i’m so bad at this lol
Answers
Answer:
its 8
Step-by-step explanation:
<3
5. 63 = -7x
so you solve this like any other equation, divide both sides by -7
your answer to question 5 is x = -9
6. 18 = 6x - 12
you will add your 12 over
30 = 6x
x = 5
7. -4 = 1/2x - 3
-1 = 1/2x
x = -2
i cant read what 8 is but just repeat
What is the slope of the line represented by the equation y- 6 = 2(x+3)?
Answers
And m is slope and b is the y-intercept -
In this equation y-6=2(x+3)
m is 2 because if u multiply 2 to (x+3) it will become 2x+6
since m is 2 and b is 6
The slope is 2
And the y-int. is 6
So the slope is 2.
Answer: 2
Step-by-step explanation:
remember y=mx^2+c
where m = gradient = slope
∴ expand and simplify y-5=2(x+3)
y-5=2x+6
y=2x+11
m=2 ∴ slope: 2
A store is offering a 20% discount on all items. Write an equation for the relationship between the discount price of an item, d , and it's original price, p.
Answers
Answer:
80%p=d
Step-by-step explanation:
20% discount is 80% of original.
An equation for the relationship between the discount price of an item, d, and it's original price, p is p = 0.8d.
What is a percentage?A ratio or value that may be stated as a fraction of 100 is called a percentage. And it is represented by the symbol '%'.
Given:
A store is offering a 20% discount on all items.
Let the discount price of an item, d, and it's original price, p.
The original price = (100 - 20%)discount price
p = 80%d
p = 0.8d
Therefore, p = 0.8d is the required equation.
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What is an equation of the line that passes through the points (-4,-2)
and (8,1)?
Answers
Answer:
y=0.25x-1
Step-by-step explanation:
y=mx+b
slope = 0.25
y = 0.25x+b
the y intercept = -1
y = 0.25x+-1
y = 0.25x-1
What is the value of x in this triangle?
Enter your answer in the box.
x =
Answers
Answer:
61
Step-by-step explanation:
37+82+x=180 (Sum of triangle is 180)
119+x=180
x=180-119
x=61
Are the points (1,4) and
(2,5) symmetrical about this
line?
(Yes or no question)