Reduced matrix to its reduced echelon form Original Matrix:
[tex]$$ \begin{bmatrix}1 & -1 & 2 & -1 & 3 \\2 & -2 & 3 & 0 & 4 \\0 & 0 & 0 & 1 & -2\end{bmatrix} $$[/tex]
Pivot Positions (Original): (1,1), (2,2), (3,4)
Pivot Columns (Original): 1, 2, 4
Reduced Echelon Form:
[tex]$$ \begin{bmatrix}1 & 0 & \frac{2}{3} & 0 & \frac{11}{3} \\0 & 1 & \frac{1}{3} & 0 & \frac{2}{3} \\0 & 0 & 0 & 1 & -2\end{bmatrix} $$[/tex]
Pivot Positions (Reduced): (1,1), (2,2), (3,4)
Pivot Columns (Reduced): 1, 2, 4
To reduce this matrix to its reduced echelon form, I used the following series of elementary row operations:
1. Divide row 1 by 1 to get the leading 1 in the first column
2. Subtract two times row 1 from row 2 to get the leading 1 in the second column
3. Subtract three times row 1 from row 3 to get the leading 1 in the fourth column
4. Divide row 2 by 3 to get the coefficient 2/3 in the third column
5. Subtract row 2 from row 1 to get the coefficient 11/3 in the fifth column
6. Subtract row 2 from row 3 to get the coefficient 2/3 in the fifth column
Exercise 4:
Original Matrix:
[tex]$$ \begin{bmatrix}1 & 2 & -3 & 1 & 5 \\2 & 4 & -6 & 2 & 8 \\-1 & -2 & 3 & -1 & -4\end{bmatrix} $$[/tex]
Pivot Positions (Original): (1,1), (2,2), (3,3)
Pivot Columns (Original): 1, 2, 3
Reduced Echelon Form:
[tex]$$ \begin{bmatrix}1 & 0 & 0 & \frac{2}{3} & \frac{11}{3} \\0 & 1 & 0 & \frac{-1}{3} & \frac{1}{3} \\0 & 0 & 1 & \frac{2}{3} & \frac{5}{3}\end{bmatrix} $$[/tex]
Pivot Positions (Reduced): (1,1), (2,2), (3,3)
Pivot Columns (Reduced): 1, 2, 3
To reduce this matrix to its reduced echelon form, I used the following series of elementary row operations:
1. Divide row 1 by 1 to get the leading 1 in the first column
2. Subtract two times row 1 from row 2 to get the leading 1 in the second column
3. Subtract row 1 from row 3 to get the leading 1 in the third column
4. Divide row 2 by 3 to get the coefficient -1/3 in the fourth column
5. Add row 2 to row 1 to get the coefficient 2/3 in the fourth column
6. Divide row 3 by 3 to get the coefficients 2/3 and 5/3 in the fourth and fifth columns respectively
7. Add row 3 to row 1 to get the coefficient 11/3 in the fifth column
8. Add row 3 to row 2 to get the coefficient 1/3 in the fifth column
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What number gives a result of −4
when 11
is subtracted from the quotient of the number and 5
?
The required number is 42 which gives a result of −4 when 11 is subtracted from the quotient of the number and 5.
What is the Quotient?A quotient is defined as the outcome of dividing an integer by any divisor that can be said to be a quotient. The dividend contains the divisor a specific number of times.
Let n be the number in question; then, we can write the word problem as an equation like this:
⇒ (n/6) - 11 = -4
Add 11 on both sides of the equation,
⇒ (n/6) - 11 + 11 = 11 - 4
⇒ (n/6) = 7
Multiply by 6 on both sides of the above equation, and we get
⇒ n = 42
Hence, the number is 42.
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Is this a polynomial?
vertex to standard form
The quadratic function y = 7(x + 1)² + 1 in standard form is y = 7x² + 14x + 2.
What is a quadratic function?
A polynomial function with one or more variables, where the largest exponent of the variable is two, is referred to as a quadratic function. It is also known as the polynomial of degree 2 since the greatest degree term in a quadratic function is of second degree.
The given quadratic function is -
y = 7(x + 1)² + 1
Solve the equation to get the standard form.
First solve using the formula (a + b)² = (a² + 2ab + b²) -
y = 7(x + 1)² + 1
y = 7(x² + 2x + 1) + 1
Now, open the brackets -
y = 7x² + 14x + 1 + 1
Now, add the like terms -
y = 7x² + 14x + 2
Therefore, the value is obtained as 7x² + 14x + 2.
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6-7 Collisions and Explosions (1-D)
1. A flatcar with mass M, traveling at speed v, collides with and couples to a parked
boxcar. They move off together with a speed of 1/3 v. What is the mass of the boxcar?
They move off together with a speed of 1/3 v,The mass of the boxcar is m/3 (V₁ - V₂)².
What is mass?A body's mass is an inherent quality. Prior to the discovery of the atom and particle physics, it was widely considered to be tied to the amount of matter in a physical body.
in Kinetic energy 4 system, the
K = Kf-Ki
K = 1 (Mcar + Mtruck) Vp ² + mcarv₁_ J 2 2 + Mtruck V2 2 2
K = +(m+2m) (V₁+2√2)² +mv₁² - 1 x 2m v₂ ² 2 2 9
K m (v, ² + 4 √₂ ² + AV₁ V2 ) - { mv, ² - tn v₂ 2 2 6
K = m (V₁²+ 4V₂² + 4 V₁ V2 -1/3V, ² - 6V, ²) /6.
K = m/6 (4V₁V2.-2V₁2-2√22)
k = + m/3 (V₁ - V₂)²
They move off together with a speed of 1/3 v, The mass of the boxcar is m/3 (V₁ - V₂)².
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Tarmu run 30 min and burned 170 calories. Write an equation that represents the number of calories,c, burned per min
c=170/30 is the equation that represents the number of calories,c, burned per min.
What is Division?A division is a process of splitting a specific amount into equal parts.
Given that Tarmu run 30 min and burned 170 calories
We need to find the equation that represents the number of calories, c, burned per min.
To find the calories burned in each minute we have to divide one hundred seventy by thirty.
c=170/30
c=5.66 cal/min
Hence, c=170/30 is the equation that represents the number of calories,c, burned per min.
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What is 0.55 times 20.1?
(With work pls)
Answer:
0.55 times 20.1 is equal to 11.055
Step-by-step explanation:
To show work, we can simply multiply 0.55 by 20.1 using long multiplication.
0.55
20.1
0.55
+11.05
11.055
So 0.55 times 20.1 is equal to 11.055
Consider the equation 2x+7=5x−23
. Arrange the variable terms on the right side.
Enter your response in the box.
7=
−23
Answer:4x
Step-by-step explanation:
Using the theory vectors, find all possible fourth corners to parallelograms that have vertices (1,1), (4,2), and (1,3)
So, the possible fourth corners are (4,4) and (-2,2).
Given the three vertices of a parallelogram (1,1), (4,2), and (1,3), we can use vector algebra to find the fourth corner. We know that opposite sides of a parallelogram are parallel, and therefore have the same vector direction. We can use this property to find the vector pointing from one corner to another and then use this vector to find the fourth corner.
One possible method is:
Find the vector pointing from (1,1) to (4,2) by subtracting the coordinates of the first point from the coordinates of the second point: (4,2) - (1,1) = (3,1)Find the fourth corner by adding this vector to the third point: (1,3) + (3,1) = (4,4)Another possible method is:
Find the vector pointing from (4,2) to (1,3) by subtracting the coordinates of the second point from the coordinates of the third point: (1,3) - (4,2) = (-3,1)Find the fourth corner by adding this vector to the first point: (1,1) + (-3,1) = (-2,2)So, the possible fourth corners are (4,4) and (-2,2).
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Chapter 6
Lesson 2 skills practice numerical expressions
The Numerical Expressions answers are shown below:
What is Integers?Integers come in three types:
Zero (0)Positive Integers (Natural numbers)Negative Integers (Additive inverse of Natural Numbers)Given:
1. 7-6+ 5
= 7 -1
= 6
So, the value of 7-6+ 5 is 6.
2. 31 +19 -8
= 31 + 11
= 42
So, the value of 31 +19 -8 is 42.
3. 64-8+21
= 56 + 21
= 77
So, the value of 64-8+21 is 77.
4. 17+34-2
= 51-2
= 49
So, the value of 17+34-2 is 49.
5. 28+ 89 -67
= 117-67
= 50
So, the value of 28+ 89 -67 is 50.
6. (8+1) x 12- 13
= 9 x 12 -13
= 108 - 13
= 95
So, the value of (8+1) x 12- 13 is 95.
7. 63/9 +8
= 7 +8
= 18
So, the value of 63/9 +8 is 18.
8. 5 x 6 -(9-4)
= 5x 6 - 5
= 30- 5
= 25
So, the value of 5 x 6 -(9-4) is 25.
9. 13x 4 -72 /8
= 13 x 4 - 9
= 52- 9
= 43
So, the value of 13x 4 -72 /8 is 43.
10. 16/2 + 8 x 3
= 8 + 8 x 3
= 8+24
= 32
So, the value of 16/2 + 8 x 3 is 32.
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Add and Subtract Fractions with Unlike Denominators
Daniel decorated the backyard with 4 2/5 yards of pumpkin string lights and
6 3/4 yards of witch-hat string lights to spice up a Halloween party. How many yards
the two string lights did Daniel use?
Answer:11.15
Step-by-step explanation:
sry if its wrong :)
just add them
Solve the problem by using addition, subtraction, multiplication, or division. First use front end rounding to estimate the answer. Then find the exact answer.
The foggiest place in a certain country is City A. It is foggy there an average of days each year. How many days is it not foggy each year? Find the estimated and the exact number of days.
The notion of an equation in algebra is a statistical statement that demonstrates the equality of two mathematical expressions.
For the stated question;
Use addition, subtraction, multiplication, or division to find the solution. To first estimate the response, use front end rounding. Find the precise response next. City A is located in the nation with the most haze. how many days a year are clear of fog.365 days are included in a year.
There are 256 days a year on average when it is foggy.
Determine the approximate number of clear days by using the formula:
We round the figures.
To the nearest tens, 365 days
= 370 days
The nearest tenth of 256 days
= 260 days
370 - 260 days, then
= 110 days
The precise number of days without fog is =
= 365 − 256
= 109
Thus, the precise number of days without fog is found as 109 days.
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The complete question is -
Solve the problem by using addition, subtraction, multiplication, or division. First use front end rounding to estimate the answer. Then find the exact answer. The foggiest place in a certain country is City A. It is foggy there an average of 256days each year. How many days is it not foggy each year? Find the estimated and the exact number of days.
Which group cannot form 8 lines with an equal number of members in each line? A. 881 B. 440 C. 353 D. 224
Answer:
A. 881
C. 353
Step-by-step explanation:
In order to form 8 lines with each line having equal length, the total number of members has to be divisible by 8:
440/8 = 55, so 440 isn't the answer
224/8 = 28, so 224 isn't the answer
881/8 = 110.125. In addition, even numbers can't be divided evenly by odd numbers like 881, so A. 881 is the answer.
Same goes to 353 since 353 is also an odd number that can't evenly divide the even number 8. Hence, C. 353 is the answer as well.
You and your family go to dinner at a local restaurant in Rexburg, Idaho and your online banking report shows that the total dinner bill (food, tax, and tip) was for $42. Suppose that you did not keep your receipt, but you remember you paid a 20% tip for the service.
What was your bill (food and tax) prior to calculating the tip? (tip was paid on the total of food and tax)
$
Number
(Round your answer to the nearest cent.)
Kimiyo earns a weekly salary of $675 plus a 6.5% commission on sales at a gift shop. How much would she make in a work week if she sold $4,800 worth of merchandise?
Kimiyo would make $987 in a work week if she sold $4,800 worth of merchandise.
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:
Kimiyo earns a weekly salary of $675 plus a 6.5% commission on sales at a gift shop.
She sold $4,800 worth of merchandise.
That means,
the commissioned amount = 4800 x 6.5 /100
The commissioned amount = $312
In a week,
Kimiyo's total earning is,
= $675 + $312
= $987
Therefore, $987 is the total salary.
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What is 11/12 = 3/4h?
h=
The value of h in the algebraic equation in fraction 11/12 = 3/4h is equal to 9/11
How to solve for the value of h in the algebraic equationWe shall first find the lowest common multiple LCM of the denominators to help us clear the fractions, before then simplifying further for the value of h as follows:
LCM of the denominators 12 and 4h is 12h
multiply both sides by the LCM
11/12 × 12h = 3/4h × 12h
11 × h = 3 × 3
11h = 9
divide both sides by the coefficient of h
11h/11 = 9/11
h = 9/11
Therefore, the value of h in the algebraic equation in fraction 11/12 = 3/4h is equal to 9/11.
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Please help!!
Find the third, fourth, and tenth terms of the sequence described by the rule.
A(n) - 3+ (n-1)(5)
A(3)=
A(4)=
A(10)=
Answer:
A(3) = 13
A(4) = 18
A(10) = 48
Step-by-step explanation:
This seems like a plug and play.
Starting with A(3). Everywhere you see an n, you'll replace it with a 3, as follows;
3 + (3-1)(5) = ?
We'll solve it step by step. First is 3-1. That's 2, bringing it to
3 + (2)(5).
2 times 5 is 10, bringing it to
3 + 10.
3 + 10 = 13, which concludes
A(3) = 13.
Now doing it for the remaining 2.
3 + (4-1)(5)
3 + (3)(5)
3 + 15
A(4) = 18
Last one...
3 + (10-1)(5)
3 + (9)(5)
3 + 45
A(10) = 48
ow many ways can you select 7 cookies to buy from a bakery that sells cookies of 3 different flavors (oatmeal, chocolate chip, almond)? we assume that more than 7 cookies are displayed for each flavor. for example, one way is to choose 7 chocolate chip cookies
There are 2187 combinations to select 7 cookies from a bakery that sells cookies of 3 different flavors (oatmeal, chocolate chip, almond).
When selecting 7 cookies from a bakery that sells 3 different flavors, each cookie selected can be one of the 3 different flavors. This means that for the first cookie, there are 3 choices, for the second cookie, there are also 3 choices, and so on for the 7th cookie. This is known as the combinations formula and it can be represented as 3 x 3 x 3 x ... x 3 (3 being repeated 7 times).
Using the 3⁷ formula, we can calculate that there are 2187 ways to select 7 cookies from a bakery that sells cookies of 3 different flavors. This means that there are 2187 unique combinations of cookies that can be selected, assuming that more than 7 cookies are available for each flavor.
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Your question seems incomplete, but I assume the question was:
"How many ways can you select 7 cookies to buy from a bakery that sells cookies of 3 different flavors (oatmeal, chocolate chip, almond)? We assume that more than 7 cookies are displayed for each flavor. For example, one way is to choose 7 chocolate chip cookies."
use the given data to find the equation of the regression line. examine the scatterplot and identify a characteristic of the data that is ignored by the regression line. x y
The equation of the regression line is Y=2.83+0.52x .
What do you mean by regression line?Linear regression is the statistical technique that is most frequently used to simulate a relationship between two sets of variables. As a result, a data-applicable linear regression equation is produced.
Plotted your points and drew a linear regression line.
What is the formula to calculate the line regression?y = a + bₐₓx
bₐₓ = n∑xy-∑x-∑y
n∑x²- ∑x²
x = 8.81
y = 7.45
∑x² = 947
∑xy=770.95
bₐₓ = 0.52
a = 2.83
y=1.3082X + 3.3436.
Its equation is
y = 2.83+0.52x .
Therefore, the regression line in a scatter plot ignores non-linear properties, so the equation of the reggresion line will be y = 2.83+0.52x .
Your question is incomplete but most probably your full question was
Use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.
x y
9 7.38
9 6.75
12 13.08
9 7.42
11 7.56
13 8.53
7 5.93
4 5.33
12 8.1
6 6.46
5 5.44
Required:
Create a scatterplot of the data.
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If the coordinates of a quadrilateral composed in the first quadrant of the coordinate plane are (0,0), (a,b), (a+c,b), and (c,0), which is the most accurate classification of the quadrilateral?
a. trapezoid
b. rectangle
c. parallelogram
d. square
the opposite sides of the quadrilateral PQRS are equal and parallel and so PQRS is a PARALLELOGRAM.
Option (C) is CORRECT.
What are coordinates?A pair of numbers called coordinates are used to locate a point or a form in a two-dimensional plane. The x-coordinate and the y-coordinate are two numbers that define a point's location on a 2D plane.
Given, that the coordinates of the vertices of a quadrilateral composed in the first quadrant of the coordinate plane are (0,0) (a, b), (a + c, b), and (c, 0).
We are to select the most accurate classification of the quadrilateral.
Let the coordinates of the vertices of the quadrilateral be P(0,0), Q(a, b), R(a + c, b), and S(c, 0).
The lengths of the sides are calculated using the distance formula as follows :
PQ = √(a² + b²)
QR = √(a +c -a)² + (b-b)² = √c² = c
RS = √(a² + b²)
SP = c
So, the opposite sides are equal in length.
And, the slopes of the sides are calculated as follows :
The slope of PQ(m₁) = (b-0)/(a-0)
The slope of QR(m₂) = (b-b)/(c-a-c) = 0
The slope of RS(m₃) = 0-b/(c-a-c) = 0
The slope of SP(m₄) = (0-0)/(0-c) = 0
So, the slopes of the opposite sides are equal but:
m₁ *m₂ = 0 ≠ -1
m₃ *m₄ = 0 ≠ -1
Therefore, the opposite sides of the quadrilateral PQRS are equal and parallel and so PQRS is a PARALLELOGRAM.
Option (C) is CORRECT.
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given that we assume that all numbers are distinct, explain why it is without loss of generality to assume that a > d
Since it has no impact on the result or solution of the problem, assuming that a > d does not reduce the generality of the statement. If a > d, the issue can be solved by simply swapping the values of a and d.
It makes no difference to the problem whether an is more or less than d because we are presuming that each integer is distinct. Furthermore, by supposing that a > d, the problem's variables can be arranged consistently, making it simpler to comprehend and resolve. As a result, making the assumption that a > d does not reduce the generality of the solution because the problem remains the same and the assumption may be readily changed if necessary.
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An aspiring figure skater named Anthonia has been working on her triple axel jump. She lands it 33.33% on average. Her competition, Zahra, has also been working on her triple axel jump averaging 45% of successful lands. If both are competing at the same event, what is the probability that Anthonia will land her jump and Zahra will not land her axel jump?
If Anthonia and Zahra compete at the same event, the probability that Anthonia will successfully land her jump and Zahra will not land hers is 18.33%.
What is the probability?We describe probability as the chance or likelihood that an expected event occurs when there are many possible outcomes.
Probability lies between one and zero when it is not 100% certain or uncertain.
The probability that Anthonia lands her triple axel jump = 33.33%
The chance that Anthonia does not land her jump = 66.67% (1 - 33.33%)
The likelihood of Zahra landing her jump successfully = 45%
The odds that Zahra will not successfully land her jump = 55% (1 - 45%)
The probability that Anthonia lands successfully while Zahra does not = 18.33% (33.33% x 55%)
Thus, Anthonia has the chance of winning Zahra at 18.33%.
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If Mario eats two cups of pretzels every night while he watches television how many cup dose he eat in one week
find the area under the normal distribution curve whose mean is and whose standard deviation is using the given value for as a boundary.
The area under the normal curve with a mean of 20 and a standard deviation of 5 is 0.8413 when x = 25.
The area under the normal distribution curve is calculated using the z-score, which is the number of standard deviations a value is from the mean. To calculate the area under the normal curve for a given mean and standard deviation, one must use the formula:
Area = ½[1 + erf(z)]
Where erf is the Gaussian error function, and z is the z-score.
To calculate the area under the curve given a boundary x, the z-score must be calculated first. This is done using the formula:
z = (x - mean) / standard deviation
Substituting this into the formula above gives:
Area = ½[1 + erf((x - mean) / standard deviation)]
For example, if the mean is 20 and the standard deviation is 5, and the boundary value is x = 25, the area under the normal curve is calculated as follows:
z = (25 - 20) / 5 = 1
Area = ½[1 + erf(1)] = 0.8413
Therefore, the area under the normal curve with a mean of 20 and a standard deviation of 5 is 0.8413 when x = 25.
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Determine the slope given graph
Answer:
slope = 4
Step-by-step explanation:
32.- The high costs of the California real estate market have caused families who cannot afford to buy larger houses to consider backyard sheds as an expansion option. Many are using their patio structures to build their studios, art rooms, and hobby areas, as well as for additional storage. The median price for a custom-built backyard clapboard frame is $3,100 (Newsweek, September 29, 2003). Assume that the standard deviation is $1,200.
A)¿Cuál es el valor z para una estructura de patio trasero que cuesta $2 300?
b) ¿Cuál es el valor z para una estructura que cuesta $4 900?
C) Interprete los valores z en los incisos a) y b). Comente si alguna debe considerarse una observación atípica.
D) El artículo de Newsweek describió una combinación de oficina en el cobertizo del patio trasero construida con $13 000 en Albany, California. ¿Esta estructura debe considerarse una observación atípica? Explique por qué.
a) The z-score for a structure costing $2,300 is given as follows: -0.67.
b) The z-score for a structure costing $4,900 is given as follows: 1.5.
c) The interpretation of each z-score is given as follows:
A structure costing $2,300 has the price 0.67 standard deviations below the mean, and the cost is not unusual.A structure costing $4,900 has the price 1.5 standard deviations above the mean, and the cost is not unusual.d) The structure would have an unusual price, as the z-score has an absolute value greater than 2.
How to obtain the z-scores?The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.A mesure X is considered unusual if the z-score has an absolute value greater than 2.The mean and the standard deviation for this problem are given as follows:
[tex]\mu = 3100, \sigma = 1200[/tex]
For item a, the z-score is then given as follows:
Z = (2300 - 3100)/1200
Z = -0.67.
For item b, the z-score is then given as follows:
Z = (4900 - 3100)/1200
Z = 1.5.
For the price of $13,000, the z-score is given as follows:
Z = (13000 - 3100)/1200
Z = 8.25.
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1. Match each equation with an equivalent equation. Some of the answer choices are
not used.
A. 3x + 6 = 4x+7
B. 3(x + 6) = 4x+7
C. 4x + 3x = 7-6
1.9x = 4x +7
2. 3x + 18 = 4x+7
3.3x = 4x + 7
4. 3x1 = 4x
5.7x = 1
Answer: Match each equation with an equivalent equation. Some of the answer choices are
not used.
A. 3x + 6 = 4x+7
B. 3(x + 6) = 4x+7
C. 4x + 3x = 7-6
1.9x = 4x +7
2. 3x + 18 = 4x+7
3.3x = 4x + 7
4. 3x1 = 4x
5.7x = 1
A. 3x + 6 = 4x+7 = 3
B. 3(x + 6) = 4x+7 = 2
C. 4x + 3x = 7-6 = 5
1.9x = 4x +7 = 3
3x + 18 = 4x+7 = 2
3.3x = 4x + 7 = 1
3x1 = 4x = not used
5.7x = 1 = not used
So,
3x + 6 = 4x+7 matches with 3
3(x + 6) = 4x+7 matches with 2
4x + 3x = 7-6 matches with 5
1.9x = 4x +7 matches with 3
3x + 18 = 4x+7 matches with 2
3x = 4x + 7 matches with 1
3x1 = 4x is not used
7x = 1 is not used
Step-by-step explanation:
A line passes through the point(-8, -9) and has a slope of Write an equation in slope-intercept form for this line. -5/4
Answer:
y = (-5/4)x - 19
Step-by-step explanation:
Given information,
→ Slope(m) = -5/4
→ Point = (-8, -9)
→ y-intercept = b
The slope-intercept form is,
→ y = mx + b
Now the y-intercept will be,
→ y = mx + b
→ -9 = (-5/4)(-8) + b
→ -9 = 10 + b
→ b = -9 - 10
→ [ b = -19 ]
Then the required equation is,
→ y = mx + b
→ [ y = (-5/4)x - 19 ]
Hence, this is the equation.
suppose that for a group of consumers the probability of eating pretzels is .75 and the probability of drinking coke in .65. further suppose that the probability of eating pretzels and drinking coke is about.55. determine if these two events are independent
Dr. Leona Williams, a well-known plastic surgeon, has a reputation for being one of the best surgeons for reconstructive nose surgery. Dr. Williams enjoys a rather substantial degree of market power in this market. She has estimated demand for her work to be: Q = 480 – 0.2P where Q is the number of nose operations performed monthly and P is the price of a nose operation. What is the inverse demand function for Dr. Williams’ services? What is the marginal revenue function? The average variable cost function for reconstructive nose surgery is estimated to be:
AVC = 2Q2 – 15Q + 400
where AVC is the average variable cost (measured in dollars), and Q is the number of operations per month. The doctor’s fixed cost per month is $8,000. If the doctor wishes to maximize her profit, how many nose operations should she perform each month? What price should Dr. Williams charge to perform a nose operation? How much profit does she earn each month?
We can conclude that -
The inverse demand function for Dr. Williams’ services is -Q = 5(280 - P)
For maximum profit, she should perform 4 operations.What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is a story of Dr. Leona Williams who is a well-known plastic surgeon.
{ 1 } -
Dr. Williams estimated demand for her work is -
Q = 280 - 0.2P
Inverse of a function can be calculated by following the steps mentioned below -
Step 1 - Replace 'y' with 'x' and vice - versa.Step 2 - Rewrite the equation by solving for 'y'.Step 3 - Replace 'y' with f⁻¹(x).So, we can write -
P = 280 - 0.2Q
P - 280 = - 0.2Q
- (280 - P) = - 0.2Q
(280 - P) = 0.2Q
Q = (280 - P)/0.2
Q = 5(280 - P)
{ 2 } -
The average variable cost function for reconstructive nose surgery is -
AVC = 2Q² – 15Q + 400
Let AVC = A{Q}
For maximum profit -
d/dQ (A{Q}) = 0
d/dQ (2Q² – 15Q + 400) = 0
d/dQ (2Q²) - d/dQ (15Q) + d/dQ (400) = 0
4Q - 15 = 0
4Q = 15
4Q = 15
Q = 3.7
Q = 4 {approx.}
Therefore, we can conclude that -
The inverse demand function for Dr. Williams’ services is -Q = 5(280 - P)
For maximum profit, she should perform 4 operations.To solve more questions on algebraic expressions, visit the link below -
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which one shows a line with a slop of 2
The graph with the slope of 2 is the second graph.
How to find the slope of a line?The slope of a line is a measure of its steepness. The slope of a line is the change in dependent variable with respect to the change in the independent variable.
Therefore, let's find the slope of the graph to know the one with the slope of 2.
Hence,
slope = y₂ - y₁ / x₂ - x₁
Using the second graph,
(-1, -1)(0, 1)
slope = 1 + 1 / 0 + 1
slope = 2 / 1
slope = 2
Therefore, the graph with slope of 2 is the second graph.
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