If an augmented matrix [ A b ] is transformed into [ C d ] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets.
true false
If an augmented matrix [ A b ] is transformed into [ C d ] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets is true.
Elementary row operations are specific operations that can be performed on the rows of a matrix. These operations include:
Swapping two rows.Multiplying a row by a non-zero scalar.Adding a multiple of one row to another row.When we apply elementary row operations to an augmented matrix [ A b ], we are essentially performing the same operations on the corresponding system of linear equations Ax = b. The purpose of these operations is to simplify the matrix or system of equations, but they do not change the solution set.
Each elementary row operation corresponds to an equivalent algebraic operation on the system of equations. For example, swapping two rows corresponds to rearranging the order of equations, multiplying a row by a scalar corresponds to multiplying both sides of the equation by the same scalar, and adding a multiple of one row to another row corresponds to adding or subtracting equations.
Since elementary row operations preserve the equivalence between the matrix and the system of equations, any solution that satisfies the original system Ax = b will also satisfy the transformed system Cx = d. Similarly, any solution that satisfies Cx = d will also satisfy Ax = b.
Therefore, the solution sets of the original system Ax = b and the transformed system Cx = d are exactly the same.
Therefore, If an augmented matrix [ A b ] is transformed into [ C d ] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets is true.
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please help 6th grade math
Answer:
B
Step-by-step explanation:
Soon Yee was putting together her notebook after the first day of
school. She bought a pack of 200 sheets of loose-leaf notebook
paper and some dividers. She wanted to put the same amount of
paper in each of the four sections for her core subjects: math,
history, English and science. She figured that herſtwaelective
courses (art and music) each only needed half bf the amount of
paper that a core subject needed. How many sheets of paper did
she put into the art section of her notebook?
Answer: Since her art and music sections each only had half the number of sheets of paper as a core subject, together the two sections had the same amount of paper as a core subject. Therefore, it is almost like her notebook had five core subjects, rather than four core subjects and two electives. If she divided the 200 sheets equally among the five core subjects, there would be 200 ÷ 5 = 40 sheets in each section. Now we can see that art would actually have half of this amount, or 20 sheets of paper.
May someone please help me with this :)
Answer: It would be option D :)
Step-by-step explanation: the formula for volume in a cylinder is π r squared times height. You plug those in and that’s your answer!
The total cost (in dollars) of producing x food processors is C(x)=2500+20x-0,1x². (A) Find the exact cost of producing the 71st food processor. (B) Use the marginal cost to approximate the cost of producing the 71st food processor.
(A) The exact cost of producing the 71st food processor is $3415.90. (B) Using the marginal cost approximation, the cost of producing the 71st food processor is approximately $5.80.
The cost of producing the 71st food processor can be found by substituting x = 71 into the cost function C(x) = 2500 + 20x - 0.1x². (A) The exact cost of producing the 71st food processor is C(71) = 2500 + 20(71) - 0.1(71)² = 2500 + 1420 - 504.1 = $3415.90.
The marginal cost represents the change in cost when one additional unit is produced. It can be approximated by calculating the difference between the cost of producing x+1 units and x units and dividing it by the change in quantity. In this case, the marginal cost is the derivative of the cost function, which is given by C'(x) = 20 - 0.2x. To approximate the cost of producing the 71st food processor using the marginal cost, we can evaluate C'(x) at x = 71. (B) C'(71) = 20 - 0.2(71) = 20 - 14.2 = $5.80. Therefore, the approximate cost of producing the 71st food processor using the marginal cost is $5.80.
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Ashley got 18 out of 42 questions correct on the test. How many more questions would she have
needed to get comect to receive a 70% on the testi
9) A carton contains 2 litres of tropical juice. The juice consists of 25% mango juice, 30% pineapple juice, and the rest is guava juice. Find how many millilitres of each juice went into the carton of tropical juice.
The carton of tropical juice contains 500 ml of mango juice, 600 ml of pineapple juice, and 900 ml of guava juice.
To find the amount of each juice that went into the carton of tropical juice, we'll calculate the quantities based on the given percentages.
Let's denote the volume of mango juice as M (in milliliters), the volume of pineapple juice as P, and the volume of guava juice as G. Since the total volume of the carton is 2 liters, we have 2000 milliliters as the total volume.
According to the given information, mango juice accounts for 25% of the total volume, pineapple juice accounts for 30%, and the remaining volume is guava juice.
We can express these relationships as equations:
M + P + G = 2000 (Equation 1)
M = 0.25 * 2000 (Equation 2)
P = 0.30 * 2000 (Equation 3)
G = 2000 - (M + P) (Equation 4)
From Equation 2, we have M = 0.25 * 2000 = 500 ml.
From Equation 3, we have P = 0.30 * 2000 = 600 ml.
Substituting these values into Equation 4, we get:
G = 2000 - (500 + 600) = 900 ml.
Therefore, the quantities of each juice that went into the carton of tropical juice are as follows:
Mango juice: 500 ml
Pineapple juice: 600 ml
Guava juice: 900 ml
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a hollow copper wire with an inner diameter of 1.1 mm and an outer diameter of 2.2 mm carries a current of 10 a.
A hollow copper wire with an inner diameter of 1.1 mm and an outer diameter of 2.2 mm carries a current of 10 A.
The given scenario describes a hollow copper wire with an inner diameter of 1.1 mm and an outer diameter of 2.2 mm that carries a current of 10 A. To determine certain characteristics of the wire, we can utilize the concepts of current, diameter, and material properties.
Firstly, the current of 10 A represents the magnitude of electric current flowing through the wire. This current is a measure of the rate at which electric charges are moving through the conductor.
The dimensions of the wire, specifically the inner and outer diameters, provide insights into its cross-sectional area. By calculating the difference between the outer and inner radii (1.1 mm and 2.2 mm, respectively), we can determine the thickness of the wire's walls. This thickness directly affects the wire's resistance and conductivity.
Copper is known for its excellent electrical conductivity, which allows for efficient current flow. Its resistivity is relatively low compared to other materials. This property ensures that the wire experiences minimal losses and heat generation while carrying the current of 10 A.
In conclusion, the provided information about the hollow copper wire's dimensions and current helps us understand key aspects such as current flow, wire thickness, and the conductive properties of copper. These factors are crucial in determining the wire's electrical behavior and efficiency.
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a
252 cm sq.
b
324 cm sq.
c
144 cm sq.
d
21 cm sq.
Solve the linear congruence 2x + 6 = 4(mod 8).
The solution to the linear congruence 2x + 6 ≡ 4 (mod 8) is x ≡ 3 (mod 8).To solve the linear congruence 2x + 6 ≡ 4 (mod 8), we need to find the values of x that satisfy the congruence equation.
First, let's simplify the equation:
2x + 6 ≡ 4 (mod 8)
Subtracting 6 from both sides:
2x ≡ 4 - 6 (mod 8)
2x ≡ -2 (mod 8)
Next, we can simplify -2 (mod 8) by adding 8 to it:
-2 ≡ 6 (mod 8)
Now the congruence equation becomes:
2x ≡ 6 (mod 8)
To solve for x, we need to find the multiplicative inverse of 2 modulo 8. The multiplicative inverse of 2 is 4, as 2 * 4 ≡ 1 (mod 8).
Multiplying both sides of the congruence equation by 4:
4 * 2x ≡ 4 * 6 (mod 8)
8x ≡ 24 (mod 8)
Simplifying the equation:
8x ≡ 24 (mod 8)
x ≡ 3 (mod 8)
Therefore, the solution to the linear congruence 2x + 6 ≡ 4 (mod 8) is x ≡ 3 (mod 8).
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Use the figure to find the measures of the numbered angles. PLZZZ HELPPPP
angle 1 = 90°
angle 2 = 90°
angle 3 = 90°
angle 4 = 90°
angle 5 = 90°
angle 6 = 90°
angle 7 = 90°
Answer:
All of them are 90 degrees.
Step-by-step explanation:
From Monday through Friday Sydney works in the bookstore on one day and in the tutoring center where on another doing on Saturday and Sunday Sydney cooks food 50% of the days how many days does Sidney work in a week what percent of Monday through Friday does Sidney work
Answer: The other 50%
Step-by-step explanation:
Consider the following. x=3cos(θ),y=4sin(θ),−π/2≤θ≤π/2 (a) Eliminate the parameter to find a Cartesian equation of the curve.
The Cartesian equation of the curve is 4x² + 3y² = 36.
We can eliminate the parameter θ by using the trigonometric identity cos²(θ) + sin²(θ) = 1, which gives us:
x²/9 + y²/16 = cos²(θ) + sin²(θ) = 1
Multiplying both sides by 16 to get rid of the fraction, we have:
16(x²/9 + y²/16) = 16
4x² + 3y² = 36
Therefore, the Cartesian equation of the curve is 4x² + 3y² = 36.
The Cartesian equation 4x² + 3y² = 36 represents an ellipse centered at the origin with semi-axes lengths of 2√3 and 2√2. The standard form for the equation of an ellipse centered at the origin is:
(x²/a²) + (y²/b²) = 1
where a and b are the semi-axes lengths of the ellipse. To convert the given equation to this standard form, we need to divide both sides by 36:
(4x²/36) + (3y²/36) = 1
Simplifying:
x²/9 + y²/12 = 1
Comparing with the standard form, we have:
a² = 9 and b² = 12
Therefore, the semi-axes lengths are √9 = 3 and √12 ≈ 3.464.
So, the equation 4x² + 3y² = 36 represents an ellipse centered at the origin with semi-axes lengths of 3 and approximately 3.464.
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Use a known Maclaurin series to obtain a Maclaurin series for the given function. f(x) = xe^7x [infinity]
f(x) = ∑
n=0
Find the associated radius of convergence, R. R =
Using the known Maclaurin series expansion for e^x: e^x = 1 + x + (x^2 / 2!) + (x^3 / 3!) + ...
We can obtain the Maclaurin series for the function f(x) = xe^(7x) by multiplying the Maclaurin series for e^x by x:
f(x) = x(1 + 7x + (7x)^2 / 2! + (7x)^3 / 3! + ...)
Simplifying further:
f(x) = x + 7x^2 + (49x^3 / 2!) + (343x^4 / 3!) + ...
Therefore, the Maclaurin series expansion for f(x) is:
f(x) = x + 7x^2 + (49x^3 / 2!) + (343x^4 / 3!) + ...
The associated radius of convergence, R, can be determined by applying the ratio test. In this case, we have an infinite power series with terms involving powers of x. The ratio test states that if the limit of |a_(n+1) / a_n| as n approaches infinity is L, then the radius of convergence is R = 1 / L.
In our case, applying the ratio test yields a limit of 0 as n approaches infinity. Therefore, the radius of convergence R is infinite, indicating that the Maclaurin series for f(x) converges for all values of x.
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Find the number of subsets in the set (x l x is a day of the week)
a. 124
b. 256
c. 127
d. 128
PLEASE GRAPH THE ANSWER AND HAVE A PICTURE ATTACHED. I WILL REALLY APPRECIATE IT :)
-Also no links please...
Answer:
The first is (-1, 1) and the second is (2, -2)
Step-by-step explanation:
A paint color is made using 4 drops of red and 5 drops of blue for each 5 gallons of paint. How many gallons of paint are being colored when 45 drops of color are used?
simplify the expression ((-1)^5/(-2)-3)^2
Answer:
Sure! Let's simplify the expression step by step:
((-1)^5/(-2)-3)^2
= (-1/2 - 3)^2 // Since (-1)^5 = -1
= (-7/2)^2
= 49/4
Therefore, ((-1)^5/(-2)-3)^2 simplifies to 49/4.
Use the inverse matrices to find and (24) ? 7 - 2 2 2 A-1 B-1 = 2 1 3 4 4. 3 4 2 11 (b) (47) b 100 -100 co (24) JUD .11
To find the value of matrix A given A⁻¹ and B⁻¹, we can use the formula A = (A⁻¹)⁻¹ * B⁻¹.
Given A⁻¹ = [[2, 7], [-2, 2]] and B⁻¹ = [[2, 1], [3, 4], [4, 3], [2, 11]], we can calculate A as follows:
Step 1: Calculate the inverse of A⁻¹: (A⁻¹)⁻¹
(A⁻¹)⁻¹ = [[2, 7], [-2, 2]]⁻¹
To find the inverse of a 2x2 matrix, we use the following formula:
[[a, b], [c, d]]⁻¹ = (1 / det([[a, b], [c, d]])) * [[d, -b], [-c, a]]
Step 2: Calculate the determinant of A⁻¹: det([[2, 7], [-2, 2]])
det([[2, 7], [-2, 2]]) = (2 * 2) - (7 * -2) = 4 + 14 = 18
Step 3: Calculate the inverse of A⁻¹: (A⁻¹)⁻¹
(A⁻¹)⁻¹ = (1 / 18) * [[2, -7], [2, 2]] = [[2/18, -7/18], [2/18, 2/18]]
Step 4: Calculate A = (A⁻¹)⁻¹ * B⁻¹
A = [[2/18, -7/18], [2/18, 2/18]] * [[2, 1], [3, 4], [4, 3], [2, 11]]
Performing matrix multiplication, we get:
A = [[2/18 * 2 + (-7/18) * 3, 2/18 * 1 + (-7/18) * 4],
[2/18 * 2 + 2/18 * 3, 2/18 * 1 + 2/18 * 4]]
Simplifying the calculations, we find:
A = [[-1/3, -2/3],
[2/9, 4/9]]
Therefore, the value of matrix A is [[-1/3, -2/3], [2/9, 4/9]].
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The average low temperature in Tobermorey in October is 5 C. In February it is 23 C lower. What is the average low temperature in Tobermorey in February?
please show your answer
(It’s a math question about negative numbers and positive numbers on the number line)
Answer:
-18 C
Step-by-step explanation:
we have to do this operation
5 - 23
we can proceed like this
if we do 5-5 we reach the 0
23 - 5 = 18
now we have to do 18 steps to the left of the zero, that it means reach -18, that is the answer of the question
Can someone help me with this. Will Mark brainliest.
Answer:
1. True
2. True
3. True
An author of a book gets 3% of the price of each book sold. He received $24,380. How many copies of the book were sold, if each book was sold for $32.00?
Please Explain
divide the number by 32 then divide what you get by 3 and last times 100 equals 25,395.83333333 round it if you want to to 25,395.8
1 - m∠QRU = _____ °.
2 - m∠VUW = _____ °.
3 - m∠TUW = _____ °.
Answer:
Step-by-step explanation:
74
Please help! Due tonight.
The lateral area of the pyramid is 126 metres squared.
How to find the lateral area of a pyramid?The diagram above is a pyramid. The lateral area of the pyramid can be found as follows:
lateral area of the pyramid = 1 / 2 pl
where
p = perimeter of the basel = height of the pyramidTherefore,
p = 7 × 3 = 21 metres
l = 12 metres
Hence,
lateral area of the pyramid = 1 / 2 × 21 × 12
lateral area of the pyramid = 21 × 6
lateral area of the pyramid = 126 metres²
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Jenica bowls three games and scores an average of 116 points per game. She scores 105 on her first game and 128 on her second game. What does she need to score on her third game to get a mean score of 116?
Answer:
115 points on her third game.
Step-by-step explanation:
To find the answer add the first and second games together.
128 + 105 = 233 ÷ 2 = 116.5 (then divide by two because there is two games).
This is the average of two games. To find the average of THREE games try different numbers to see which number works.
233 + 110 = 343 ÷ 3 = 114.3
(so we have to try a higher number than 110).
233 + 112 = 345 ÷ 3 = 115 on average (still not enough).
233 + 115 = 348 ÷ 3 = 116. (we found the answer!)
233 + 115 = 348 ÷ 3 = 116
Convert: 24 yards = _____ feet
72 feet
48 feet
76 feet
8 feet
Answer:
24 yards = 72 feet
Step-by-step explanation:
3 feet per yard
24 x 3 = 72
Questions are in the image above.
Answer:
hope this helps
you will find all answer in the photo i sent
Which is greater? 40 mm or 3 m
and how do I show my work for that?
correct=brainliest
Answer:
The answer is 1 meter
Step-by-step explanation:
One meter is equal to 1000 mm
A store has 125 scientific calculators.
Brand x sells for $20 each; Brand y sells
for $15 each. If the total selling price
of the calculators is $2,275, how many
of each calculator does the store have?
Answer:
Brand x: 80
Brand y: 45
Step-by-step explanation:
x + y = 125
20x + 15y = 2275
-20x - 20y = -2500
(+) 20x + 15y = 2275
--------------------------------
-5y = -225
y = 45
x + y = 125
x + 45 = 125
x = 80
Answer:
Brand x: 80
Brand y: 45
Chloe has a bag of marbles with 5 blue marbles, 1 white marbles, and 3 red marbles.
Find the following probabilities of Chloe drawing the given marbles from the bag if the first marble(s) is(are) returned to the bag after they are drawn.
a) A Blue, then a red =
b) A red, then a white =
c) A Blue, then a Blue, then a Blue =
a) The probability of Chloe drawing a blue marble, then a red marble, with replacement is (5/9) * (3/9) = 15/81 ≈ 0.185.
b) The probability of Chloe drawing a red marble, then a white marble, with replacement is (3/9) * (1/9) = 3/81 ≈ 0.037.
c) The probability of Chloe drawing three blue marbles in a row, with replacement, is (5/9) * (5/9) * (5/9) = 125/729 ≈ 0.172.
For each probability, we multiply the probabilities of drawing each marble consecutively, taking into account that the marble is returned to the bag after each draw. The denominator is 9 because there are a total of 9 marbles in the bag. The numerators reflect the number of marbles of each color in the bag.
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