Given the following system of equations:
6X1 - 6x2 - 4x3 = 0
X1 - 7x2 - 6x3 = 2
X1 +5x2 + nx3 = -2
Rewrite the system in Ax = b format and determine the following:
a. By reduction of the augmented matrix [A|b] to ref, find a value for n such that the system is consistent with an infinite number of solutions.
b. Based on your solution in part A, identify the rank of matrix A and rank of the augmented matrix [A|b].
c. Based on the value of the rank, how many equations (the row vectors of the augmented matrix [Ab]) are linearly independent?
d. Using your solution in part A, solve the system of equations using Gauss-jordan elimination.
Answers
Answer:
Step-by-step explanation:
Given:-
- The following system of equations is given:
\(6x_1 - 6x_2 -4x_3 = 0\\\\x_1 - 7x_2 -6x_3 = 0\\\\x_1 - 5x_2 -nx_3 = 0\\\)
Solution:-
- The matrix equation consists of coefficient matrix "A" and a variable matrix " x ". These two matrices undergo multiplication to yield a solution column vector "b".
- The matrix A, is a symmetrical square matrix with its elements representing the coefficients of each variable as follows:
\(A = \left[\begin{array}{ccc}a_1_1&a_1_2&a_1_3\\a_2_1&a_2_2&a_2_3\\a_3_1&a_3_2&a_3_3\end{array}\right]\)
- Where the elements first subscript denotes the equation number and second subscript denotes the variable number.
\(A = \left[\begin{array}{ccc}6&-6&-4\\1&-7&-6\\1&5&n\end{array}\right]\)
- Similarly, the variable matrix " X " is a column vector that lists all the variables in the the system of equations in a ascending order.
\(X = \left[\begin{array}{c}x_1&x_2&x_3\end{array}\right]\)
- The solution vector " b " is the corresponding solution or any number written on the right hand side of the equals to sign " = " :
\(b = \left[\begin{array}{c}0&2&-2\end{array}\right]\)
- Now, we can express the given system in the asked format:
\(A*X = b\\\\\left[\begin{array}{ccc}6&-6&-4\\1&-7&-6\\1&5&n\end{array}\right]*\left[\begin{array}{c}x_1&x_2&x_3\end{array}\right] = \left[\begin{array}{c}0&2&-2\end{array}\right]\)
- The augmented matrix is a matrix that combines the coefficient matrix " A " and the solution vector " b ". A solution vector "b" as an extra column to the coefficient matrix:
\([ A | b ]\\\\ \left[\begin{array}{ccccc}6&-6&-4&|&0\\1&-7&-6&|&2\\1&5&n&|&-2\end{array}\right]\)
- Now we will perform row reduction operation such that the system is consistent and has infinite number of solution.
- Row operation: R3 - R2 & R1/6
\(\left[\begin{array}{ccccc}1&-1&-\frac{2}{3} &|&0\\1&-7&-6&|&2\\0&12&n+6&|&-4\end{array}\right]\)
- Row operation: R2 - R1 & R3 / 12
\(\left[\begin{array}{ccccc}1&-1&-\frac{2}{3} &|&0\\0&-6&-\frac{16}{3} &|&2\\0&1&\frac{n+6}{12} &|&-\frac{1}{3}\end{array}\right]\)
- Row operation: R2 / 6
\(\left[\begin{array}{ccccc}1&-1&-\frac{2}{3} &|&0\\0&-1&-\frac{8}{9} &|&\frac{1}{3} \\0&1&\frac{n+6}{12} &|&-\frac{1}{3}\end{array}\right]\)
For the above system to be consistent and have infinite many solution then the coefficient of " x3 " for the 2nd and 3rd row must be equal:
\(-x_2 - ( \frac{n+6}{12})*x_3 = \frac{1}{3}\)
\(-x_2 - ( \frac{8}{9})*x_3 = \frac{1}{3}\)
The coefficient of " x_3 " must be equal:
\(( \frac{n+6}{12}) = \frac{8}{9} \\\\\\\n = \frac{14}{3}\)
- The augmented matrix in reduced form becomes:
\(\left[\begin{array}{ccccc}1&-1&-\frac{2}{3} &|&0\\0&1&\frac{8}{9} &|&-\frac{1}{3} \\0&0&0 &|&0\end{array}\right]\)
Answer: Rank = Number of non-zero rows = 2
- The number of linearly independent rows are equal to the rank of the augmented matrix.
Hence,
Answer: Number of linearly independent rows = 2
Row operation: R1 + R2
\(\left[\begin{array}{ccccc}1&0&\frac{2}{9} &|&-\frac{1}{3} \\0&1&\frac{8}{9} &|&-\frac{1}{3} \\0&0&0 &|&0\end{array}\right]\)
- The variable "x_3" will take any arbitrary value for which the solution holds infinitely many solutions.
\(x_2 + \frac{8}{9}*x_3 = -\frac{1}{3} \\\\x_2 = - ( \frac{8}{9}*x_3 + \frac{1}{3} )\\\\x_1 + \frac{2}{9}*x_3 = -\frac{1}{3} \\\\x_1 = - ( \frac{2}{9}*x_3 + \frac{1}{3} )\\\)
- Taking x_3 = α:
Answers:
\(x_1 = -\frac{1}{3} + \frac{2}{9} \alpha \\\\x_2 = -\frac{1}{3} + \frac{8}{9} \alpha\)
Related Questions
On a piece of paper, graph f(x) = (4x≤3. Then determine which answer
12 if x > 3
choice matches the graph you drew.
Answers
The graph of the piecewise function is the one in option D-
Which one is the graph of f(x)?First, we know that:
f(x) = 4 when x ≤ 3
So, we want to have a horizontal line in the left side of the graph, such that it ends in a closed circle at x = 3 (we need to have a closed circle because the symbol ≤ is used).
From the given graphs, the only one with that form is D.
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Select the correct answer.
Evaluate the following expression when x = -4 and y = 4.
x
6
−
x
4
y
A.
1
,
025
4
B.
1
,
023
4
C.
16
,
385
4
D.
−
1
,
023
4
Answers
Answer:
1023/4
Step-by-step explanation:
shown in the picture
Which inequality is true if p = 3.4? A. 3p < 10.2 B. 13.6 ≤ 3.9p C. 5p > 17.1 D. 8.5 ≥ 2.5p
Answers
Answer:
D 8.5 ≥ 2.5p
Step-by-step explanation:
How can you eliminate the y-terms in this system?
Answers
To eliminate the y-terms in this system of equations, we can multiply the second equation by 2 and add it to the first equation.
We have,
To eliminate the y-terms in this system of equations, we can multiply the second equation by 2 and add it to the first equation.
This will result in the y-terms canceling out, and we will be left with only x-terms on one side of the equation, which we can then solve for x.
So, multiplying the second equation by 2, we get:
2(3x + 4y) = 2(5)
6x + 8y = 10
Now, adding this to the first equation, we get:
4x - 8y + 6x + 8y = 20 + 10
10x = 30
Dividing both sides by 10, we get:
x = 3
Now, we can substitute this value of x back into either of the original equations to solve for y.
4x - 8y = 2
So,
4(3) - 8y = 20
12 - 8y = 20
-8y = 8
y = -1
Therefore,
The solution to the system of equations is (x, y) = (3, -1).
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Is 3x-18 equivalent to 3.(x-6)
Answers
Answer:
Yes it is.
Step-by-step explanation:
If you distribute you will get the same answer, 3x-18 because 3 times x is 3x and 3 times 6 is 18.
Hope this helps!!
I couldn't type it so I added a file.
Answers
Answer
First factor Second factor
1 45
3 15
5 9
Step-by-step explanation:
So in order to find the factors, you might need to multiply or divide.
1 * 45 = 45
3 * 15 = 45
5 * 9 = 45
Laura typed 550 words during a 5-minute test in her keyboarding class. Which equation can be used to find t, the time required for Laura to type 2,000 words, typing at the same rate?
Answers
Answer:
Step-by-step explanation:
This Bar Chart shows the number of DVDs sold at a local music store during one week.
Which measure(s) of central tendency can be used to determine the average number of DVDs sold each day?
A. the median
B. the mean
C. the mean and the median
D. the mode
Answers
Answer:
B. the mean
Step-by-step explanation:
According to the Bar Chart shown, the number of DVDs sold at a local music store during one week are displayed.
Therefore, the measure(s) of central tendency that can be used to determine the average number of DVDs sold each day is the mean.
This is because the mean is the sum total of the number of DVDs sold, divided by the number, which gives the average.
Answer:
The Mean
Step-by-step explanation:
I took the test
A rhombus has side lengths of 25. What could be the lengths of the diagonals?
A. 22 and 40
B. 26 and 36
C. 26 and 48
D. 30 and 40
Answers
Answer:
The correct option is;
D. 30 and 40
Step-by-step explanation:
Here, we have that the rhombus is a quadrilateral with equal and parallel sides hence the length of the diagonals will be
2 × 25×sinθ and 2 × 25× cosθ
Therefore tanθ = (2 × 25×sinθ)/(2 × 25× cosθ) = (sinθ)/(cosθ)
Therefore, the root of the sum squares of both diagonals = 50
Therefore, we analyze each of the options as follows
For A. we have √(22² + 40²) = 45.65 ≠ 50 therefore these are not the length of diagonals of the rhombus in question
For B. we have √(26² + 36²) = 44.41 ≠ 50 therefore these are not the length of diagonals of the rhombus in question
For C. we have √(26² + 48²) = 55.59 ≠ 50 therefore these are not the length of diagonals of the rhombus in question
For D. we have √(30² + 40²) = 50 therefore these are possible lengths of diagonals of the rhombus in question.
A Christmas tree is supported by a wire that is 2 meters longer than the height of the tree. The wire is anchored at a point whose distance from the base of the tree is 23 meters shorter than the height of the tree. What is the height of the tree?
Answers
Answer:
The height of the tree is 35 meters.
Step-by-step explanation:
Let the height of the tree be x meters, then the length of the wire is
x+2 meters.
The distance from the tree to the anchor is x-23 meters.
So using the pythagoras theorem:
(x + 2)^2 = x^2 +(x -23) ^2
x^2 + 4x + 4 = x^2 + x^2 - 46x + 529
x^2 - 50x + 525 = 0
( x - 15)(x - 35) = 0
x = 15, 35
x cannot be 15 as this would make the distance 15-23 which is negative so x mustr be 35 meters,
5 What is the solution for y in the following system of equations?
7x + 2y = 8
-
7x+6y= -88
Answers
Answer:
working on a answer one second
Step-by-step explanation:
The probability that Lexie is on time for a given class is 95%. If there are 32 classes during the semester, what is the best estimate of the number of times out of 32 that Lexie is on time to class? Round your answer to the nearest integer.
Answers
Answer:
between 31 and 29 classes
Step-by-step explanation:
The best estimate of the number of times out of 32 that Lexie is on time to class is 30.4.
What is probability?It is a branch of mathematics that deals with the occurrence of a random event.
Given:
The probability that Lexie is on time for given class is 95% and there are 32 classes during the semester.
So,
=32 * 0.95
= 30.4
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Bandhan Bank employee salary after 10 years
Answers
Answer:
- Banking Operations salary in India with less than 1 year of experience to 10 years ranges from ₹ 1.4 Lakhs to ₹ 7 Lakhs with an average annual salary of ₹ 3.1 Lakhs based on 261 latest salaries
The function f(x)=−3x+2 is defined over the domain −1
Answers
The domain of the function f(x) = -3x + 2 is (-∞, +∞), representing all real numbers, and the range is (-∞, 2], representing all real numbers less than or equal to 2.
The function f(x) = -3x + 2 is a linear function defined by a straight line. To determine the domain of this function, we need to identify the range of values for which the function is defined.
The domain of a linear function is typically all real numbers unless there are any restrictions. In this case, there is no explicit restriction mentioned, so we can assume that the function is defined for all real numbers.
Therefore, the domain of the function f(x) = -3x + 2 is (-∞, +∞), which represents all real numbers.
Now, let's analyze the range of the function. The range of a linear function can be determined by observing the slope of the line. In this case, the slope of the line is -3, which means that as x increases, the function values will decrease.
Since the slope is negative, the range of the function f(x) = -3x + 2 will be all real numbers less than or equal to the y-intercept, which is 2.
Therefore, the range of the function is (-∞, 2] since the function values cannot exceed 2.
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Need help with this equation 8x²+10x-3
Answers
Answer:
Step-by-step explanation:
You have to use the quadratic formula
x = -b +/- \(\sqrt{(b^{2} -4ac)/2a\)
and then you can simplify both answers so the first one would be -13/2 and the second would be -27/2
you usually have to reject an answer but since there is no equal sign in the question, you should be fine
Sheila and Carla enter into Dillards with $25.00 between them they want to purchase a few items together. Sheila has $10.00 dollars, and Carla has $15.00 dollars. Sheila's purchases come to $12.21. Carla's purchases come to $12.79. How much change does Dillards owe them? What does Sheila owe Carla?
Answers
The amount of money Dillards owe Sheila and Carla is $0, while Sheila owe Carla $2.21
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Sheila has $10.00 dollars, and Carla has $15.00 dollars. Sheila's purchases come to $12.21. Carla's purchases come to $12.79. Hence:
Amount of money Dillards owe them = 25 - (12.21 + 12.79) = $0
Amount of money Sheila owe Carla = 12.21 - 10 = $2.21
The amount of money Dillards owe Sheila and Carla is $0, while Sheila owe Carla $2.21
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Use: 0, 4, 6, 11, 9, 8, 9, 1, 5, 9, 7 to construct a box-and-whisker plot. List the maximum, minimum, and quartiles below. has to be a sentence
Answers
A box-and-whisker plot which represent the given data set is shown in the image attached below.
The five-number summary for the given data set have been listed correctly below.
What is a box-and-whisker plot?In Mathematics, a box-and-whisker plot is sometimes referred to as a box plot and it can be defined as a type of chart that is used for the graphical or visual representation of the five-number summary of a data set with respect to locality, skewness, and spread.
By using an online box-and-whisker plot calculator, the five-number summary for the given data set include the following:
Minimum = 0.First quartile = 4.Median = 7.Third quartile = 9.Maximum = 11.By critically observing the box-and-whisker plots (see attachment), we can logically deduce that all of the five-number summary for the given data set are correctly listed.
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i need help please lol
Answers
Answer:
that's a right angle and its equals 90° but I think you add all of them 56°,31°, and x°
Answer:
x should equal 93
Step-by-step explanation:
180° - 56° - 31 degrees = 93°.
Let me if its wrong and I will correct it if is.
A colony of bacteria grows 5% every hour.How long does it take for the colony to double in size ? with show working
Answers
5%=1hour
100% =?
100×1/5
20 hours
for it to double it needs to replicate itself by100%
Which method would determine the volume of prism with dimesions 2×21/4×4 shown below?
Answers
Answer:
42
Step-by-step explanation:
10[25-{8-6(16-13) }]÷5
solve it
Answers
Answer:
70
Step-by-step explanation:
10[25-{8-6(16-13) }]÷5
10[25-{8-6*3)}]/5
10[25-{8-18}]/5
10[25-(-10)]/5
10[25+10]/5
10*35/5
70
Step-by-step explanation:
please mark me as brainlest
![10[25-{8-6(16-13) }]5solve it](https://i5t5.c14.e2-1.dev/h-images-qa/answers/attachments/sdnIvtk8s79XJBqh20RMpRN88ribocLC.jpeg)
What is the equation of the line that passes through the point (-5,-7)and has a slope of 2/5
Answers
Answer:
6.33333 is the answer <3
What is the non-negative zero of the function f, where f(x) = 6x^2-9x-6?
Answers
Answer:
The non-negative zero of the function f(x) is x = 2.
Step-by-step explanation:
For a given function f(x), the "zeros" of the function are the values of x such that:
f(x) = 0
In this case, we have the function:
f(x) = 6*x^2 - 9*x - 6
If we want to find the zeros of this function, we need to solve:
f(x) = 0 = 6*x^2 - 9*x - 6
To solve this, we can use the Bhaskara's formula, which says that for a general quadratic equation:
0 = a*x^2 + b*x + c
The zeros are:
\(x = \frac{-b \pm \sqrt{b^2 - 4*a*c} }{2*a}\)
In this case our equation is:
0 = 6*x^2 - 9*x - 6
then, in the above notation, we have:
a = 6
b = -9
c = -6
Replacing these in our general formula, we get:
\(x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4*6*(-6)} }{2*6} = \frac{9 \pm 15 }{12}\)
Then we have two zeros:
x = (9 + 15)/12 = 24/12 = 2
x = (9 - 15)/12 = -6/12 = -1/2
But we want only the non-negative, so we can discard the second one.
Concluding, the non-negative zero of the function f(x) is x = 2.
Desi owes her landlord last month's rent and utilities. The total bill is $825. Utilities cost $575 less than rent. a) Let x represent the cost of b) Then represents the cost of c) Write an equation and find the cost of each. (Show your work).
Answers
From the given information
x represents the rental cost
y represents the utility cost
Cost of rent is $700
Cost of Utility is $125
Given :
The total bill is $825. Utilities cost $575 less than rent.
We need to find the cost of months rent and utilities cost
So lets assume variables for each
Let x represents the cost of rent
and y represents the utilities cost
Total bill is 825
\(x+y=825\)
Utilities cost $575 less than rent.
Utility cost \(y=x-575\)
Replace the second equation in first equation
\(x+y=825\\x+x-575=825\\2x-575=825\\2x=825+575\\2x=1400\\x=700\)
The cost of rent is $700
\(y=x-575\\y=700-575\\y=125\)
Utility cost is $125
The cost of rent is $700
Utility cost is $125
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Nicholas put 1,012 baseball cards into boxes. He put 22 cards in each box. How many boxes did Nicholas need for these baseball cards?
Answers
Answer:
46 boxes
Step-by-step explanation:
1012 / 22 = 46
Which quadratic function has an axis of symmetry of x = 5?y = x^2 + 3x + 5y = -x^2 + 6x + 2y = x^2 + 6x + 3y = x^2 + x + 3
Answers
Problem Statement
The question tells us to find the equation of the quadratic function with an axis of symmetry of x = 5.
Method
To solve this question, we need to check each option for which has its vertex at the axis of symmetry.
To find this vertex, we need to follow these steps:
1. Add and subtract half of the square of the coefficient of x.
2. Factorize the result.
3. Find the vertices.
Implementation
Option A:
\(\begin{gathered} y=x^2+3x+5 \\ y=x^2+3x+\frac{9}{4}-\frac{9}{4}+5 \\ y=(x+\frac{3}{2})^2+\frac{11}{4} \\ \\ \text{vertex is:} \\ (-\frac{3}{2},\frac{11}{4}) \end{gathered}\)Option B:
\(\begin{gathered} y=-x^2+6x+2 \\ y=-x^2+6x+9-9+2 \\ y=-(x-3)^2+9+2 \\ y=-(x-3)^2+11 \\ \\ \text{Vertex:}(3,11) \end{gathered}\)Option C
\(\begin{gathered} y=x^2+6x+3 \\ y=x^2+6x+9-9+3 \\ y=(x+3)^2-6 \\ \\ \text{vertex: }(-3,-6) \end{gathered}\)Option D:
\(\begin{gathered} y=x^2+x+3 \\ y=x^2+x+\frac{1}{4}-\frac{1}{4}+3 \\ y=(x+\frac{1}{2})^2+\frac{11}{4} \\ \\ \text{Vertex: }(-\frac{1}{2},\frac{11}{4}) \end{gathered}\)Final Answer
THEREFORE, NONE OF THE FUNCTIONS HAVE THEIR AXIS OF SYMMETRY AT X = 5
(Will give BRAINLIEST ANSWER If you got it correct)
Perform the following binary multiplication.
1111X111
Answers
ANS : 12 3 321
Refer the attachment for your reference !!
A ladder of length (2x+6) feet is positioned x feet from a wall. If the ladder reaches a height of (2x+4) feet along the wall. Find the longest leg.
A. 10ft
B. 24ft
C. 26ft
D. 13cm
Answers
Using the Pythagoras theorem, the longest leg has the length of 24 feet.
Given that,
A ladder of length (2x+6) feet is positioned x feet from a wall.
Height of the ladder = (2x + 6) feet
Distance of ladder from the wall = x feet
Height of the wall that the ladder is placed = (2x + 4) feet
These three lengths form s right triangle where (2x + 6) feet is the hypotenuse.
Longest leg is (2x + 4) feet
Using the Pythagoras theorem,
(2x + 6)² = (2x + 4)² + x²
4x² + 24x + 36 = 4x² + 16x + 16 + x²
4x² + 24x + 36 = 5x² + 16x + 16
x² - 8x - 20 = 0
(x - 10) (x + 2) = 0
x = 10 or x = -2
x = 2 is not possible.
So x = 10
Longest leg = 2x + 4 = 20 + 4 = 24 feet
Hence the length of the longest leg is 24 feet.
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Casey gets paid $2.55 per hour waiting tables. On Friday she earned $72 in tips and made a total of $82.20. How many hours did she work Friday?
Answers
First you would subtract 72 from 82.20 which is 12.20 and then you would divide 12.20 by 2.55 which is 4.78431373 and you round that to the nearest whole :)
In two or more complete sentences, identify the parent function, and describe the transformations that were applied to obtain the graph, f(x)=√2x+6+1.
Answers
Answer:
The parent function is f(x)=√x. This graph has been transformed by a translation left 6 units, a translation up 2 units, and a stretch by a factor of 2. Adding a number at the end of a function results in a vertical translation. Adding a number inside of a function, in this case under the square root, translates the graph horizontally. Multiplying the variable by a number before another operation results in a stretch or shrink of the graph.
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Step-by-step explanation:
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