The solution of Ax = b is unique precisely when Ax = 0 has only the trivial solution, because Ax-b is consistent, its solution set is obtained by translating the solution set of Ax-0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax 0 is a single vector, and that happens if and only if Ax 0 has only the trivial solution.
The correct answer is an option (b)
Let us assume that the equation Ax = b has a solution.
Now we need to show the solution is unique when Ax = 0 has only the trivial solution.
As we know Ax = 0 represents homogenous equation.
If Ax = b has a solution, it is unique if and only if every column of A is a pivot column. If every column of A is a pivot column, there are no free variables, and therefore the homogeneous equation has only the trivial solution.
Since Ax-b is consistent, its solution set is obtained by translating the solution set of Ax=0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax= 0 is a single vector, and that happens if and only if Ax 0 has only the trivial solution.
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The complete question is:
Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. Choose the correct answer.
A. Since Ax = b is inconsistent, its solution set is obtained by translating the solution set of Ax-0. For Ax-b to be inconsistent, Ax = 0 has only the trivial solution.
B. Since Ax-b is consistent, its solution set is obtained by translating the solution set of Ax-0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax 0 is a single vector, and that happens if and only if Ax 0 has only the trivial solution.
C. Since Ax = b is inconsistent, then the solution set of Ax = 0 is also inconsistent. The solution set of Ax = 0 is inconsistent if and only if Ax = 0 has only the trivial solution.
D. Since Ax = b is consistent, then the solution is unique if and only if there is at least one free variable in the corresponding system of equations. This happens if and only if the equation Ax 0 has only the trivial solution.
Find the area of quadrilateral, if it's four sides are 27cm, 25cm, 28cm and 32cm and a diagonal is 30cm
The area of quadrilateral ABCD = 4√6314 sq cm. + 15√663 sq. cm
Define the features of quadrilateral?A polygon with four sides, 4 angles, and 4 vertices is called a quadrilateral. The Latin words quadri, which indicates four, and latus, which means side, were combined to form the English word quadrilateral.For this given question:
four sides are 27cm, 25cm, 28cm and 32cm. diagonal is 30cmIn ΔABD , using Heron's formula.
semi perimeter s = a + b + c / 2
s = 27 + 25 + 30 / 2
s = 41 cm
Now,
Area ( ΔABD ) = √s(s - a)(s - b)(s - c)
Area ( ΔABD ) = √41(41 - 27)(41 - 25)(41 - 30)
Area ( ΔABD ) = √41x14x16x11
Area ( ΔABD ) = 4√6314 sq cm.
Now,
In ΔBCD
semi perimeter s = a + b + c / 2
s = 32 + 28 + 30 / 2
s = 45cm
Area ( ΔBCD ) = √s(s - a)(s - b)(s - c)
Area ( ΔABD ) = √45(45 -32)(45 - 28)(45 - 30)
Area ( ΔABD ) = 15√663 sq. cm
Thus, the area of quadrilateral ABCD = 4√6314 sq cm. + 15√663 sq. cm
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A plumber is cutting pieces of pipe that are 3.27 meters long. The pipe must be within 0.07 meters of the required length. Write and solve an absolute value inequality to find the range of the length of pipe.
The range of the length is 3.2 meters to 3.34 meters
How to determine the range of the lengthFrom the question, we have the following parameters that can be used in our computation:
Length of pipe = 3.27 meters
Relative error = 0.07 meters
Represent the range with x
So, we have
|x - Length| = Relative error
substitute the known values in the above equation, so, we have the following representation
|x - 3.27| = 0.07
Remove the bracket
x - 3.27 = 0.07 or x - 3.27 = -0.07
Evaluate
x = 3.2 or x = 3.34
Hence, the range is 3.2 meters to 3.34 meters
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Find two positive and two negative angles that are coterminal with the quadrantal angle 0= -540° such that each angle lies between -1080° and 720°.
The two positive angles are
(Use a comma to separate ariswers as needed.)
The two positive coterminal angles are 180° and 540°, the two negative ones are -900° and -180°.
How to find the coterminal angles?
For any given angle A, the family of coterminal angles can be written as:
B = A + n*360°
Where n can be any integer number.
Here we want to find angles coterminal to -540°, so we have the relation:
B = -540° + n*360°
if n = -1 we get:
B = -540° - 1*360° = -900°
if n = 1
B = -540° + 1*360° = -180°
if n = 2
B = -540° + 2*360° = 180°
if n = 3
B = -540° + 3*360° = 540°
These are the 4 coterminal angles on the given range.
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How long does it take Emil to ride his skateboard
42 meters at 6 meter/minute?
7 hours
7 minutes
7 days
Answer:
42/6 = 7 Minutes
Step-by-step explanation:
(/8-/+)+(/++/4) find the sum plsss
The sum of the fraction given for the sum of 7/8 and 1/4 will be 1 1/8.
How to calculate the fraction?A fraction simply means a piece of a whole. In this situation, the number is represented as a quotient such that the numerator and denominator are split. In this situation, in a simple fraction, the numerator as well as the denominator are both integers.
It should be noted that the sum of 7/8 and 1/4 will be:
= 7/8 + 1/4
= 7/8 + 2/8
= 9/8
= 1 1/8
Therefore, the sum is 1 1/8.
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Complete question
7/8 + 1/4. Find the sum.
Choose the two equations you would use to solve the absolute value equation below. Then solve the two equations.
|6x – 4| = 6
A.
6x + 4 = 6 and 6x – 4 = –6;
B.
6x + 4 = 6 and 6x – 4 = –6;
C.
6x – 4 = 6 and 6x – 4 = –6
D.
6x – 4 = 6 and 6x – 4 = –6;
The solution to the absolute value equation |6x – 4| = 6 is 6x - 4 = 6 and 6x - 4 = -6
What is an equation?An equation is an expression that shows the relationship between numbers and variables.
The absolute value of a real number x is the non-negative value of x without regard to its sign.
Given the absolute value equation:
|6x - 4| = 6
Therefore, the absolute equation can be represented by:
6x - 4 = 6 and -(6x - 4) = 6
Simplifying:
6x - 4 = 6 and 6x - 4 = -6
adding like terms:
6x = 10 and 6x = -2
x = 5/3 and x = -2/3
The solution to |6x – 4| = 6 is x = 5/3 and x = -2/3
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Help me!!!! can you please help I really need it by tomorrow please please please please help me
Answer: I believe it's 30 vehicles
Step-by-step explanation:
If you divided by 40 you get 30. There's for 4 rows of 10 which is 40 so you would divide, I think? I hope this helps! Been so long since I did this math.
2. a farmer has 2,000 feet of fencing to enclose a pasture area. the field will be in the shape of a rectangle and will be placed against a river where there is no fencing needed. what is the largest area field that can be created and what are its dimensions?
The correct answer is 500 yards. The largest area for the field will be when the cube is a forecourt. With 2000 yds. of fencing, each side would be 2000/4 = 500 yds.
Long, or 500 yds. If there were no swash also, the largest area would be given by a square configuration since adding the length and dwindling the range of a square by the same value, t will drop the area by a square with sides of length. Now add the swash on one side. The swash acts like a glass.
As we guard in a cube on one side of the swash, imagine an identical cube on the other side. The maximum area is given when the two blocks form a square, that's when we have a cube doubly as long as wide.
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The value of the largest area of the field will be 5000 sq. ft.
From the given data,
Total pasture area of fencing =2,000 feet
The shape of the field =rectangular
As the area of the rectangle = l*b
So, we can write it as;
2000=x+2y, (As the area of surrounded by a river from one side, so we need to only fence twice of width and only one length (Side) of the rectangle.
Now applying the formula of the area of the rectangle
we will get it as:
[tex]$$\begin{aligned}& A(x)=x\left(\frac{2000-x}{2}\right)=\frac{1}{2}\left[2000 x-x^2\right] \\& A^{\prime}(x)=\frac{1}{2}[2000-2 x]=0 \\\end{aligned}$$[/tex]
Solving the value,
We will get: x=100 ft y=50 ft
So, determining the value of the largest area,
We will get it as:
A_{max}=(100)(50)=5000 Sq. ft.
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An anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows: 875 drank alcohol regularly, 665 smoked cigarettes, 207 used illegal drugs, 422drank alcohol regularly and smoked cigarettes, 113 drank alcohol regularly and used illegal drugs,126 smoked cigarettes and used illegal drugs, 97 engaged in all three behaviors, and217 engaged in none of these behaviors.
a) How many students were surveyed?
nothing
According to the following Venn diagram, the number students surveyed is 2063
The term Venn diagram in math is defined as overlapping circles or other shapes to illustrate the logical relationships between two or more sets of items
Here we have given that anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs.
While we looking into the Venn diagram we have identified that 875 drank alcohol regularly, 665 smoked cigarettes, 207 used illegal drugs, 422drank alcohol regularly and smoked cigarettes, 113 drank alcohol regularly and used illegal drugs,126 smoked cigarettes and used illegal drugs, 97 engaged in all three behaviors, and 217 engaged in none of these behaviors.
Then the number of students is calculated as,
=> 875 + 667 + 207 + 97 + 217
=> 2063
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Al repartir pastelitos entre 7 niños cada uno le tocaron un pastelito completo y 3/7 de otro cuanto pastelitos reparto?
Answer: 10 magdalenas
espero que esto ayude
What is the area of the roof top garden?
Total area of the roof top garden is 272 ft² .
What is the garden's square foot size?Rooftop planters can have soil depths ranging from six inches at the very least, to allow for little groundcovers and shrubs, to three to four feet, which is necessary for huge shade and coniferous trees. Divide the length by the width of a rectangle or square to find its area.
The square footage of the area is calculated by multiplying the length and width. Please maintain the same unit size (feet or inches). For a rectangular garden area, multiply the length by the width to get the square footage. A suitable garden is often one that is roughly 10 feet by 10 feet, if your landscape permits it (3 x 3 meters).
Given, the figure comprises of triangle and rectangle,
Area of rectangle = length × breadth
= 12 ft × 20 ft
= 240 ft²
Area of triangle = 1/2 × base × height
= 1/2 × 8 × (20 - 12)
= 1/2 × 8 × 8
= 32 ft²
Total area of the figure = Area of rectangle + Area of triangle
= 240 + 32
= 272 ft²
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What is the GCF of 8xy and 12x all i want is the answer
If the perimeter of triangle ABC is 27. 6cm, what is the perimeter, in centimeters of triangle BCD. AB = 9. 6cm BC = 6 cm
The perimeter of the triangle BCD is 17.25 centimeters.
A closed route that covers, encircles, or outlines a one-dimensional length or a two-dimensional form is called a perimeter. A circle's or an ellipse's circumference is referred to as its perimeter. There are several uses in real life for perimeter calculations.
so, perimeter of the triangle is the sum of all its side length.
Perimeter of one triangle / perimeter of another triangle= side of one triangle by side of another triangle.
let us suppose that the perimeter of triangle BCD is equal to X
27.6/x = 9.6/6
x = 17.25 cm
therefore, the perimeter of triangle BCD is equal to 17.25 centimeters.
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Plsss helppp!!!!!!!!
Answer: (7)^-2 and 1/49 are equivalent
Step-by-step explanation:
What 2 numbers are equivalent to 35%
35% can be represented in equivalent decimal form and fraction form 0.35 and 7/20
How is this calculated?35% is equivalent to 0.35 in decimal form. Divide the percentage by 100 to get the decimal equivalent. 35 / 100 = 0.35.
35% is equivalent to 7/20 as a fraction. A percentage must be multiplied by 100 before being simplified to become a fraction. 35/100 = 0.35, or 7/20, can be written more simply.
Although both forms are equal, they are employed in various contexts. The fraction form is more frequently used in probability and statistics while the decimal form is more frequently used in calculations.
It's also important to keep in mind that percentages are a technique to express numbers as fractions of 100. For example, 35% is the same as 35/100, or 7/20.
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A figure is rotated 180 degrees clockwise about the origin. Which statement is true about the rotated figure?
(a) It is a different shape and size from the figure.
(b) It is the same shape as the figure but is larger.
(c) It is the same shape and size as the figure.
(d) It is the same shape as the figure but is smaller.
The shape and size remains the same represents the correct alternative.
What is image rotation?A rotation is a type of transformation which is a turn.A figure can be turned clockwise or counterclockwise on the coordinate plane.In both transformations the size and shape of the figure stays exactly the same.Given is that a figure is rotated 180 degrees clockwise about the origin.
When a coordinate (x, y) is rotated by 180 degrees clockwise about the origin, the new coordinates become ( - x, - y). But only the coordinates change, the shape and size remains the same.
Therefore, the shape and size remains the same represents the correct alternative.
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Solve for a. a> a+4 ○ a > 21/1/20 O O a -7
The inequality expression has no solution for a
How to determine the solution of aFrom the question, we have the following parameters that can be used in our computation:
a > a + 4
Subtract the variable a from both sides of the equation
So, we have the following representation
a - a > a + 4 - a
Evaluate the difference
This gives
0 > 4
The above expression is false
hence, the inequality has no solution
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What is the slope of the line ( 3 ,5 ) and (-6,-6)
What are the domain and range of the function y=f(x) as shown on the graph?
For the graphed function we have:
The domain
D: {-3, -2, -1, 0, 1, 2}
And the range
R: {0.25, 0.5, 1, 2, 4, 8}
What are the domain and the range of the function?For a function y = f(x), we define the domain as the set of possible inputs and the range as the set of possible outputs.
Remember that the notation in point form is (input, output).
Now, the graphed points are:
(-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 0.5), (2, 0.25)
The domain is the set of the first values:
D: {-3, -2, -1, 0, 1, 2}
And the range is the set of the second values:
R: {0.25, 0.5, 1, 2, 4, 8}
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Recently, students in a marketing research class were interested in the driving behavior of students. Specifically, the marketing students were interested in finding out if exceeding the speed limit was related to social activity. They collected the following responses from 100 randomly selected students: Exceeds the Speed Limit Does Not Exceed the Speed Limit Male 40 25 Female 10 25 Based on the analysis, what can be concluded?a. Driving behavior and gender are correlated. B. Driving behavior and gender are not related. C. Driving behavior and gender are related. D. No conclusion is possible
Based on the analysis, we can be conclude that (C) Driving behavior and gender are related.
The link between variables is described by correlation. It can be viewed positively or negatively, as well as strong or weak. A correlation does not prove a cause. When the variable on the x-axis rises as the variable on the y-axis rises, there is a positive linear association. An upward-sloping straight regression line illustrates this.
When one variable rises while the other falls, there is a negative linear correlation. A straight regression line with a falling slope illustrates this. When there is a relationship between variables, but the relationship is not linear, there is a non-linear correlation (straight).
Hence, according to correlation theory, Driving behavior and gender are related.
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state the converse, contrapositive, and inverse of each of these conditional statements. a) if it snows today, i will ski tomorrow. b) i come to class whenever there is going to be a quiz. c) apositive integer is a prime only if it has no divisors other than 1 and itself.
a) If it snows today, I will ski tomorrow.
Converse: If I ski tomorrow, it snows today.
Contrapositive: If I don't ski tomorrow, it doesn't snow today.
Inverse: If it doesn't snow today, I won't ski tomorrow.
b) I come to class whenever there is going to be a quiz.
Converse: Whenever I come to class, there is going to be a quiz.
Contrapositive: Whenever I don't come to class, there isn't going to be a quiz.
Inverse: Whenever there isn't going to be a quiz, I don't come to class.
c) A positive integer is a prime only if it has no divisors other than 1 and itself.
Converse: If a positive integer has no divisors other than 1 and itself, then it is a prime.
Contrapositive: If a positive integer is not a prime, it has divisors other than 1 and itself.
Inverse: If a positive integer has divisors other than 1 and itself, it is not a prime.
A mathematical conditional statement is a statement that presents a relationship between two statements, where one statement (the "condition") is only true if the other statement (the "conclusion") is also true.
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how many variables are in the data set? b. which of the following variables are categorical, and which are quantitative? overall score - select your answer - recommended - select your answer - owner satisfaction - select your answer - overall miles per gallon - select your answer - acceleration () sec - select your answer - c. what percentage of these vehicles are recommended? round your answer to one decimal place. d. what is the average of the overall miles per gallon across all vehicles? round your answer to one decimal place.
All parts of the problem given have been explained and provided below.
Variables in any particular dataset are defined as any characteristics, numbers, or quantity that can be measured or counted to give information about any object, on which the dataset is based.
So, the variables in this given dataset are;
Overall score
Recommended
owner satisfaction
Overall miles per gallon
Acceleration (0-60) sec
Categorical or qualitative variables are those which can't be quantified or measured. Their values are defined by an order relation between the different categories.
the categorical variables in this dataset are,
Recommended
Owner satisfaction
Quantitative variables are those which can be measured in an order to provide information about an object.
the quantitative variables in this dataset are,
Overall score
Overall miles per gallon
Acceleration (0-60) sec
In the given dataset, 7 vehicles are recommended,
So their percentage would be,
7/15 × 100 = 46.67%
The average overall miles per gallon of these 15 vehicles is 25.4.
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In the figure, ABFG and CDEF are two
parallelograms such that the sum of their areas
702m square. AB = CD = EF = FG = 1/2BC find the
area of the shaded region
,
The area of the shaded region is 251 m^2.
the solution is attached with this,
parallelograms
A parallelogram is a two-dimensional geometrical shape whose sides are parallel to each other.
The Sum of adjacent angles of a parallelogram is equal to 180 degrees.
It is a type of polygon having four sides, where the pair of parallel sides are equal in length.
polygon
A polygon can be defined as a flat or plane, two-dimensional closed shape bounded with straight sides. It does not have curved sides. The sides of a polygon are also called its edges.
The points where two sides meet are the vertices of a polygon.
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Tammy is selling tickets for the school jazz band concert. Adult tickets cost $5, and student tickets cost $3. If
Tammy sells a total of 130 tickets and collects $462, how many of each kind ofticket did she sell?
Answer:
36 adult and 94 student tickets
Step-by-step explanation:
x = adult ticket y = student ticket
5x + 3y = 462
x + y = 130
x + y = 130
y = 130 - x
5x + 3y = 462
5x + 3(130 - x) = 462
5x + 390 - 3x = 462
2x + 390 = 462
2x = 72
x = 36
x + y = 130
36 + y = 130
y = 94
Which set of side lengths can be used to form a triangle?
O 3 ft, 3 ft, 7 ft
O 5 ft, 9 ft, 3 ft
O 6 ft, 7 ft, 7 ft
O 9 ft, 7 ft, 2 ft
so we have hmmm 4 sets above, hmm let's check say the second set, the triangle with sides 5, 9, 3. Check the picture below.
we have three sides, let's look at the two smaller sides first, 5 and 3.
if we move the sides closer and ever closer to each other, to the extent that one is right on top of the other, what is the length of the red side? Well, assuming the two smaller sides are one pancaked on top of the other, the red side will be as long as 5 - 3 = 2. However, the sides can't be on top of each other, because if that's so, we have a flat-line, and thus we wouldn't have a triangle. So whatever the third side may be, it must be greater than 2.
Now, if we move the sides away from each other, farther and farther to the extent that one is parallel to the other, then the third side will just be as long as 3 + 5 = 8. However, we can't do that, because if that were to happen, we again will have a flat-line and not a triangle. So whatever the third side may be, it must be less than 8.
so for those reasons the third side can't be 9, it won't work for a triangle, even 8 is not good, so it must be less.
Now, if we look at the 1st set and 4th set, we end up with the same dilemma, the longest side is too long.
If we take a looksie at the 3rd set however with small sides of 6 and 7, it gives us a third side that must be more than 1 but less than 13, so 7 fits just nicely, so that one works.
dr. edward jenner realized that cows have a disease called cowpox, which is like a disease that infects humans called smallpox; jenner noticed that milkmaids whose hands were infected with cowpox were not contracting smallpox. jenner infected a child with the pus from a cowpox blister, and found that the child did not contract smallpox. which statement represents a supporting hypothesis?
The cowpox infection will prevent the child from being infected by the small pox virus.
What is hypothesis?Hypothes is a tentative assumption made in order to draw out and test its logical or empirical consequences
A simple hypothesis suggests only the relationship between two variables: one independent and one dependent. Examples: If you stay up late, then you feel tired the next day. Turning off your phone makes it charge faster.
Given paragraph is "dr. edward jenner realized that cows have a disease called cowpox, which is like a disease that infects humans called smallpox; jenner noticed that milkmaids whose hands were infected with cowpox were not contracting smallpox. jenner infected a child with the pus from a cowpox blister, and found that the child did not contract smallpox."
In this supporting hypothesis is "The cowpox infection will prevent the child from being infected by the small pox virus."
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Simplify the expression below.
12.8 - 3s + 15.5 + 8s
Answer: 5s + 28.3
Step-by-step explanation:
The cost of a stereo cable has dropped to $11 today. Yesterday's cost was $17 . Find the percentage decrease. Round your answer to the nearest tenth of a percent.
11 is a 35.3% decrease of 17.
17 - 11/((17 + 11)/2) = 6/14 = 0.42857142857143 = 42.857142857143%
The functions f(x)=−34x+2 and g(x)=(14)x+1 are shown in the graph. What are the solutions to −34x+2=(14)x+1? Select each correct answer. Responses −1 negative 1 0 0 1 1 2 2 3
The solutions to the equations is the point of intersection of graphs and the points are ( 0 , 2 ) and ( 1 , 1.25 )
What is Equation of Graph of Polynomials?Graphs behave differently at various x-intercepts. Sometimes the graph will cross over the x-axis at an intercept. Other times the graph will touch the x-axis and bounce off.
Identify the even and odd multiplicities of the polynomial functions' zeros.
Using end behavior, turning points, intercepts, and the Intermediate Value Theorem, plot the graph of a polynomial function.
The graphs cross or are tangent to the x-axis at these x-values for zeros with even multiplicities. The graphs cross or intersect the x-axis at these x-values for zeros with odd multiplicities
Given data ,
Let the first equation be represented as A
Now , the value of A is
y = ( -3/4 )x + 2 be equation (1)
Let the second equation be represented as B
Now , the value of B is
y = ( 1/4 )ˣ + 1 be equation (2)
Now , to calculate the solutions to the equations , plot the equations in the graph
The point of intersection of equation of lines represents the solution to the equation
Substituting the values in the equation , we get
( -3/4 )x + 2 = ( 1/4 )ˣ + 1
The first point of intersection is ( 0 , 2 )
when x = 0 , y = 2
( -3/4 ) (0) + 2 = ( 1/4 )⁰ + 1
On simplifying the equation , we get
0 + 2 = 1 + 1
2 = 2
So , the solution is ( 0 , 2 )
Hence , the equations are solved
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Calculate the indicated length in the figure.