b. The system has a unique solution when k is not equal to -2 or 10.
c. The system has infinitely many solutions when k = 10.
d. The system has no solution when k = -2.
The augmented system for the system is:
[1 0 2 -2]
[1 1 k 2]
[3 k -2 2]
The system to have a unique solution, the rank of the coefficient matrix must be equal to the rank of the augmented matrix.
Using row reduction to reduce the augmented matrix to echelon form, we get:
[1 0 2 -2]
[0 1 k+2 4]
[0 0 (k-10)/(k+2) 10]
So, the system has a unique solution when k is not equal to -2 or 10.
The system to have infinitely many solutions, the rank of the coefficient matrix must be less than the rank of the augmented matrix, and the last row of the echelon form of the augmented matrix must be all zeros.
This occurs when:
(k-10)/(k+2) = 0
which happens when k = 10.
So, the system has infinitely many solutions when k = 10.
The system to have no solution, the last row of the echelon form of the augmented matrix must have a non-zero constant on the right-hand side.
This occurs when:
(k-10)/(k+2) ≠ 0
True for all values of k except k = -2. So, the system has no solution when k = -2.
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(a) The augmented matrix for the system is: [1 0 2 | -2] [1 1 k | 2] [3 k -2 | 2] (b) The system has a unique solution when the determinant of the coefficient matrix is nonzero.
In this case, the determinant is 2k + 3. Therefore, the system has a unique solution for any value of k except k = -3/2. (c) The system has infinitely many solutions when the determinant of the coefficient matrix is zero, and the system is consistent (i.e., the right-hand side of each equation is consistent with the others).
In this case, when k = -3/2, the determinant becomes zero, and the system has infinitely many solutions.
(d) The system has no solution when the determinant of the coefficient matrix is zero, and the system is inconsistent (i.e., the right-hand side of at least one equation is inconsistent with the others). In this case, there are no specific values of k that make the system inconsistent.
To determine the unique solution, infinitely many solutions, or no solution for the system, we analyze the determinant of the coefficient matrix. If the determinant is nonzero, there is a unique solution. If the determinant is zero and the system is consistent, there are infinitely many solutions. If the determinant is zero and the system is inconsistent, there is no solution.
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Jarrod helped his teachers pack textbooks into boxes for the summer. He packed 6 boxes of science textbooks with 14 books per box. He also packed math textbooks with 12 books per box. If x represents the total number of boxes he packed, which of the following equations can be used to find the total number of books that Jarrod packed into boxes?
The total number of books that Jarrod packed into boxes can be represented by 84 + x*12.
The total number of books that Jarrod packed into boxes can be found by multiplying the number of boxes he packed with the number of books per box.
In this case, Jarrod packed 6 boxes of science textbooks with 14 books per box, so the number of books in the science boxes is 614 = 84. He also packed math textbooks with 12 books per box, so the number of books in the math boxes can be represented by x*12, where x represents the total number of boxes he packed.
So the total number of books that Jarrod packed into boxes can be represented by 84 + x*12.
Therefore, the equation that can be used to find the total number of books that Jarrod packed into boxes is: 84 + x*12 = Total number of books.
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Algebra i need help asap
Answer:
Below
Step-by-step explanation:
Solutions are where the graphs intersect....they are equal at these points
(2,-3) and (4,-1)
There is likely one more point of intersection down and to the left which is not shown on the graph
Which method of solving for the variable could be used instead of cross multiplication?.
Answer: multiplying by denominators
Step-by-step explanation: not sure if this is the right answer in this context but i hope it helps
technically cross multiplying basically is multiplying by denominators, but depending on the math level they are sometimes considered different methods
again, this may not be correct so if you want a more clear answer could you provide a bit of context? sorry, hope it helps anyway though.
i need help, i feel like i know how to do it but when i try it makes no sense. can someone explain this to me with steps?
Completing a square is a method for transforming a quadratic equation’s form so that its LHS becomes a perfect square, commonly known as vertex form. This is accomplished by rearranging the phrase.
What exactly is an X2?X2 is x multiplied by itself, which is symbolized by and may be expressed as xx or x(x) as an algebraic term. 2 is an exponent in. It denotes that x has been multiplied by itself twice. x 2 = x .
heres the answer is, X squared is a helpful notation that we use in math. It can be used in different kinds of algebraic expressions. It simplifies our calculations and allows us to develop formulas to solve problems quicker!
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the length of a rectangle is 6 feet longer than the width. the perimeter is 28 feet. find the dimensions of the regtangle
The length of rectangle :10 feet
The width of rectangle : 4 feet
Given
perimeter of the rectangle :28 feet
let width of the rectangle be :x
and length of the rectangle be : 6+x (as per given condition)
therefore
perimeter = 2(length +width)
28 = 2(6+x+x)
28 = 2(6+2x)
28 = 12+4x
28-12 = 4x
16 = 4x
x = 4
as width of rectangle is : 4
similarly length : 6+x
6+4
:10
The dimensions of the rectangle are : length =10
width = 4
verification :
perimeter =2(l+w)
28 = 2(10+4)
28 = 28
a function pairs each teacher with 30 students
On solving the provided question we can say that when each teacher is matched with 30 students, t is not a function.
what is function?The subject of mathematics includes quantities and their variations, equations and related structures, shapes and their locations, and places where they can be found. The term "function" refers to the relationship between a set of inputs, each of which has an associated output. A connection between inputs and outputs in which each input leads to a single, distinct result is known as a function. Each function is given a domain and a codomain, or scope. Usually, f is used to denote functions (x). input is an x. There are four main types of functions accessible. based on the following factors: on functions, one-to-one functions, many-to-one functions, inside functions, and on functions.
When each teacher is matched with 30 students, t is not a function.
This is true since it demonstrates a one-to-many link.
Therefore, this is the case because it shows a many-to-one relationship and multiple weights can have the same cost.
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Can someone help me with this please?
A: Classify the triangle above by its sides
B: Classify the triangle above by its angles
C: Explain your answer to part a and b
Mrs. Peterson had lunch at Lunchee's. She paid $12.99 for her meal. She left a tip of 2.60. What percent did she leave?
answer = _____%
The percentage of the tip that she leaves will be 20.02%.
What is the percentage?The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates for one hundred percent. It is symbolized by the character '%'.
The percentage is given as,
Percentage (P) = [Final value - Initial value] / Initial value x 100
Mrs. Peterson had lunch at Lunchee's. She paid $12.99 for her meal. She left a tip of 2.60.
Then the percentage is given as,
P = [2.6 / 12.99] x 100
P = 0.2002 x 100
P = 20.02%
The percentage of the tip that she leaves will be 20.02%.
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Find the sum of the measures of the angles in quadrilateral ABCD. (Hint: Draw diagonal AC. How can you use the figures you have formed to find the sum?)
The measure of four angles of quadrilateral is 360°.
What is quadrilateral?
A closed quadrilateral has four sides, four vertices, and four angles.
It is a type of polygon.
In order to create it, four non-collinear points are joined.
Quadrilaterals always have a total interior angle of 360 degrees.
A polygon with four sides, four angles, and four vertices is called a quadrilateral.
The Latin words quadri, which means four, and latus, which means side, were combined to create the English word quadrilateral.
A quadrilateral is shown in the above image as an example.
A quadrilateral is a polygon with four vertices and four sides that each contain four angles, the sum of which is 360 degrees.
Two triangles are created when we draw a diagonal to a quadrilateral. These two triangles have a 180° angle sum.
As a result, the quadrilateral's total angle sum is 360°.
Hence, the measure of four angles of quadrilateral is 360°.
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Pls, Help!! My assignment is due tomorrow!
Mina will receive $13.33 after converting her 18 Pounds GBP to Dollars USD
What is conversion?A conversion factor is a number used to change one set of units to another, by multiplying or dividing. When a conversion is necessary, the appropriate conversion factor to an equal value must be used
To change 18GBP to USD at the current exchange rate of 1.35GBP to 1 USD
Divide the Amount being converted by the exchange rate
that is, 18GBP/1.35
=13.33USD
Hence, 13.33USD is the amount that Mina will receive after converting her 18 GBP
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In ΔMNO, o = 8 inches, m m∠O=48° and m m∠M=28°.
The length of side m is, 5.08 inches.
What is mean by Triangle?A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.
Given that;
In ΔMNO,
⇒ o = 8 inches, m ∠M = 28° and m ∠O = 48°.
Now, Let the length of side m = x
So, By sine rule, we get;
⇒ sin M / m = Sin O / o
Substitute all the values, we get;
⇒ sin28°/ x = sin 48° / 8
⇒ 0.47 / x = 0.74 / 8
⇒ x = 0.47 × 8 / 0.74
⇒ x = 5.08 inches
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If f(x)=6x-4 and f(x)=14, what is the value of x?
Answer:
Since, 6x -4 and 14 both are values of f(X)then,
6x-4=14
or,6x=14+4
or,X=18/6
:.X=3
*mark me brainliest
Answer: X=3 :)
Step-by-step explanation:
f(x)= 6x-4 and f(x)=14
6x-4=14
6x=4+14
6x=18
Divide both by 6
x=3
What is the formula for finding the area of a sector of a triangle?
The formula for a sector's area is: 1. A = (sector angle / 360 ) * (pi *r2)
2. A = (sector angle / (2*pi)) * (pi * r2) 3. A = (fraction of the circle) * (pi * r2)
The ratio used to find the area of the sector is that of how to know what fraction of the circle is the sector.
Ratio = (θ/360°)
θ is the angle subtended at the center of the circle by the arc, measured in degrees.
For this question,
θ = 60°
Ratio = (θ/360°) = (60°/360°) = (1/6)
Area of a circle is given as
Area of a circle = πr²
where
π = pi
r = radius of the circle = 8
Area of the circle = πr²
Area of the circle = π (8²) = 64π square units
Area of the sector is given as
Area of the sector = (Ratio) × (Area of the circle)
Ratio = (1/6)
Area of the circle = 64π square units
Area of the sector = (Ratio) × (Area of the circle)
Area of the sector = (1/6) × (64π) = (64π/6) = (32π/3) square units
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What is the answer of X² 6x 16?
The solution of (x^2 + 6x - 16) is ( x+8 )( x−2 ).
Lets solve the equation (x^2 + 6x - 16) in order to find its solution:
The equation is ( x^2 + 6x − 16 )
Now factorizing it
= ( x^2 + 8x − 2x − 16)
Taking x common from the first two terms and -2 common from the last two terms:
= [x (x+8) −2 (x+8) ]
Now,
= [ (x+8) (x−2) ]
Therefore, it is shown that the solution of the equation "(x^2 + 6x - 16)" is (x+8)(x−2) which is obtained through factorization.
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Write and solve an equation to find the value of x below.
Answer:
Equation:
[tex]7x + 10 = 150[/tex]
Value of x:
[tex]x = 20[/tex]
Step-by-step explanation:
In the figure, we are given 2 angles, 150° and (7x+10)°.
These 2 angles are vertical angles because they are directly opposite of each other and formed by intersecting lines.
Vertical angles are congruent, which means the same.
Thus, to make an equation to solve for x, we need to set the angles equal to each other.
Equation: [tex]7x + 10 = 150[/tex]
Solving for x:
[tex]7x + 10 = 150[/tex]
⇒ [tex]7x = 140[/tex] . . . . . . . . . subtract 10 from both sides of the equation
⇒ ∴ [tex]x = 20[/tex] . . . . . . . . . divide both sides by 7 to isolate x
if this response helped you, please mark it the "brainliest"!
Answer:
Step-by-step explanation:
Using VAT:
Angles opposite each other are consider equal.
Therefore : 7x+10 = 150
Solve for x: 7x= 150-10
x= 140/7
x= 20
To check : 7x+10 = 150
7(20)+10=150
140+10=150
150=150 (correct)
The nearest whole number.
12) Huong enlarged the size of a painting to a
width of 12 cm. What is the new height if
it was originally 4 cm wide and 3 cm tall?
The nearest whole number to which the new height of the image is enlarged is 9cm.
What is scaling?By scaling, we can create a drawing of an object that corresponds to the object's true size. In geometry, scaling refers to either growing or reducing figures in order to preserve their fundamental shape. Similar figures are what we call scaled figures.
Given that the original size was in the ratio 4: 3.
When the image is enlarged the enlarged image will be in the similar ratio to the original size.
Hence, the ratio in which the enlarged image will be increased is 4x : 3x.
The new width is given as 12,
The value of 4x = 12
x = 3
Then the new height of the image is:
3 (3) = 9cm.
Hence, the new height of the image is 9cm.
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Is dilation horizontal or vertical?
Dilation is neither horizontal nor vertical. Dilation is a transformation that changes the size of a shape but does not change its orientation.
It can be represented by a formula in coordinate geometry: (x, y) → (kx, ky), where k is a scale factor (greater than 0). The scale factor determines the amount of scaling that is applied to the shape. For example, if k = 2, then the coordinates of the original point (x, y) would be multiplied by 2, resulting in a new point (2x, 2y). This would result in the shape being twice as big as the original. Similarly, if k = 0.5, then the coordinates of the original point (x, y) would be divided by 2, resulting in a new point (0.5x, 0.5y). This would result in the shape being half as big as the original. Therefore, dilation can be used to scale shapes both up and down in size, but it does not alter their orientation.
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y= 2/3x -3 and x=-3 solution
PLEASE HELP!!! I NEED HELP WITH MY HOMEWORK!!!!
(GEOMETRY)
WILL MARK BRAINLIEST!!!!!
Answer:U: 1.98 V: 1.4 And also the other side, square root symbol: 1.4
Step-by-step explanation:
I hope this helps man, i havnt done this stuff in a while...
How long do eyes stay dilated?
Depending on the patient and the strength of the eye drop, eye dilation caused by eye drops used for eye examinations typically lasts between 4 and 24 hours.
What is eyes dilation?
There are causes of dilated pupils besides underlying disease. Examples include a pitch-black room, sleep, fear, or adverse drug reactions.
When you get your eyes dilated, your doctor uses special drops to open your pupils wide.
This let them get a look at your whole eye, including all the nerves and blood vessels at the back, to check for signs of diseases
When dilated, the pupil is unable to get smaller, and thus block out light in bright areas. This causes glare in your vision so you can't see things clearly.
The drops also make it difficult to focus on things up close, but usually don't affect distance vision.
So, Your eyesight will be impaired, so some experts say you should play it safe and not get behind the wheel will leads to affecting your driving
Hence, depending on the patient and the strength of the eye drop, eye dilation caused by eye drops used for eye examinations typically lasts between 4 and 24 hours.
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What can you conclude about the following figure?
What we can conclude by the given triangle about similarity is;
ΔPQR ∼ ΔPST
How to Identify Triangle similarity theorem?Similar triangles are defined as triangles that have the same shape, but their sizes may vary. Thus, if two triangles are similar, then it means that their corresponding angles are congruent and corresponding sides are in equal proportion.
Now, there are three types of similarity theorems for triangles namely;
Angle Angle(AA) Similarity
Side Angle Side(SAS) Similarity
Side Side Side(SSS) Similarity
We see that both triangles PQR and PST have equal angle R and angle T which are both 36 degrees. Thus, they are both similar by the AA Similarity theorem
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What is 14/10 reduced??
The reduced form of 14/10 is 7/5. The answer is obtained using the method of simplifying fractions.
What is meant by simplifying fractions?
Making a fraction as simple as possible is referred to as simplifying (or reducing) a fraction.
To do this, we divide the denominator and numerator by the biggest number that can be divided exactly into both numbers. In other words, we choose the biggest common number between the top and bottom and divide it by that.
In a fraction, the numerator and denominator are referred to as the "terms." Thus, when we simplify a fraction, we cast it in its most basic mathematical form.
Now, to reduce 14/10, we will find the highest common factor of 14 and 10.
The highest common factor is 2 i.e. both numerator and denominator are divisible by 2.
So, on dividing both the terms by 2, we get
(14÷2)/(10÷2)= 7/5
Hence, the reduced form of 14/10 is 7/5.
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You can work a total of no more than 40 hours each week at your two jobs. Housecleaning pays $4 hour and your sales job pays $8 per hour. You need to earn at least $120 each week to pay your bills. Write and graph a system of inequalities that shows the various numbers of hours you can work at each job.
To make a specific hair dye a hairstylist uses a ratio of one and 1/8 oz of red tone 3/4 oz of Gray tone and 5/8 oz of Brown tone if the stylist needs to make 20 oz of dye how much of each dye color is needed?
Answer: To make 20 oz of hair dye, the hairstylist needs to use 1 and 1/8 oz of red tone, 3/4 oz of gray tone and 5/8 oz of brown tone.
We can use proportionality to figure out how much of each dye color is needed.
Let's call the amount of red dye needed x, the amount of gray dye needed y and the amount of brown dye needed z.
According to the proportionality, we can write the following equation:
(1 + 1/8)x + (3/4)y + (5/8)z = 20
To find the amount of each dye color needed, we can solve this equation for x, y and z.
First, we can clear the fractions by multiplying all terms by 8:
(8 + 1)x + (6)y + (5)z = 160
Now we can solve for x:
9x + 6y + 5z = 160
x = (160 - 6y - 5z)/9
Similarly, we can solve for y and z:
y = (160 - 9x - 5z)/6
z = (160 - 9x - 6y)/5
Now we can substitute the desired amount of 20 oz into the equation and solve for x, y and z
Step-by-step explanation:
Aira got 76, 85, 81 and 77 in her four quizzes. What average score must she get in her next two quizzes to obtain an average of 83?.
Aira should must score a total of 179 combined in her next two quizzes in order to make the average of all the marks of the quizzes as 83.
Aira has got 76, 85, 81 and 77 marks in her 4 quizzes.
I have to give two more quizzes and the marks obtained in those two quizzes should be such that the average of all the quizzes should be 83.
We know that we can find the average of the data by taking the sum of all the given values and dividing that particular value by the number of data given to us.
Here in this case let assume that she obtained X marks in her next to test combined. Now this way total number of observation are 6.
Now, we can simply apply the formula to find the average of the data.
Average = (76 + 85 + 81 + 77 + X)/6
83 = (76 + 85 + 81 + 77 + X)/6
X = 179
So, Aira should obtain 179 marks in her next to quizzes in order to make the average of all the score equal to 83.
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What do you call the total distance around a polygon, or 2-D shape?
A .area
B .perimeter
C .volume
D .width
Answer:
It is called the perimeter
Step-by-step explanation:
What are the 7 steps to solve a math problem?
Answer:
sometimes there are more, sometimes there are less
Step-by-step explanation:
for example 1+1 only requires 1 step
hope this helps
Answer:
STEP 1: The Right Problem to Solve
STEP 2: Analyse the Problem
STEP 3: Define the Problem
STEP 4: Develop Opportunities (Possible Solutions)
STEP 5: Select the Best Solution
STEP 6: Implement the Solution
STEP 7: Evaluate and Learn
Step-by-step explanation:
Hope It Helps!
Do all exponential functions pass through 0 1?
All exponential functions pass through (0, 1)
We know that the exponential function f with base b is defined as:
f(x) = b^x
where b > 0 and b is not equal to 1
and x is any real number.
For x = 0,
f(0) = b^0
f(0) = 1
This means, the graph of function f(x) passes through (0, 1)
For example consider an exponential function f(x) = 2^x
For x = 0,
f(0) = 2^0
f(0) = 1
the graph of f(x) = 2^x passes through (0, 1)
so, the characteristics of the graphs of all exponential functions:
The graphs contain the point (0, 1).
The x-axis is an asymptote.
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Which of the following represents the diameter of the circle below?
A. XY
B. O
C. X
D. YO
XY is the diameter because the diameter is the entire side to side while radius is from the middle to the side.
How to find the diameter of a circle?The diameter is the entire side to side while radius is from the middle to the side.
The diameter of a circle is represented by the letter d.
here O is the center of the circle
radius of the circle is OX and YO
diameter of the circle is 2×radius
then here diameter of the circle is XY
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What is the formula for square units?
The formula for the area of a square is A = s², where A is the area of the square and s is the length of one of its sides.
To calculate the area of a square, multiply the length of one side by itself. For example, if the side of a square is 5 meters, the area of the square would be 25 square meters (5² = 25). If the side of a square is 8 feet, the area of the square would be 64 square feet (8² = 64). The area of a square is always equal to the length of one side multiplied by itself.
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