Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', 'a = ', or 'a ≠', then specify a value or comma-separated list of values. 5x1+10X2 = 0
-x1+3x2+10x3 = 0
x1+x2+ax3=0
Answers
- For "a ≠ -2", the system has a unique solution.
- For "a = -2", the system has infinitely many solutions.
To determine the values of "a" for which the given system of linear equations has no solutions, a unique solution, or infinitely many solutions, we can use the concept of matrix row operations and the determinant of the coefficient matrix.
The given system of equations can be represented in matrix form as:
\(\left[\begin{array}{ccc}5&10&0\\-1&3&10\\1&1&a\end{array}\right]\)\(\left[\begin{array}{ccc}x_1\\x_2\\x_3\end{array}\right]\) =\(\left[\begin{array}{ccc}0\\0\\0\end{array}\right]\)
Let's analyze the determinant of the coefficient matrix to determine the solutions:
1. If the determinant is non-zero (det ≠ 0), the system has a unique solution.
2. If the determinant is zero (det = 0) and one of the rows has all zeros in the augmented matrix, the system has no solutions.
3. If the determinant is zero (det = 0) and none of the rows have all zeros in the augmented matrix, the system has infinitely many solutions.
The coefficient matrix is:
\(\left[\begin{array}{ccc}5&10&0\\-1&3&10\\1&1&a\end{array}\right]\)
Calculating the determinant of this matrix:
det = 5(3a - 10) - 10(-a - 10) + 0(-1 - 3)
Simplifying:
det = 15a - 50 + 10a + 100 + 0
det = 25a + 50
Now let's analyze the cases:
1. If det ≠ 0, the system has a unique solution. Therefore, we can say "a ≠ -2".
2. If det = 0 and one of the rows has all zeros in the augmented matrix, the system has no solutions. Since the third row has all zeros in the augmented matrix for any value of "a", we can say "always".
3. If det = 0 and none of the rows have all zeros in the augmented matrix, the system has infinitely many solutions. This occurs when 25a + 50 = 0. Solving for "a":
25a = -50
a = -2
Therefore, for "a = -2", the system has infinitely many solutions.
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Related Questions
Which instruments belong to the brass family?
Answers
Answer:
Baritone/Euphonium
Trumpet
Tuba
Trombone
Good luck!!
Sam has a bag of 11 tiles representing each letter in the word MATHEMATICS and another bag containing 15 tiles numbered 1 through 15. If Sam draws 1 tile from each bag at random, find P(letter M or T, number less than 10)? Give answers as a reduced fraction OR a decimal rounded to thousandths
place.
Answers
the probability of getting a letter M or T and a number smaller than 10 is P = 0.2182
How to find the probability?Here we want to get the probability of getting a letter M or T and a number smaller than 10.
First, let's find the probability for the letter.
On the bag, there are 11 tiles, and on the word mathematics, we have two letters T and two letters M (4 in total). So the probability of getting one of these is:
p = 4/11
For the other bag, there are 15 tiles, and 9 of these have numbers smaller than 10, so the probability of getting one of these is:
q = 9/15 = 3/5
The joint probability is then:
P = p*q = (4/11)*(3/5) = 0.2182
the probability of getting a letter M or T and a number smaller than 10 is P = 0.2182
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a mass weighing 16 pounds is attached to a spring whose spring constant is 9 lb/ft. what is the period of simple harmonic motion?
Answers
The period of simple harmonic motion for the given mass and spring constant is approximately 8.3836 seconds.
To find the period of simple harmonic motion, we can use the formula:
T = 2π√(m/k) where,
T is the period, m is the mass, and k is the spring constant.
Given,
m = 16 pounds
k = 9 lb/ft
To use the formula, we need to convert the mass to the corresponding units of the spring constant. Since the spring constant is in lb/ft, we need to convert the mass from pounds to slugs (1 slug = 32.2 pounds).
m = 16 pounds / 32.2 pounds/slug
≈ 0.4975 slugs
Now we can substitute the values into the formula:
T = 2π√(0.4975 slugs / 9 lb/ft)
To simplify the units, we convert the units of the spring constant to slugs/ft:
k = 9 lb/ft / 32.2 pounds/slug
≈ 0.2795 slugs/ft
Substituting the values into the formula:
T = 2π√(0.4975 slugs / 0.2795 slugs/ft)
T = 2π√(1.7806 ft)
T ≈ 2π(1.3357 ft)
T ≈ 8.3836 ft
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Two numbers have a sum of 124
Write in algebraic expression
Answers
Answer:
Step-by-step explanation:
Let one number = x
Let the other number = y
x + y = 124
You can't go any further than that without more information.
The good news is that no number is excluded.
x = 248
y = -124 for example
Everyone using the PACED decision-making model will end up making the same decision
True
False
Answers
The statement "Everyone using the PACED decision-making model will end up making the same decision" is false because it does not guarantee that everyone who uses it will end up making the same decision
In the first step, the problem is defined and the decision-making team identifies the goals they wish to achieve. Next, a range of alternatives are generated and evaluated based on a set of criteria that have been agreed upon.
Now, the question is whether everyone who uses the PACED decision-making model will end up making the same decision or not. The answer to this question is False.
To understand this better, let us consider a simple example. Suppose a group of students needs to decide on the best way to raise money for a charity event. They use the PACED model to evaluate three alternatives: a bake sale, a car wash, and a sponsored walk.
They agree on a set of criteria such as cost, time, and potential revenue, and evaluate each alternative based on these criteria. After careful consideration, the group decides that the bake sale is the best option.
In mathematical terms, we can say that the PACED model provides a systematic approach to decision making, but the final decision depends on the subjective preferences of the decision makers. Different decision makers may assign different weights to the criteria used to evaluate alternatives, leading to different decisions.
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Find the measure of the arc.
?
55°
46°
Answers
Answer:
79
Step-by-step explanation:
What are two ways to show the mixed number: 4 and two-tenths?
Answers
the two different forms of mixed numbers can be represented are \(4 \frac{2}{10}\) and 42/10.
What is mixed fraction?
A mixed fraction is a fraction that is created by fusing a whole number with a fraction. A mixed fraction is a fraction that is created by fusing a whole number with a fraction. A mixed fraction is a form of fraction that consists of both a whole number part and a fractional part. A mixed fraction would be, for instance, 3 1/10. It is also known as a mixed number.
the mixed number: 4 and two-tenths
= \(4 \frac{2}{10}\)
this can also be written as 4*10 +2
= 42/10
Hence the two different forms of mixed numbers can be represented are \(4 \frac{2}{10}\) and 42/10.
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Could someone help me Plz and Ty <3
Answers
\(\bold{Heya~!}\)
Your answer will be is :
\(\bold{\underline{A.}}\) \(\mathfrak{3\sqrt[3]{3}}\)
EXPLANATION :Because its the square root of \(\mathfrak{3\sqrt[3]{3.}}\)
Hope It Helps ! ~
#LearnWithBrainly
\(\underline{Answer~:}\)
\(\mathfrak{\underline{Dollixna}}\)
How much sales tax will be charged at Store B?
Answers
Answer: $94.50 is how much sales tax will be charged at store B asumming you are talking about the image attached to this answer, if not could you attach an image?
Solve the inequality 4x -5 greater than or equal to 7
Answers
to 3.
Explanation: Add 5 on both sides and then after divide 4 on both sides. The sign stays the same.
The table below shows a proportional relationship. True or False?
Answers
According to the image, the table shows a relationship that is not proportional. First of all, you cannot multiply a number by 0 and get something other than 0. Anything multiplied by 0 is always 0.
Second, the rule is different for each row.
2x5=10
4x4=16
6x3.6=22
A proportional relationship must increase by the same amount, and this does not, which means it is not proportional.
Find common ratio
0+4(1)=42+4(2)=104+4(3)=166+4(4)=22Yes they are proportional
factor each completely
n^2 - 11n + 10
Answers
\( {n}^{2} - 11n + 10 = \)
\((n - 1)(n - 10)\)
.................................................
You survey your class and 4 like vanilla ice cream
and 14 like chocolate ice cream. Write the ratio of
people who like vanilla to people who like chocolate ice
cream, then write the ratio in simplest form.
Answers
Answer: 2:7
Step-by-step explanation:
Vanilla icecream = 4
Chocolate icecream = 14
4:14=2:7
2X2=4
7X2=14
Step-by-step explanation:
a ratio is simply relating 2 (or more) groups and their sizes.
in our case here the ratio is
4/14 = 2/7
this is often written as
4:14 = 2:7
it means that for every 2 people liking (I assume "preferring") vanilla ice cream there are 7 people liking (preferring) chocolate ice cream.
FYI
in total, a ratio also tells us about the size of the sample for the observation or analysis. 4/14 tells us there are 18 (4+14) people in the survey. and the simplified form 2/7 tells us that we can split the whole survey group into units of 9 (2+7) people with the same behavior as the total group.
a ratio is a fraction (incl. all the calculation and comparison rules) with a special meaning.
so, don't mix the ratio up with e.g. the fractions of the different behaviors in the group :
4/18 = 2/9 of the whole group prefer vanilla.
14/18 = 7/9 of the whole group prefer chocolate.
the ratio is now the relative size relationship between both : 2/9 / 7/9 = 2/7
What’s the answer to this question?
x² =
12
16
48
Answers
"Suppose you want to design a model that links child obesity and diabetes and your model predicts that overweight children have an 80% higher risk of suffering from diabetes in their adult life. If data shows that overweight children do not suffer from diabetes as predicted in your model (i.e., data shows a lower than 80% probability), which would your next step be?"
Answers
If data shows that overweight children do not suffer from diabetes as predicted in the model that links child obesity and diabetes (i.e., data shows a lower than 80% probability), then the next step would be to review the model's assumptions and revise the model as necessary.
This may involve analyzing additional data or changing the model's structure or parameters, to better reflect the observed relationships between child obesity and diabetes. It is important to ensure that the model is based on accurate and reliable data, as this will help to ensure that the model's predictions are useful and informative.
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What is the value of 0.52 +
3
5
Answers
Answer: 35.52
Step-by-step explanation:
Answer:
35.52
Step-by-step explanation:
ILL BRAINLIEST YOU IF YOU GET IT RIGHT
WHICH DIAGRAM SHOWS A PAIR OF ANGLE MEASURE THAT PROVIDES LINES c and d are parallel?
Answers
Answer:
It's C.
Explanation:
I took a test on this
I have the answers I just need someone to work it out I’ll mark as brainlesttt
Answers
Answer:
Look Below
Step-by-step explanation:
The formula for a circle is πr².
Therefore, we can divide the area of the circle by 3.14, and then square root it. 490.63/3.14=156.25... If you square root that, you get 12.5. Same thing with the other circle. 219.45/3.14=69.888... If you square root that, you get 8.36. Now that we have the radius, we can multiply that by two to find the diameter. And we're done.
Feel free to tell me if I did anything wrong! :)
Which statements prove that a quadrilateral is a parallelogram?
A - Quadrilateral DEFG has two sets of consecutive angles that are complementary.
B - Quadrilateral DEFG has one set of opposite sides that are both congruent and parallel.
C - Quadrilateral DEFG has diagonals that are congruent.
D - Quadrilateral DEFG has opposite angles that are congruent.
Answers
Answer:
B - I HOPE THIS IS RIGHT. ...
Simplify the following:
a) 3^2x3^2
b) 7x7^4
c) 10^9/10^7
d) 5^8/5
Answers
Answer:
see attached
Step-by-step explanation:
pleass see attached
translate phrases and sentences into expressions, equations and inequalities
Answers
translating phrases and sentences into expressions, equations, and inequalities is an important skill in mathematics. It involves identifying key words and phrases that indicate mathematical operations and relationships. By understanding these key words, we can represent real-world problems using mathematical language and symbols, making it easier to solve and analyze them.
translating phrases and sentences into expressions, equations, and inequalities
When solving mathematical problems, it is often necessary to translate phrases and sentences into mathematical expressions, equations, and inequalities. This process involves identifying key words and phrases that indicate mathematical operations and relationships.
For example, let's consider the following sentence:
'The sum of a number and 5 is equal to 12.'
In this sentence, the key words 'sum,' 'number,' 'equal to,' and '12' indicate that we need to perform addition and find the value of the unknown number. We can represent this sentence as the following equation:
x + 5 = 12
Here, 'x' represents the unknown number.
Similarly, let's consider the following sentence:
'Twice a number decreased by 7 is greater than 15.'
In this sentence, the key words 'twice,' 'decreased by,' 'greater than,' and '15' indicate that we need to perform multiplication, subtraction, and comparison. We can represent this sentence as the following inequality:
2x - 7 > 15
Here, 'x' represents the unknown number.
By understanding key words and phrases, we can effectively translate real-world problems into mathematical language and solve them using expressions, equations, and inequalities.
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1) An inequality to represent the given phrase is 5q>30.
2) An equation to represent the given phrase is 2b²=8.
1) The given phrase is "The product of 5 and q is greater than 30."
Here, the product of 5 and q = 5×q
= 5q
Now, the product is greater than 30.
Thus, 5q>30
q>30/5
q>6
2) The given phrase is "Twice the square of b is equal to 8."
Here, the square of b = b×b
= b²
Twice the square of b = 2b²
So, the product is equal to 8.
That is. 2b²=8
b²=4
b=±2
Therefore,
1) An inequality to represent the given phrase is 5q>30.
2) An equation to represent the given phrase is 2b²=8.
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"Your question is incomplete, probably the complete question/missing part is:"
Translate phrases and sentences into expressions, equations and inequalities.
1) The product of 5 and q is greater than 30.
2) Twice the square of b is equal to 8.
Please help with these 3! Don’t need the steps, just the answer
Answers
Bottom left is y=-x+2
The right one is y=2/3x-4
Drag each number to show if it is equal to 0.75, 7100, or neither. clear check 0.75 7100 other
Answers
In this question, number 0.75 is equal to 0.75, 7100 is neither equal to 0.75 nor to 7100.
0.75: This number is equal to 0.75, as it matches the value exactly.
7100: This number is neither equal to 0.75 nor to 7100. It is a different value altogether.
Other: This category includes any number that is not equal to 0.75 or 7100. It could be any other number, positive or negative, fractional or whole, but it is not specifically equal to 0.75 or 7100.
By categorizing the numbers into "Equal to 0.75," "Equal to 7100," and "Other," we can determine whether each number matches one of the given values or is something different.
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The area of a square is 49 square inches or less. Which set describes the domain for s, the
length of one side of the square?
Answers
Step-by-step explanation:
Any positive number can. be a side length. It cannot pass 7 because a square with 49 sqr lnches as a area has side lengths with the measures of 7.
\(0 \leqslant x \leqslant 7\)
Trinity Rubidoux
Angle Terminology with Equations
Sep 19, 3:36:40 PM
?
ZA and ZB are vertical angles. If mA = (5x +24)° and m/B= (6x - 5)°,
then find the measure of ZB.
Answers
The measure of ∠B = 189°
Here it is given that
∠A = (5x + 24)° and ∠B = (6x - 5)°
∠A and ∠B are vertical angles
Since vertical angles are always equal so
∠A = ∠B
5x + 24 = 6x - 5
add +5 on both sides, we get
5x + 24 + 5 = 6x - 5 + 5
5x + 29 = 6x
add -5x on both side, we get
5x - 5x + 29 = 6x - 5x
x = 29
Now put the value of x in 6x - 5
6 × 29 - 5 = 189
The measure of ∠B = 189°
What is vertical angle theorem?According to the vertical angle theorem, the vertical angles that are created when two lines intersect are congruent. There are equal amounts of vertical angles. For instance, if a, b, c, and d are the four angles created by two intersecting lines and a, b, and c are vertically opposed to d and c, respectively, then a, b, and c are congruent to d.
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What is the answer
To 7/9 divided by 1/8
Answers
To get to the answer you flip the second fraction 1/8 to 8/1. Then multiply 7/9 x 8/1
...............................
Answers
Answer:
\(2 \sqrt{10} \)Option C is the correct option.
Step-by-step explanation:
√ 8 • √ 5
Calculate the product
\( = \sqrt{40} \)
Simplify the radical expression
\( = \sqrt{2 \times 2 \times 10} \)
\( = 2 \sqrt{10} \)
Hope this helps...
Best regards!!
helppp its in the picture
Answers
Answer:
$220
Step-by-step explanation:
$5.50*8= $44
$44*5= $220
Hope it helps!
(d) Given that n(§) = 96, n(A) = 50 and n(B) = 60. Find the maximum and minimum values for n(An B).
how to get the answer? I want the solution
Answers
Step-by-step explanation:
the max. value is when the smaller set (A) is completely contained in the larger set (B).
then n(A n B) is n(A) = 50.
the set intersection between A and B cannot get bigger than that. or A gets bigger ...
after all, the intersection means it is a set of all elements that exist in BOTH sets.
but then there must be other elements besides A and B in the universal set too, because n(universal set) = 96, and n(A u B) would be only 60.
the min. value could be the empty set or 0. but because n(universal set) = 96, and n(A) + n(B) = 110 and larger than 96, it means that there have to be some shared elements. at least 110 - 96 = 14 elements.
in this case there cannot be other elements in the universal set than A and B. and n(universal set) = n(AuB) = 96.
Help me with that white square please
Answers
Answer:
3 or 6 but best one would be 6 though
Step-by-step explanation: