We have proved that 1*1! + 2*2! + ... + n*n! = (n + 1)! -1 is true for all positive integers n.
Given expression is: 1*1! + 2*2! + ... + n*n! = (n + 1)! -1
We need to prove this equation for n positive integers.
Let's prove this by induction, Base Case:
For n=1,
1*1! = (1+1)! - 1
=> 1 = 2 - 1
=> 1 = 1
The base case is true.
Inductive Step:
Let's assume that the given expression holds for some k, that is:
1*1! + 2*2! + ... + k*k! = (k + 1)! - 1
Now we will prove that the same equation holds for k+1, that is:
1*1! + 2*2! + ... + (k+1)*(k+1)! = (k + 2)! - 1
By using the inductive hypothesis, we can replace the sum up to k with (k+1)! - 1, that is:
(k+1)! - 1 + (k+1)*(k+1)! = (k+1)! - 1 + (k+1)!
We can take (k+1)! common from the above equation. That is:
(k+1)![(k+1) - 1 + 1] = (k+1)! * (k+2) => (k+2)! - (k+1)!
= (k+1)! * (k+2) - (k+1)!
= (k+1)!(k+2-1)
= (k+1)!*k+1
That is: 1*1! + 2*2! + ... + (k+1)*(k+1)! = (k + 2)! - 1
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Somebody can help me please I don’t understand:(
Answer:
1) x=67, y=83, z=80 2) x=5
Step-by-step explanation:
1) 28+85=113 a triangle is 180 degrees so 180-113=67 so x=67
x and y are supplementary meaning both sides equal 150 so 150-67=83
83+17=100 now going back how much did i say a triangle should equal? 180 so z is 80
2)a triangle = 180 since the triangle is a right angle it includes the angle 90 so make the equation to solve for x
180=90+8x+2+9x+3
take 90 to other side so you are subtracting 180-90
90=8x+2+9x+3
combine like terms
90=17x+5
subtrcat 5 from 90
85=17x
now divide 85 by 17 and you get x=5
Select all the true statements. *
1 point
A. The opposite of -5 is greater than the absolute value of -1.
B. The opposite of the opposite of -10 is equivalent to - |-10|.
C. -2.7 and -27 are both examples of integers.
D. 5.1 is an example of a rational number.
E. The absolute value of a number is its distance from zero on a number line.
Membership to a music club costs $165. Members
pay $25 per music lesson and nonmembers pay $40
per music lesson. How many music lessons would
have to be taken for the cost to be the same for
members and nonmembers?
plz my time is running out
Answer:
11 music lessons.
Step-by-step explanation:
We know that membership costs $165 and members pay $25 per music lesson.
So, we can write the following expression:
[tex]165+25m[/tex]
The 165 represents the one-time membership fee and the 25m represents the cost for m music lessons.
We know that non-members pay no membership fee but their cost per lesson is $40. So:
[tex]40m[/tex]
Represents the cost for non-members for m music lessons.
We want to find how many music lessons would have to be taken for the cost to be the same for both members and non-members. So, we can set the expressions equal to each other:
[tex]165+25m=40m[/tex]
And solve for m. Let's subtract 25m from both sides:
[tex]165=15m[/tex]
Now, divide both sides by 15:
[tex]m=11[/tex]
So, at the 11th music lesson, members and non-members will pay the same.
Further Notes:
This means that if a person would only like to take 10 or less lessons, the non-membership is best because there is no initial fee.
However, if a person would like to take 12 or more lessons, than the membership is best because the membership has a lower cost per lesson than the non-membership.
And we're done!
Jackson can swim 38 laps in 95 minutes. Write an equation relating the laps swam s to the number of minutes m. At this rate, how many laps can he swim in 115 minutes?
Answer:
46 laps
Step-by-step explanation:
Set up a ratio:
38/95 = s/115
cross multiply
4370 = 95s
divide both sides by 95
s= 46
During the current year, merchandise is sold for $114,000 cash and $548,100 on account. The cost of the goods sold is $423,700. What is the amount of the gross profit? $fill in the blank 1
During the current year, merchandise was sold for $662,100. Of this amount, $114,000 was cash sales and $548,100 was on account. The cost of goods sold was $423,700. Therefore, the gross profit was $238,400.
The gross profit is calculated by subtracting the cost of goods sold from the total sales. The cost of goods sold is the direct cost of producing the goods that were sold. It includes the cost of materials, labor, and overhead.
The gross profit is an important measure of a company's profitability. It shows how much money the company makes from its core business activities, before deducting other expenses.
In this case, the company's gross profit margin is 35.8%. This means that for every $100 of sales, the company makes $35.80 in gross profit. A high gross profit margin is generally considered to be a sign of a healthy business.
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find the points that are equidistant from both axes and the point (3 6)
==============================================
Explanation:
The term "equidistant" means "equally distant".
We want (x,y) to be the same distance to the x axis, the y axis, and the point (3,6).
Let n represent this unknown equal distance. Negative distance doesn't make sense, so we'll have n > 0.
Start at the origin and move n units to the right to arrive at (n,0). We are n units from the y axis.
Now move n units up to arrive at (n,n). This point is guaranteed to be n units away from each axis.
That is one potential point. Other potential points are:
(-n,n)(n,-n)(-n,-n)One point for each quadrant.
Let's assume that the answer is in the 1st quadrant since (3,6) is located here.
We need to find an expression that represents the distance from the point (n,n) to (3,6). We'll set that distance expression equal to n so we can solve.
Part 1 of those steps
[tex]d = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\\\\n = \sqrt{(n-3)^2+(n-6)^2}\\\\n^2 = (n-3)^2+(n-6)^2\\\\n^2 = (n^2-6n+9)+(n^2-12n+36)\\\\n^2 = 2n^2-18n+45\\\\[/tex]
Part 2 of those steps
[tex]0 = 2n^2-18n+45-n^2\\\\0 = n^2-18n+45\\\\n^2-18n+45 = 0\\\\(n-3)(n-15) = 0\\\\n-3 = 0\ \text{ or } \ n-15 = 0\\\\n = 3\ \text{ or } \ n = 15\\\\[/tex]
The quadratic formula could be used as an alternative pathway.
Therefore, the mystery point could be located at (3,3) or at (15,15)
Use the distance formula to verify each answer. I'll let you do this part.
---------
Extra info:
If you were to perform similar steps with the point (-n,n), then you'll find that no real numbered solutions exist. The two solutions would be complex numbers.
The cases (n,-n) and (-n,-n) have no solutions at all.
Therefore, we have confirmed only case (n,n) would work.
The sum of 3 consecutive even integers is 18
Answer:
4,6,8
Step-by-step explanation:
Let x = 1st even integer
x+2 =2nd even integer
x+4 =3nd even integer
x+ x+2 + x+4 = 18
Combine like terms
3x+6 = 18
Subtract 6 from each side
3x+6-6=18-6
3x= 12
Divide by 3
3x/3 =12/3
x = 4
x+2 = 6
x+4 = 8
4. Prove that to every A E L (R", R¹) corresponds a unique y € R such that Ax = x.y. Prove also that || A|| = |y|. Hint: Under certain conditions, equality holds in the Schwarz inequality.
For every matrix A in the space of linear transformations from [tex]R^n[/tex] to [tex]R^1[/tex], there exists a unique vector y such that Ax = x.y
The norm of A is equal to the absolute value of y. This is proven using the Schwarz inequality.
Let A be a matrix in the space of linear transformations from [tex]R^n[/tex] to [tex]R^1[/tex]. For any vector x in [tex]R^n[/tex], we can define Ax as a dot product between A and x. Let y be a vector in [tex]R^1[/tex] such that Ax = x.y for all x in [tex]R^n[/tex] . This means that the dot product of A and x yields a scalar multiple of x, where the scalar is y.
To prove uniqueness, assume there are two vectors [tex]y_{1}[/tex] and [tex]y_{1}[/tex] in [tex]R^1[/tex] such that Ax = x.[tex]y_{1}[/tex] and Ax = x.[tex]y_{2}[/tex] for all x in [tex]R^n[/tex]. Then, x.[tex]y_{1}[/tex] = x.[tex]y_{2}[/tex] for all x in [tex]R^n[/tex], which implies [tex]y_{1}= y_{2}[/tex]. Therefore, there exists a unique vector y such that Ax = x.y.
To prove that ||A|| = |y|, we can use the Schwarz inequality. The Schwarz inequality states that for any vectors u and v in [tex]R^n[/tex], |u.v| ≤ ||u|| * ||v||, with equality holding if and only if u and v are linearly dependent. Applying this inequality to the vectors A and x, we have |Ax.x| ≤ ||Ax|| * ||x||. Since Ax = x.y and ||Ax|| = ||x.y|| = |y| * ||x||, the inequality becomes |x.y.x| ≤ |y| * [tex]||x||^2[/tex]. Since the dot product x.y.x is equal to y * (x.x) = y * [tex]||x||^2[/tex], we have |y| *[tex]||x||^2[/tex] ≤ |y| * [tex]||x||^2[/tex], which holds with equality. Hence, ||A|| = |y|.
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can you explain what proportional means in this instance and how to determine whether or not this is proportional
Answer:
It can divide equally
Step-by-step explanation:
Find the Distance between (-1,4) and (3,1)
Answer:
5. This question is hard to explain, sorry.
Two companies offer jane sales position. Company a pays $3,000 per month plus $250 per car sold.. Company B pays 2,500 per month plus $350 per car sold. Which equation could be used to determine the number of cars sold in a month, x, that would result in the same total income from each company?
Given :
Two companies offer jane sales position. Company a pays $3,000 per month plus $250 per car sold.. Company B pays 2,500 per month plus $350 per car sold.
To Find :
Which equation could be used to determine the number of cars sold in a month, x, that would result in the same total income from each company.
Solution :
Let, x car are sold for same income.
Mathematical equation of company one :
[tex]T_1 =3000+250x[/tex]
Mathematical equation of company two :
[tex]T_2 =2500+350x[/tex]
For same income :
[tex]T_1=T_2\\\\3000 + 250x = 2500 + 350x\\\\100x = 500\\\\x=5[/tex]
So, for same total income from each company the number of cars sold in a month should be 5.
Hence, this is the required solution.
what is the answer?
-14 - 8x = -2 (-3x + 7)
Answer:
x=0
Step-by-step explanation:
-14-8x=-2(-3x+7)
-14-8x=6x-14
-8x=6x+0
-2x=0
x=0
Anyone know the answer ???
Answer:
The answer to what?
Step-by-step explanation:
Verify if f(x) = 1/2x - 2 and g(x) = 2x + 4 are inverses.
Answer:
Step-by-step explanation:
Try using g(x)
Let g(x) = y
y = 2x + 4
Interchange x and y
x = 2y + 4
Subtract 4 from both sides
x - 4 = 2y
Divide by 2
x/2 - 4/2 = 2y/2
y = x/2 - 2
This is exactly what f(x) is
So g(x) and f(x) are inverses.
for the function f(x)= x2-4 evaluate f(x+h)
Answer:
x² + 2hx + h² - 4
Step-by-step explanation:
To evaluate f(x + h), substitute x = x + h into f(x), that is
fx + h)
= (x + h)² - 4 ← expand (x + h)² using FOIL
= x² + 2hx + h² - 4
What is this number in standard notation? 9.41 x 106
Answer:
9997.46
Step by step Explanation
Quiz 3
The following data points represent the number of remote controllers each student in Tria's video game
club owns.
Sort the data from least to greatest.
0
0
0
1
2
2
4
5
7
8
10
Find the interquartile range (IQR) of the data set.
remote controllers
A
Report a problem
The interquartile range (IQR) of the data set is 7.To find the interquartile range (IQR) of a data set, we need to first determine the values of the first and third quartiles.
The first quartile (Q1) represents the 25th percentile of the data, while the third quartile (Q3) represents the 75th percentile of the data. We can find these values by first sorting the data set in ascending order:
0, 0, 0, 1, 2, 2, 4, 5, 7, 8, 10
Next, we can find the median (Q2) of the entire data set, which is the value that separates the lower 50% of the data from the upper 50%:
0, 0, 0, 1, 2, | 2, 4, 5, 7, 8, 10
Q2 = 2
Now we can find Q1 and Q3. To find Q1, we take the median of the lower half of the data set:
0, 0, 0, | 1, 2,
Q1 = 0
To find Q3, we take the median of the upper half of the data set:
2, 4, 5, 7, 8, 10
Q3 = 7
Finally, we can calculate the interquartile range (IQR) as the difference between Q3 and Q1:
IQR = Q3 - Q1
= 7 - 0
= 7
Therefore, the interquartile range (IQR) of the data set is 7.
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Answer: 7
Step-by-step explanation:
What is a coefficient in this equaiton?
3x + 2 = -10
Answer:
3 is the coefficient
Step-by-step explanation:
3x + 2 = -10
The coefficient is the number that is multiplied by the variable
3 is the coefficient
Answer:
3
Step-by-step explanation:
the 'a' coefficient is 3
3 is the number infront of x (the first one)
a coefficient is the number infront of the unknown value
10x+4x-1=0
10= coefficient a
4= coefficient b
-1= coefficient c
Suppose a baseball is thrown at 89 miles per hour. The ball will travel 324 feet when hit by a bat swung at 51 miles per hour and will travel 444 feet when hit by a boat swung at 81 miles per hour. Let y be the number of feet traveled by the ball when hit by a bat swung at x miles per hour. How much farther will a ball travel for each mile-per-hour increase in the speed of the bat?
Given:
Baseball is thrown at 89 miles per hour.
The ball will travel 324 feet when hit by a bat swung at 51 miles per hour.
It will travel 444 feet when hit by a boat swung at 81 miles per hour.
To find:
The addition distance travelled by the ball if the speed of bat increased by 1 unit.
Step-by-step explanation:
Let y be the number of feet traveled by the ball when hit by a bat swung at x miles per hour.
The ball will travel 324 feet when hit by a bat swung at 51 miles per hour. So, the point is (51,324).
It will travel 444 feet when hit by a boat swung at 81 miles per hour. So, the point is (81, 444).
Equation of line passes through two points is
[tex]y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
[tex]y-324=\dfrac{444-324}{81-51}(x-51)[/tex]
[tex]y-324=\dfrac{120}{30}(x-51)[/tex]
[tex]y-324=4(x-51)[/tex]
[tex]y-324=4x-204[/tex]
Add 324 on both sides.
[tex]y=4x-204+324[/tex]
[tex]y=4x+120[/tex]
Using the slope intercept form,
[tex]y=mx+b[/tex]
where, m is slope.
The slope of equation [tex]y=4x+120[/tex] is 4. It means, the ball will travel 4 feet farther for each mile-per-hour increase in the speed of the bat.
The following data set represents the average day temperature in Orange County, CA for December. Find the
probability that a randomly selected day will have an average temperature of at least 60 0F.In the 20 days data, only 6 days avarege of 60*F. How do I find the probability? is it based on the 20 days or the 31 days of the month?
The probability that a randomly selected day in December will have an average temperature of at least 60°F is approximately 0.1935 (or 19.35%).
To calculate the probability of randomly selecting a day with an average temperature of at least 60°F, you need additional information. Specifically, you need to know the total number of days in December in Orange County, CA.
Since you mentioned that there are 31 days in the month of December, we'll assume that this is accurate. In that case, the probability can be calculated based on the 31-day period.
Given that the dataset consists of 20 days and only 6 of them have an average temperature of at least 60°F, you can calculate the probability as follows:
Probability = (Number of days with average temperature ≥ 60°F) / (Total number of days in December)
Probability = 6 / 31
So, the probability that a randomly selected day in December will have an average temperature of at least 60°F is approximately 0.1935 (or 19.35%).
Please note that if the actual number of days in December differs from 31 or if you have access to a larger dataset, the probability calculation would change accordingly.
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What is the product of the complex numbers 3 4i and 6 + i, where V-1
The product of the complex numbers (3 + 4i) and (6 + i) is 14 + 27i, where i² = -1.
To find the product of the given complex numbers: (3 + 4i)(6 + i)There are different methods to solve this multiplication problem, but one common way is to use the FOIL method:
First, Outer, Inner, Last. This method involves multiplying each term in the first complex number by each term in the second complex number.
FOIL method: First: (3)(6) = 18Outer: (3)(i) = 3iInner: (4i)(6) = 24iLast: (4i)(i) = -4
Using this method, we can write the product as: (3 + 4i)(6 + i) = 18 + 3i + 24i - 4i²
The term i² equals -1, so we can substitute -1 for i² and simplify: (3 + 4i)(6 + i) = 18 + 27i - 4= 14 + 27i
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On simplifying (-3y2 − 8) − (-5y2 + 1), we get
Answer:
-3y2-8+5y2-1=
2y2-9
Step-by-step explanation:
Answer:
[tex]\huge{\boxed{\sf{ 2y^{2} - 9 }}}[/tex]
Step-by-step explanation:
[tex]( - 3y^{2} - 8 ) - ( - 5y^{2}+ 1 )[/tex]
While subtracting , sign of each term of the second expression changes.
⇒[tex]- 3y^{2} - 8 + 5y^{2} - 1[/tex]
Combine like terms : [tex]-3y^{2}[/tex] and [tex]5y^{2}[/tex]
Like terms are those which have the same base.
⇒[tex]2y^{2} - 8 - 1[/tex]
Remember : The negative integers are always added but posses the negative ( - ) sign.
⇒[tex]2y^{2} - 9[/tex]
Hope I helped!
Best regards !
-[tex]\sf{ TheAnimeGirl}[/tex]
a number cube with faces labeled from to will be rolled once. the number rolled will be recorded as the outcome. give the sample space describing all possible outcomes. then give all of the outcomes for the event of rolling an odd number. if there is more than one element in the set, separate them with commas.
The probability of rolling an odd number is 1/2, assuming that the number cube is fair.
The sample space is the set of all possible outcomes of an experiment. In this case, the experiment is rolling a number cube once and recording the outcome. Since the number cube has six faces labeled from 1 to 6, the sample space can be represented as:
S = {1, 2, 3, 4, 5, 6}
The event of rolling an odd number consists of the outcomes 1, 3, and 5.
Therefore, the set of all outcomes for the event of rolling an odd number is: {1, 3, 5}
The probability of rolling an odd number can be found by adding the probabilities of the individual outcomes in the event:
{P(odd)} = P(1) + P(3) + P(5) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2
Therefore, the probability of rolling an odd number is 1/2.
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-3x -4.6 = 5.9
x = ?
Answer:
x=-3.5
Step-by-step explanation:
-3x=10.5 divide both sides by -3
Answer:
x = -3.5
Step-by-step explanation:
-3x - 4.6 = 5.9 ( add 4.6 on both sides)
-3x = 10.5 (divide by -3 on both sides)
x = -3.5
You plan to purchase a house for 12.5 million baht, using a 30-year mortgage. You will make a down payment of 15 percent of the purchase price and you will not pay off the mortgage early. Ignore taxes in your analysis. Your bank offers you the following two options for payment.
Option 1: Mortgage rate of 7.25 percent and zero points.
Option 2: Mortgage rate of 6.75 percent and 4 points.
You would choose option ------- (1 or 2) because the Net Present Value of choosing option 2 is------ . (The answer can be negative. Do not round intermediate calculations and round your answer to two decimal places, e.g., 32.16.)
Option 2 has a lower NPV and thus should be chosen as it results in less total payment and better savings. Hence, you would choose option 2 as the Net Present Value of choosing option 2 is -9,903,615.48.
Given:
Purchase price = 12.5 million baht
Down payment = 15%30-year mortgage
We have to calculate the option which is best for us.
Option 1: Mortgage rate = 7.25%
Option 2:Mortgage rate = 6.75%
(a) Monthly Payment of Option 1:
Loan Amount = 12.5 * (100-15)%
= 10,625,000 baht
n = 30 * 12
= 360
i = 7.25%/12
= 0.006042857
Yearly Interest Paid = i * 10,625,000
= 64,515.48 baht
Monthly Interest Paid = 64,515.48 / 12
= 5,376.29 baht
EMI =[tex][P x R x (1+R)^n] / [(1+R)^n-1][/tex]
where P = Loan Amount, R= Interest Rate per month, n= total number of payments.
EMI = [tex][10,625,000 x 0.006042857 x (1+0.006042857)^360] / [(1+0.006042857)^360-1][/tex]
EMI = 69,677.31 baht
(b) Monthly Payment of Option 2:
Loan Amount = 12.5 * (100-15)%
= 10,625,000 baht
n = 30 * 12 = 360
i = 6.75%/12 = 0.005625
Yearly Interest Paid = i * 10,625,000
= 59,765.63 baht
Monthly Interest Paid = 59,765.63 / 12
= 4,980.47 baht
Point cost = 4% of Loan Amount
= 10,625,000 x 4/100
= 425,000 baht
Loan Amount after Deducting Points = 10,625,000 - 425,000
= 10,200,000 baht
EMI = [tex][P x R x (1+R)^n] / [(1+R)^n-1][/tex]
where P = Loan Amount, R= Interest Rate per month, n= total number of payments.
EMI =[tex][10,200,000 x 0.005625 x (1+0.005625)^360] / [(1+0.005625)^360-1][/tex]
EMI = 66,128.92 baht
(c) We have to calculate the NPV of the payments to decide which option is better.
We use the formula to calculate NPV:-
NPV = - [tex][B + (PMT / i) * (1 - (1 + i)^-n)] / ((1 + i)^t)[/tex]
Where B = amount borrowed, PMT = monthly payment, i = interest rate, n = total number of payments, t = year.
NPV =[tex]- [10625000 + (69677.31 / 0.006042857) * (1 - (1 + 0.006042857)^-360)] / ((1 + 0.006042857)^30)[/tex]
= -9,892,571.85 baht
NPV = [tex]- [10200000 + (66128.92 / 0.005625) * (1 - (1 + 0.005625)^-360)] / ((1 + 0.005625)^30)[/tex]
= -9,903,615.48 baht
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You just bought a new drone. You are standing in the middle of a park flying it, and know that you are 1500 feet from the building you live in. You also know that the building you live in is 150 feet tall. You decide you do not want to carry the drone back with you and can land it on the roof of your building instead. How far would your drone need to fly to go in a straight line from your current position to the roof of your building? Also, what angle should your drone take in its flight path?
To land the drone on the roof of the building, it would need to fly approximately 1505.35 feet in a straight line. The angle between the ground and the drone's flight path would be approximately 5.74 degrees.
To determine the distance the drone needs to fly, we can use the Pythagorean theorem. The horizontal distance between your current position in the park and the building is given as 1500 feet, and the height of the building is 150 feet. We can consider this as a right-angled triangle, with the horizontal distance as the base and the height of the building as the perpendicular side.
Using the Pythagorean theorem, we can find the hypotenuse (flight distance) of the triangle, which represents the straight-line distance the drone needs to travel. The equation is:
hypotenuse^2 = base^2 + perpendicular^2
Substituting the values, we get:
hypotenuse^2 = 1500^2 + 150^2
hypotenuse^2 = 2250000 + 22500
hypotenuse^2 = 2272500
Taking the square root of both sides, we find:
hypotenuse ≈ √2272500 ≈ 1505.35 feet
Therefore, the drone would need to fly approximately 1505.35 feet in a straight line to reach the roof of your building.
To determine the angle of the flight path, we can use trigonometry. The angle (θ) can be calculated using the tangent function:
tan(θ) = perpendicular/base
Substituting the values, we get:
tan(θ) = 150/1500
tan(θ) = 0.1
Taking the inverse tangent (arctan) of both sides, we find:
θ ≈ arctan(0.1) ≈ 5.74 degrees
Hence, the drone should take an angle of approximately 5.74 degrees in its flight path to reach the roof of your building.
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Consider two equations describing the relationship between three variables X, Y, and Y₂: X-a+bY (1) X-e-dYs (2) where a, b, c, and d, are positive, known constants (numbers). The objective is to find values of X, Y₁. and Y: for which both equations are satisfied and as well Y₁ - Yz. a. How many unknown variables are there? Clearly identify them. How many equations are there? Clearly identify them
Our goal is to find values of X, Y₁, and Y₂ that make both equations equal to zero.The unknown Variables: X, Y₁, Y₂,The equations: Equation (1) and Equation (2)
In the given equations, we have three unknown variables: X, Y₁, and Y₂. These variables represent the values we are trying to find.
We also have two equations, labeled as Equation (1) and Equation (2). These equations provide the relationships between the variables.
To solve for the unknown variables, we need to find values of X, Y₁, and Y₂ that satisfy both equations simultaneously.
Let's restate the equations for clarity:
Equation (1): X - a + bY₁ = 0
Equation (2): X - e - dY₂ = 0
Our goal is to find values of X, Y₁, and Y₂ that make both equations equal to zero.
So, in summary:
- The unknown variables: X, Y₁, Y₂
- The equations: Equation (1) and Equation (2)
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the standard form of -36/-216
Answer:
[tex]\frac{1}{6}[/tex]
Step-by-step explanation:
[tex]\frac{-36}{-216} \\=\frac{36}{216} \\=\frac{1}{6}[/tex]
Which situation fits the following Inequality 7x+15≥50
Answer: x ≥ 5
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Q. 2) (10 p.) Answer each of the following: (a) A matrix polynom is given by P(x) = x². By considering the 2 by 2 matrix as A = - 1, 0 (b) For a given matrix A = [2], calculate A-, where k is any integer. -2. find P(A).
(a) The matrix polynomial P(x) = x² can be evaluated by substituting the given 2 by 2 matrix A = [-1, 0; -2, 1] into P(x).
(b) To calculate A^-, where k is any integer, we need to find the inverse of the matrix A. Additionally, P(A) can be obtained by substituting matrix A into the matrix polynomial P(x) = x².
(a) To evaluate the matrix polynomial P(x) = x² using the given 2 by 2 matrix A = [-1, 0; -2, 1], we substitute A into P(x):
P(A) = A² = [-1, 0; -2, 1]² = [(-1)(-1) + (0)(-2), (-1)(0) + (0)(1); (-2)(-1) + (1)(-2), (-2)(0) + (1)(1)] = [1, 0; 0, -3].
(b) To calculate A^-, where k is any integer, we need to find the inverse of the matrix A = [2, -2; -2, 4]. The inverse of a 2 by 2 matrix can be found using the formula:
A^-1 = (1/det(A)) * adj(A),
where det(A) is the determinant of A and adj(A) is the adjugate of A. Calculating the determinant and adjugate of A, we have:
det(A) = (2)(4) - (-2)(-2) = 12 - 4 = 8,
adj(A) = [4, 2; 2, 2].
Therefore, A^-1 = (1/8) * [4, 2; 2, 2] = [1/2, 1/4; 1/4, 1/4].
To find P(A), we substitute A into the matrix polynomial P(x) = x²:
P(A) = A² = [2, -2; -2, 4]² = [2, -2; -2, 4] * [2, -2; -2, 4] = [4 - 4, -4 + 8; -4 + 4, 4 + 16] = [0, 4; 0, 20].
Therefore, A^-1 = [1/2, 1/4; 1/4, 1/4] and P(A) = [0, 4; 0, 20].
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