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To find the ordered quintuplet (a, b, c, d, e) that satisfies the given system of equations, I can try solve the system using matrix operations. see below...

[ 1 2 3 4 5 ] [ a ] [ 41 ]
[ 2 3 4 5 1 ] * [ b ] = [ 15 ]
[ 3 4 5 1 2 ] [ c ] [ 34 ]
[ 4 5 1 2 3 ] [ d ] [ 68 ]
[ 5 1 2 3 4 ] [ e ] [ 57 ]

To solve this system, we can use matrix inversion. We'll calculate the inverse of the coefficient matrix and multiply it by the column matrix on the right-hand side to obtain the solution.

The sum of the infinite series 1/7 + 2/7^2 + 1/7^3 + 2/7^4 + ... is 9/48.

This can be found using the formula for the sum of a geometric series:

S = a/(1-r)

where

a = first term in the series

r = common ratio

S = sum of the series

In this case,

a = 1/7

r = 2/7

S = ?

S = 1/7 / (1 - 2/7) = 9/48

Therefore, the sum of the infinite series is 9/48.

[10a + b] - [10b + a] ==36

a =2*b  +  3, solve for a, b

a==5   and   b ==1

Sophie's favorite number ==51

sumfor(n, 1, 10000, (1/7^(2*n - 1) + 2/(7^(2*n)))==3 / 16

Let's start by writing the equation as a geometric series:

1 + x + x^2 + x^3 + ... = 4

We can then factor out a 1 from the left-hand side:

1(1 + x + x^2 + x^3 + ...) = 4

We can then divide both sides by 1 - x to get:

1/(1 - x) = 4

This is an equation of the form a/b = c, where a, b, and c are all real numbers. The solution to this equation is:

x = -b/a

In this case, a = 1, b = 1, and c = 4. Plugging these values into the solution formula, we get:

x = -b/a = -1/1 = -1

However, this is not the solution we are looking for. The series 1 + x + x^2 + x^3 + ... diverges when x = -1. This is because the absolute value of the common ratio (x) is greater than 1.

We need to find a solution that makes the series converge. To do this, we need to find a value of x for which the absolute value of the common ratio is less than 1. The only real number x for which this is true is x = -1/3.

When x = -1/3, the common ratio is x = -1/3. The absolute value of this is less than 1, so the series converges. Therefore, the solution to 1 + x + x^2 + x^3 + ... = 4 is x = -1/3.

Find the coefficient of y^4 in the expansion of (2y-5+3y^2)^6.  

The 375 in Answer #1 is correct.  

Check it at  https://www.symbolab.com/solver/expand-calculator  

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Find the number of ways of arranging the numbers 1,2 ,3, 4, 5, 6, 7, 8, 9   

in a row so that the product of any two adjacent numbers is even.  

For two numbers to have an even product, at least one of them must be even.  

That is, an odd number can never be adjacent to another odd number.  

So, odds and evens have to alternate, such as the example shown in the problem.  

Furthermore, since there are more odd numbers, the first number has to be odd.  

There are five odd numbers, and five positions into which they can be situated.  

So, there are 5 • 4 • 3 • 2 • 1 = 120 ways the odd numbers can be arranged.  

And, there are four even numbers, and four positions where they can be located.  

So, there are 4 • 3 • 2 • 1 = 24 ways that the even numbers can be arranged.  

Any even number arrangement can be combined with any odd number arrangement.  

Therefore, the number of possible mixes is 120 • 24 = 2880.   

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The answer is 375

Sorry, but that's incorrect.

I drove to the beach at a rate of 40 miles per hour.  If I had driven at a rate of 50 miles per hour instead, then I would have arrived 45 minutes later.  How many miles did I drive?  

You mean 45 minutes earlier.  Obviously, if you drive faster, you get there faster.  

This problem makes use of

the following relationship:                   Distance = Velocity x Time 

                                                           D  =  V • T  

case 1                                                 D  =  (40) • (T)  

case 2                                                 D  =  (50) • (T – 45)  

Since the Distance, D, is the  

same for both cases, let's set       

the "V•T"s equal to each other.             (50)(T – 45)  =  (40)(T)  

                                                               50T – 2250  =  40T  

Subtract 40T from both sides                  10T – 2250  =  0  

Add 2250 to both sides                                       10T  =  2250  

Divide both sides by 10                                           T  =  225   (this is in minutes)  

Divide minutes by 60 to get hours          225 minutes  =  3.75 hours  

Plug this T back into original equation                     D  =  (40 mi/hr) • (3.75 hr)  =  150 miles  

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An aquarium tank is 1/6 full of water. When 2 gallons of water are added, the tank becomes 1/2 full. What is the total capacity of the aquarium tank, in gallons?  

Note that 1/2 is the same as 3/6.  

So 2 gallons took the tank to 3/6 from 1/6.  

So 2 gallons added 2/6.  That means 1 gallon is 1/6.  

Therefore, the total capacity of the aquarium is 6 gallons.  

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So you are blatantly copying from homework and are shameless enough to admit it?

This question makes no sense. Are you r*tarded?

Emily Ward is married and has a non-network family health insurance plan with the benefits shown in the

figure on p. 10. Her recent non-network health care costs include 5 physician visits and 2 specialist visits.

She made co-payments for 6 physical therapy visits at $110 each. Emily has one ambulance trip to the

emergency room and a subsequent hospital bill of $14,680. Emily has no deductible. What amount did she

pay?

There are b3 three-digit numbers in base b. If all of the digits are distinct, then the first digit can be chosen in b−1 ways, the second digit can be chosen in b−2 ways, and the third digit can be chosen in b−3 ways. So, there are (b−1)(b−2)(b−3) three-digit numbers in base b with distinct digits. We are given that (b−1)(b−2)(b−3)=100. Solving for b, we find b=13​.

A candy bar and a piece of bubble gum together cost 1.50.  

The candy bar costs a dollar more than the bubble gum.  

How much does the piece of bubble gum cost, in cents?  

Call the cost of a candy bar "C"  

Call the cost of a bubble gum "B"  

                                                                      C + B  =  1.50  

                                                                      C – B  =  1.00  

Just subtract the second from the first               2B  =  0.50  

                                                                             B  =  0.25  =  25¢  

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I drove to the beach at a rate of 40 miles per hour.  If I had driven at a rate of 42 miles per hour instead, then I would have arrived 45 minutes later.  How many miles did I drive?  

You mean 45 minutes earlier.  

                                                        D  =  V * T  

case 1                                              D  =  40 * T  

case 2                                              D  =  42 * (T – 45)  

Set the two equal                             42 * (T – 45)  =  40 * T  

                                                         42T – 1890  =  40T  

                                                            2T  =  1890  

                                                              T  =  945 minutes  =  15.75 hours  

Plug T back into original D = V * T        D  =  40 mi/hr * 15.75 hr  =  630 miles  

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See answer #2 and #3 here:  https://web2.0calc.com/questions/proportion_82#r3

(8 C 4) * ((8 -4) C 3) * (1 C 1) = 280 ways to allocate the pens.

All members of our painting team paint at the same rate.  If 20 members can paint a 2000 square foot wall in 40 minutes, then how long would it take the 20 members to paint a 6000 square foot wall, in minutes?  

Same team, same speed, same paint, but three times as much wall.  

Therefore, it'd take three times as much time:   3 * 40 min  =  120 min  

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To find BC and BZ in triangle AB, we can use the angle bisector theorem.

The angle bisector theorem states that in a triangle, if a line divides one side into two segments, the ratio of the lengths of those segments is equal to the ratio of the lengths of the other two sides.

In triangle AB, let's label BC as "x" and BZ as "y".

According to the angle bisector theorem, we can set up the following ratios:

AY / CY = AB / BC 16 / 16 = 16 / x

Simplifying the equation: 1 = 16 / x

Cross-multiplying: x = 16

So, BC = 16.

Now, let's find BZ using the same theorem:

AY / CZ = AB / BZ 16 / 16 = 16 / y

Simplifying the equation: 1 = 16 / y

Cross-multiplying: y = 16

So, BZ = 16.

Therefore, BC = 16 and BZ = 16 in triangle AB.

The angle bisector theorem states that in a triangle, the angle bisector divides the opposite side into segments that are proportional to the lengths of the other two sides.

In triangle PQR, we have PQ = 9, QR = 9, and PR = 9. Since PR is the longest side, angle P is the largest angle in the triangle.

Let's label the length of PX as a and QX as b. Since the angle bisector of angle P intersects QR at point X, we can use the angle bisector theorem to set up the following equation:

PX / QX = PR / QR

a / b = 9 / 9

a / b = 1

Since a and b are in the ratio of 1:1, we can conclude that PX = QX.

Now, let's consider triangle PXY. Since PX = QX, triangle PXY is an isosceles triangle with PX = QX. Additionally, Y is the foot of the perpendicular from X to line PR, which means XY is the altitude of triangle PXY.

In an isosceles triangle, the altitude from the vertex angle bisects the base. Therefore, XY will bisect PR, and we can conclude that PY = YR.

Since PR = 9, PY + YR = PR implies PY + PY = 9, which simplifies to 2PY = 9.

Thus, PY = YR = 9 / 2 = 4.5.

Finally, we can use the Pythagorean theorem in right triangle PXY to find the length of XY:

XY^2 = PX^2 + PY^2

Since PX = PY = 4.5, we have:

XY^2 = 4.5^2 + 4.5^2

XY^2 = 20.25 + 20.25

XY^2 = 40.5

Taking the square root of both sides, we get:

XY = √40.5

Therefore, the length of XY is approximately 6.36 (rounded to two decimal places).

Hello,

To prove that PX = 12√6/5 and PY = 12√6/7, we'll use similar triangles and the properties of altitudes in a trapezoid.

Since CP/PD = 1, CP = PD. Let Q be the intersection point of AD and BC.

Consider triangles CPX and DQX. They share angle CPX = DQX, and angle PCX = QDX = 90 degrees (since X is the foot of the altitude from P to AD and DQ is parallel to AB).

Therefore, by AA similarity, triangles CPX and DQX are similar.

We know that AD = 5 and AB = 6, so DQ = (7/5) * AD = 7.2.
Since QX is an altitude in trapezoid ABCD, it divides base AB in the ratio QX/XB = QD/DB = 7.2/4.8 = 3/2.

One ordered pair (a,b) satisfies the two equations ab^4 = 48 and ab = 72. What is the value of b in this ordered pair?  

To find b, consider                                                  ab⁴  =  48  

We will divide both sides by ab.  

Since ab=72, we will divide the left side  

by "ab" and the right side by its equal 72.  

                                                                              ab⁴         48  

                                                                             ——   =   ——  

                                                                              ab           72  

Note that ab⁴ = (ab) * (b³)  

Cancel ab out of the left side.  

Reduce 48/72 on the right side.  

                                                                               b³           2  

                                                                             ——   =   ——  

                                                                                1            3  

                                                                                 b   =   cube root of (2 / 3)  

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