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D=2*N  - 5,     

5N + 10D ==225, solve for D, N

Use substitution to get:

D==17 - number of dimes

N==11 - number of nickels.

D==Distance

T ==Time taken to drive to the beach

D / 40 ==T

D / 36 ==T + 20/60, solve for D, T

Use substitution to get:

D==120 miles - distance travelled

T==3 hours - time taken to drive to the each at 40 mph.

120 / 36 ==3 + 1/3 hours -time to get to the beach driving at 36 mph.

1, 3, 5, 7, 9 ==5 odd numbers

2, 4, 6, 8 ==4 even numbers.

The 9-number arrangements must begin with an odd number and end with an odd number:

Therefore, there are: 5!  *  4! ==120  *  24 ==2,880 such arrangements possible.

[10a + b] - [10b + a] ==36,     

a=2b + 3, solve for a, b

Use substitution to get: 

a =5 and b=1

And Sophie's favorite numbers is: 51

There are 1364 possible arrangements.

Sophie's favorite number is 85.

 128 oz

Can you tell me how you got this?

\(a^{3} + 3ab^{2}=679, \\ 3a^{2}b+b^{3}=615.\)

Subtract the second equation from the first,

\(a^{3}-3a^{2}b+3ab^{2}-b^{3}=64, \\ (a-b)^{3}=64, \\ a-b = 4.\)

Let x be the number of nickels. We know that the number of dimes is 5 less than twice the number of nickels, so the number of dimes is 2x−5. We also know that Kaitlen has $2.25 in nickels and dimes. This means that 0.05x+0.10(2x−5)=2.25. Solving for x, we get x=9. Therefore, Kaitlen has 9​ nickels.

Since f(x) is divisible by x^3, f(x) is of the form ax^5 + bx^4 + cx^3.

You then want ax^5 + bx^4 + cx^3 + 2 to be divisible by (x + 1)^3.  Using long division, you get the equations

-10a  + 6b - 3c = 0

4a - 3b + 2c = 0

-a + b - c + 2 = 0

==> a = 6, b = 16, c = 12

So f(x) = 6x^5 + 16x^4 + 12x^3.

I think you wrote the problem wrong because the second line has r=1 and the third line has r=9, and that can't be right.

The general term of the series can be taken to be 1/(n*(n + 3)) where n proceeds by 3 for each term, n = 1, 4, 7, 19, ... , 94, 97.

(It could also be taken to be 1/((3n - 2)(3n + 1)) where n proceeds in steps of 1 from 1 to 33, which was the choice of #2.)

Expanding 1/(n*(n + 3) ) into the partial fractions fractions 1/(3n) - 1/(3(n + 3)),

n = 1  : 1/(1*4)     = 1/3 - 1/12,

n = 4  : 1/(4*7)     = 1/12 - 1/21,

n = 7  : 1/(7*10)   = 1/21 - 1/30,

n = 10: 1/(10*13) = 1/30 - 1/39,

.............

.............

n = 94 : 1/(94*97)   = 1/282 - 1/291,

n = 97 : 1/(97*100) = 1/291 - 1/300.

Adding all of the equations,

1/(1*4) + 1/(4*7) +1/(7*10) + ... + 1/(97*100) = 1/3 - 1/300 = 100/300 - 1/300 = 99/300 = 33/100.

Suppose that the area of the triangle is A, then (half base times height)

A = AB*CF/2, so AB = 2*A/CF,

A = BC*AD/2, so BC = 2*A/AD = 2*A/12,

A = CA*/BE/2, so CA = 2*A/BE = 2*A/15.

For the triangle to exist,

AB + BC > CA, so 1/CF + 1/12 > 1/15,

BC + CA > AB, so 1/12 + 1/15 > 1/CF,

CA + AB > BC, so 1/15 + 1/CF > 1/12.

The third of these conditions leads to CF < 60 meaning that the largest integer value for CF is 59.

One of the other two conditions gives a minimum for CF.

It would be nice to know where the numerical values for a, b and c came from.

\(( {\color{goldenrod}1}+{ \color{green}2}+{ \color{blue}3}+{ \color{Gray}4})n=180\\ 10n=180\\ n=18\\ There\ are\ {\color{blue}18\cdot 3=54}\ blue\ balls\ in\ the\ bag.\)

  !

1. AB/BC = 7/5.

6 and -3

Here are 4 integer triplets that satisfy the equation:

         a   b   c

1  = (0, 0, 0)
2  = (2, 2, 0)
3  = (2, 0, 2)
4  = (0, 2, 2)

The inverse of a function is another function that reverses the output of the original function. In this case, the inverse of g(x) would be a function that takes in the output of g(x) and returns the input.

To find the inverse of g(x), we can start by writing it as a set of three equations:

g(x) = 5x^2 if x <= -3 g(x) = 21 + x if -3 < x <= 10 g(x) = 2 - sqrt(x) if x > 10

We can then solve each equation for x in terms of g(x). This gives us the following three equations:

x = sqrt(g(x)/5) if x <= -3 x = g(x) - 21 if -3 < x <= 10 x = 1/sqrt(2 - g(x)) if x > 10

The inverse of g(x) is the function that takes in the output of g(x) and returns the corresponding input. In this case, the inverse of g(x) is a piecewise function that is made up of the three equations above.

To find g inverse of 7, we can simply plug 7 into each of the three equations and see which one gives us a valid input. In this case, the only equation that gives us a valid input is the second equation, so g inverse of 7 is 7 - 21 = -14.

There are only two integer triples that satisfy the equation ab+ac+bc=a+b+c+abc. They are:

(1, 2, 5) (2, 3, 4)

To prove this, we can first subtract abc from both sides of the equation to get:

ab+ac+bc=a+b+c

Now, we can factor the left-hand side of the equation as:

(a+b)(c-1)=a+b+c

We can see that this equation will only have integer solutions if (a+b) and (c-1) are both factors of (a+b+c).

The only positive integer pairs (a+b) and (c-1) that are factors of (a+b+c) are (1, 5) and (2, 4).

This means that the only possible solutions for a, b, and c are a=1, b=2, c=5 and a=2, b=3, c=4.

I hope this helps! Let me know if you have any other questions.

the other user forgot to divide by 12 under the k^2

C = (400-k^2) / 12

k can = 16 which makes c = 12

k can = 14 which makes c = 17 

k can = 10 which makes c = 25

k can = 8 which makes c = 28

k can = 4 which makes c = 32

k can = 2 which makes c = 33

It says that it is wrong

For a quadratic to have rational roots, its discriminant must be a perfect square. The discriminant of the given quadratic is 400 - 12c. For this to be a perfect square, we must have 400 - 12c = k^2 for some positive integer k. Solving, we get c = 400/12 - k^2. Since c < 100, we see that k must be 1, 2, 3, or 4. This gives us the four possible values of c: 25, 20, 15, and 12.

(Note: The quadratic may have other roots, such as irrational roots, that are not integers. We are only concerned with whether or not it has rational roots.)