All Answers 356467 Answers

Hello Alan,

here is your pic: https://web2.0calc.com/img/upload/c037cd1f330c754341269173/g1.jpg

Thanks very much Tiggsy, makes perfect sense.  

plug it into symbolab lol

So to do this, try setting a variable \(u=x^2\) to make it simpler. You have \(u^2=-4\) and I think you can do it.

(not trying to be cphil at all)

Binomial Theorem.

and Varxaax, I think you almost got it

now you just look at the coefficient of the term

which is 240.

but this seems VERY MUCH like an AOPS - Counting and Probability - Binomial Theorem problem.

If it is well.

good luck with your cheating career is all i can say

Let's See:

For each digit space, there can be any number, with the exception that the 1000s place cannot have a 0. We have 2 choices for the thousands place, either 1 or 3.

For the 100s place, any number is okay. 0,1, and 3 all work just fine

For the 10s place, any number can be there as well

For the 1s place, it's still 3 choices

So we have 2 choices, 3, choices, 3 choices, and 3 choices. What now? Well, we multiply our answers together, just how you would do a "clothes combination" problem. I will let you do the multiplication for yourself.

ASSUMING that Guest is correct,

\(\begin{align*} \frac{8x + 2}2 &= 7x -17\\ 8x+2&=2(7x-17)\\ 8x+2 &=14x-34\\ 2+34&=14x-8x\\ 36&=6x\\ 6&=x\\ \end{align*}\)

try y on your own!

Now it is ;)

Hmm

Khan Academy stuff I see

Although someone has given you the answer already, you shouldn't go right to that. Let's work through this.

The equation for a line is \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept.

Here, we see that \(3\) is obviously the intercept, but what is the slope? Well, to find it, we can get a set of clean points, and then apply the formula \(slope=\frac{\text{change in y}}{\text{change in x}}\). Here, we see that for every \(1\) unit across, it goes \(3\) units upwards. Using our formula, we get \(\frac{3}{1}=3\) as our slope, so our answer is \(\boxed{y=3x+3}\)

(8x + 2)/2 = 7x -17             solve for x

(12y -14)/2 = 4y + 7           solve for y

If a 7-digit number 13ab45c is divisible by 792, what is b?

\(\mathbf{792=2^3*3^2*11}\)

1.

\(\mathbf{13ab45c}\) is divisible by \(\mathbf{2}\), so \(c=\{2,4,6,8\}\)

2.

\(\mathbf{13ab45c}\) is divisible by \(\mathbf{2^3=8}\), so \(45c \equiv 0 \pmod{8}\)

\(\begin{array}{|c|r|c|} \hline c & 45c & 45c \pmod{8} \equiv 0\ ?\\ \hline 2 & 452 & \text{no} \\ \hline 4 & 454 & \text{no} \\ \hline \mathbf{\color{red}6} & 456 & \text{yes} \\ \hline 8 & 458 & \text{no} \\ \hline \end{array} \), so \(\mathbf{c=6}\)

3.

\(\mathbf{13ab45c}\) is divisible by \(\mathbf{3^2=9}\), so \(13ab456 \equiv 0 \pmod{9}\)

\(\begin{array}{|rcll|} \hline 1+3+a+b+4+5+6 &\equiv& 0 \pmod{9} \\ a+b+19 &\equiv& 0 \pmod{9} \quad | \quad 19 \equiv 1 \pmod{9} \\ a+b+1 &\equiv& 0 \pmod{9} \quad | \quad - 1 \\ a+b &\equiv& -1 \pmod{9} \\ a+b &\equiv& -1+9 \pmod{9} \\ \mathbf{a+b} &\equiv& \mathbf{ 8 \pmod{9} } \\ \hline \end{array}\)

\(\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|} \hline \mathbf{a+b}\pmod{9}= 8 \ ? \\ b \rightarrow & \mathbf{0} & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9}\\ a \downarrow \\ \hline \mathbf{0} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \color{red}8 & 0 \\ \hline \mathbf{1} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \color{red}8 & 0 & 1 \\ \hline \mathbf{2} & 2 & 3 & 4 & 5 & 6 & 7 & \color{red}8 & 0 & 1 & 2 \\ \hline \mathbf{3} & 3 & 4 & 5 & 6 & 7 & \color{red}8 & 0 & 1 & 2 & 3 \\ \hline \mathbf{4} & 4 & 5 & 6 & 7 & \color{red}8 & 0 & 1 & 2 & 3 & 4 \\ \hline \mathbf{5} & 5 & 6 & 7 & \color{red}8 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \mathbf{6} & 6 & 7 & \color{red}8 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \mathbf{7} & 7 & \color{red}8 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \mathbf{8} & \color{red}8 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7& \color{red}8 \\ \hline \mathbf{9} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7& \color{red}8 & 0 \\ \hline \end{array}\)

, so \((a,b)=\{(0,8),(1,7),(2,6),(3,5),(4,4),(5,3),(6,2),(8,0),(8,9),(9,8)\}\)

4.
\(\mathbf{13ab45c}\) is divisible by \(\mathbf{11}\), so \(13ab456 \equiv 0 \pmod{11}\)

\(\begin{array}{|rcll|} \hline 1-3+a-b+4-5+6 &\equiv& 0 \pmod{11} \\ a-b+3 &\equiv& 0 \pmod{11} \quad | \quad - 3 \\ a-b &\equiv& -3 \pmod{11} \\ a-b &\equiv& -3+11 \pmod{11} \\ \mathbf{a-b} &\equiv& \mathbf{ 8 \pmod{11} } \\ \hline \end{array}\)

\(\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|} \hline \mathbf{a-b}\pmod{11}= 8 \ ? \\ b \rightarrow & \mathbf{0} & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9}\\ a \downarrow \\ \hline \mathbf{0} & 0 & 10 & 9 & \color{red}8 & 7 & 6 & 5 & 4 & 3 & 2 \\ \hline \mathbf{1} & 1 & 0 & 10 & 9 & \color{red}8 & 7 & 6 & 5 & 4 & 3 \\ \hline \mathbf{2} & 2 & 1 & 0 & 10 & 9 & \color{red}8 & 7 & 6 & 5 & 4 \\ \hline \mathbf{3} & 3 & 2 & 1 & 0 & 10 & 9 & \color{red}8 & 7 & 6 & 5 \\ \hline \mathbf{4} & 4 & 3 & 2 & 1 & 0 & 10 & 9 & \color{red}8 & 7 & 6 \\ \hline \mathbf{5} & 5 & 4 & 3 & 2 & 1 & 0 & 10 & 9 & \color{red}8 & 7 \\ \hline \mathbf{6} & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 10 & 9 & \color{red}8 \\ \hline \mathbf{7} & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 10 & 9 \\ \hline \mathbf{8} & \color{red}8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 10 \\ \hline \mathbf{9} & 9 & \color{red}8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\ \hline \end{array}\)

, so \((a,b)=\{(0,3),(1,4),(2,5),(3,6),(4,7),(5,8),(6,9),(8,0),(9,1)\}\)

compare:

\(\mathbf{a+b}\pmod{9}= 8 : \quad (a,b)=\{(0,8),(1,7),(2,6),(3,5),(4,4),(5,3),(6,2),{\color{red}(8,0)},(8,9),(9,8)\} \\ \mathbf{a-b}\pmod{11}= 8 :\quad (a,b)=\{(0,3),(1,4),(2,5),(3,6),(4,7),(5,8),(6,9),{\color{red}(8,0)},(9,1)\}\)

,so \(a=8\) and \(b=0\)

\(\begin{array}{|lr|} \hline & \mathbf{13ab45c} = \mathbf{1380456} \\ \hline \text{check}: \\ & 1380456 : 792 = 1743 \\ \hline \end{array}\)

1,380,456 mod 792 = 0

hi guest!
 

we can start off by finding the y-coordinate which is (0,3),so that means that b=3.

next we can find the slope by using \(\dfrac{y_2-y_1}{x_2-x_1}\), and we can use (-1,0) and (0,3).

This would equal \(\dfrac{3-0}{0-(-1)}=\dfrac{3}{1}=3\).

so the answer is \(\boxed{y=3x+3}.\)

This one's for Alan or heureka...

I believe the highest range would be   -5 * -2 = 10

   the lowest would be    3 * -2 = -6

    [-6,10]

Let me know if this incorrect .... I am always learning too !

I will assume there cannot be a 0 in the first position

2 choices for first digit

3 choices for 2nd

3 choices for 3rd

3 choices for 4th

2  x  3  x  3  x  3 =....... choices

What do you know about cyclical quadrilaterals in a circle ?  ( opposite angles sum to 180^o )

    What do you know about interna angles in a pentagon?  (108^o )

Now look at this diagram:

also is there a way to delete questions?

yeah i tried that

792=8*9*11

so c=6

then, 1+3+a+b+9+6=multiple of 9

so, a+b=8 or a+b =17

for it to be divisible by 11, 1-3+a-b+4-5+6= multiple of 11, so 3+a-b= multiple of 11, which means that a-b=8 or a-b = -3.

but then i tried to plug in the equations and i got x=7 and y=10 :/

can someone please tell me what I'm doing wrong?

Try to find the prime factorization of 792. Then, use the prime factors, and see what restrictions the number puts.

Tan 70   =  opposite side /adjacent side 

tan70 = height / 12        solve for height

what about

15  1

13   3              

11   5              

  9    7             

4... I think...

Yay!