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Please do not post Art of Problem solving question here.

It is against their code of conduct.

The probability is 3/16.

The number of students is 6.

When y = 8, x = 15.

WJ/JK = 35.

The equation of the line is y = x - 1 so the inequality is y > x - 1

Standard form: -x + y + 1 > 0

That is great Cal.  I worked that out to be 52.5miles per hour.

Which is the same as I worked out my long method.

I should try and remember that. :)

Thanks very much Alan.

I am sure you do know what you are talking about.   

We know the original area is 2.5

We know  (-1,0) transforms to (-1,sqrt3) and   (0,1) transforms to (sqrt3,1)

We know that the distance between the original 2 points is  sqrt2 units

We know that the distance between the corresponding transformed two points is sqrt{8}

the ratio of new to old side lengths is  sqrt8/sqrt2 = 2

square it and we get the ratio of the areas is 4

The original area is 2.5 so the new area must be 10

NOW we agree :))

Sounds likely, would you care to prove it?

x²  -6x +8 +1-1   =    (x² -6x +9)  -1

                               =  (x-3)²  - 1 

Let's see

abcba

any digit can go in the middle,  How many is that?  10

a can't be 0

So there are 9 choices for a and 10 choices for b.

So how many altogether?

10*9*10=900  exist.

Oh dear, Bginner is right, they are asking for the sum of them

ok

sum of 100c is  100(0+1+2+...9)=100(45)=4500

sum of 10001+20002+ ..... +90009 = 10000(1+2+...9) + (1+2+...9) = 10000*45+45 = 10001*45=450045

sum of 0000+1010+2020 .... 9090 = 45(1000+10) = 45*1010 = 45450

4500+450045+45450 = 499995     that is what I get but I havn't checked it.  I could easily have miscounted.

I have not looked at any of the answers but most likely the negatives are because there is an answer with no explanation.

This is never encouraged but there is at least one troll on here that gets great pleasure from putting up answers like this that are wrong.

You should never pay much attention to unsupported answers.

yes! thank you, I really appreciate it!

Ok!

x^2 - 6x + 8

= x^2 - 6x + 36 - 36 + 8

= (x - 6)^2 - 28

You're correct!

1 lemon = 2 tablespoon lemon juice​

16 tablespoons = 1 cup​

1 pint = 2 cups​

​

​

24 * 2 = 48​

1 batch = 48 tablespoons​

Remember, 16 tablespoons = 1 cup.​

Divide 48 by 16, which equals 3.​

So 1 batch = 3 cups. We also know that 1 pint = 2 cups, so ​

It has to be 1.5 because 1.5 * 2 = 3.​

1.5 pints of lemon juice = 1 batch of lemonade​.

Hope this helps 

Nice. :))

=^._.^=

If $abc=13$ and
$\left(a+\dfrac{1}{b}\right)\left(b+\dfrac{1}{c}\right)\left(c+\dfrac{1}{a}\right)=\left(1+\dfrac{1}{a}\right)\left(1+\dfrac{1}{b}\right)\left(1+\dfrac{1}{c}\right)$,
find $a+b+c$.

$\small{ \begin{array}{|rcll|} \hline \left(a+\dfrac{1}{b}\right)\left(b+\dfrac{1}{c}\right)\left(c+\dfrac{1}{a}\right) &=&\left(1+\dfrac{1}{a}\right)\left(1+\dfrac{1}{b}\right)\left(1+\dfrac{1}{c}\right) \\ \left(\dfrac{ab+1}{b}\right)\left(\dfrac{bc+1}{c}\right)\left(\dfrac{ac+1}{a}\right) &=&\left(\dfrac{a+1}{a}\right)\left(\dfrac{b+1}{b}\right)\left(\dfrac{c+1}{c}\right) \\ (ab+1)(bc+1)(ac+1)&=&(a+1)(b+1)(c+1) \\ (ab^2c+ab+bc+1)(ac+1)= &=& (ab+a+b+1)(c+1) \\ a^2b^2c^2+ab^2c+a^2bc+ab+abc^2+bc+ac+1&=&abc+ab+ac+a+bc+b+c+1 \\ a^2b^2c^2+ab^2c+a^2bc+abc^2&=&abc+a+b+c \\ a^2b^2c^2+abc(a+b+c)&=&abc+(a+b+c) \quad | \quad \mathbf{abc=13} \\ 13^2+13(a+b+c)&=&13+(a+b+c) \\ 12(a+b+c) &=& 13 - 13^2 \\ 12(a+b+c) &=& 13(1-13) \\ 12(a+b+c) &=& -13*12 \quad | \quad :12 \\ \mathbf{ a+b+c } &=& \mathbf{-13} \\ \hline \end{array} }$

The answer is:

21

Each point connects 6 other points. But without overlapping the points, 21 points is the correct answer.

Draw a circle and try it out yourself!

I hope this helps!

Solve this too: Jai buys three pounds of lemons from Citrus Club, and Ryan buys three pounds of lemons from Fruit Mart.

-How much more money did Ryan spend to buy pounds of lemons at Fruit Mart compared to the amount of money Jai spent at Citrus Club? Show each step of your solution and explain your reasoning. This is connected to the main question

The expression $ax^2+15x+4+3x+5$ simplifies to $ax^2+18x+9$.  Since it's the square of a binomial, it must equal $(cx+d)^2$ for some constants $c$ and $d$.  Now $(cx+d)^2= c^2x^2+2cdx+d^2$ and we can equate its coefficients to that of $ax^2+18x+9$ like so:
$$\begin{eqnarray*} a &=& c^2 \\ 2cd &=& 18 \\ d^2 &=& 9 \end{eqnarray*}$$
We solve this system of 3 equations in 3 unknowns $a$, $c$, and $d$.   Note that there are two solutions but both solutions have the same value of $a$.

3931 the total     <======   What does THAT mean?   PLease clarify/proffread your question....

ax^2 + 18x + 9   =        (3x+3)^2          when a = 9