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\(\text{probably easiest to un parameterise them and solve}\\ x = 3-2t \Rightarrow t = \dfrac{3-x}{2}\\ y = -2+5\cdot \dfrac{3-x}{2} = -2+\dfrac{15}{2}-\dfrac 5 2 x = -\dfrac 5 2 x + \dfrac{11}{2}\)

\(x = -1+3s \Rightarrow s = \dfrac{x+1}{3}\\ y = 5 - 9\cdot \dfrac{x+1}{3} = 5-3x-3= -3x+2\)

\(-\dfrac 5 2 x + \dfrac{11}{2}=-3x+2\\ \dfrac 1 2 x = -\dfrac 7 2\\ x = -7\\ y = 23\\ (x,y)=(-7,23)\)

what you describe would be a hyperbola

simple! draw a Venn Diagram! Thus, the answer is \(52\) people.

Well if he has \(12\) shirts, he has \(12\) to choose from. So \(12\) for the first choice, \(11\) for the second choice (because the first has been used), and \(10\) for the third choice. Thus, he has \(12\cdot11\cdot10\) choices, but they are counted \(6\) times each, so the answer is \(\boxed{220}\).

I got 32 when I tried to solve this but it was incorrect.. ):

\(\text{The function $h(x)$ is defined as: $h(x) = \left\{ \begin{array}{cl} \lfloor 4x \rfloor & \text{if } x \le \pi, \\ 3-x & \text{if }\pi < x \le 5.2, \\ x^2& \text{if }5.2< x. \end{array}\right. $}\\ \text{Find $h\Big(h\left(\sqrt{2} \right) \Big)$.} \)

\(\begin{array}{|rcll|} \hline h\left(\sqrt{2} \right) &=& \lfloor 4\cdot \sqrt{2} \rfloor \\ &=& \lfloor 4\cdot 1.41421356237 \rfloor \\ &=& \lfloor 5.65685424949 \rfloor \\ &=& 5 \\\\ h(5) &=& 3-5 \\ &=& -2 \\\\ \mathbf{h\Big(h\left(\sqrt{2} \right) \Big)} & =& \mathbf{ -2 } \\ \hline \end{array} \)

Thank you, CPhill !

\(\text{let }\sqrt{b+\sqrt{b+\sqrt{b+\dots}}} = \beta\\ \sqrt{b+\beta}=\beta\\ b+\beta=\beta^2\\ \beta^2-\beta-b=0\\ \beta=\dfrac{1\pm \sqrt{1+4b}}{2}\\ \beta>0 \text{ as it's a square root so }\\ \beta = \dfrac{1+\sqrt{1+b^2}}{2}\\ a\bowtie b = a + \dfrac 1 2 (1+\sqrt{1+b^2})\)

\(7\bowtie 9 = 9\\ 7+\dfrac 1 2 (1+\sqrt{1+g^2})=9\\ \sqrt{1+g^2}=3\\ 1+g^2=9\\ g=\pm\sqrt{8} = \pm 2\sqrt{2}\)

14 is correct

Hmmmmm.....   wouldn't it be 14 toppings?

14 c 2 = 91

PLUS   14 single toppings = 105    PLUS NO TOPPING = 106.

Thanks!

The dump truck body represents the hypotenuse of a right triangle....one leg is 9ft ...this is the leg OPPOSITE of the angle that needs to be at least 50 degrees....

sin x = opposite/hypotenuse

sin x = 9/15        x = asin(9/15) = 36.869897645844  degrees    Not high enough yet......

Let's find out how high it needs to be:

sin 50 = x/15       x = 15*(sin(50)) = 11.490666646785 ft

Thank you! You really explained that in a way I could understand.

\(\frac{n!}{r!(n-r)!}\) is the formula you always use in these types.

N represents number of items you are choosing out of, and r represents the number of items you are choosing.

Sorry! Here I will solve it myself.

2 cases here

case 1 : 1 topping

case 2 : 2 toppings

In all there should be 106 - 1 (no toppings case) = 105 ways with 2 cases added together.

OK....

Notice how if you have x toppings with y ways there are y/x different toppings.

So we have 1 topping and 2 toppings with 105 ways. That means there are 35 different toppings

This makes sense. Because :

CASE 1 : 1 * 35 = 35 ways

CASE 2 : 2* 35 = 70 ways

CASE 3 : Not toppings (1)

So add three cases together 35 + 70 + 1 = 106

So there you have it! :D :D :D

When looking at systems of linear equations. You can either use substitution or elimination.

The question is WHICH one to use? 

Don't skim my answer for the answer

(ax with a being the constant)

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Substitution. You should use this whenever you see a variable with a constant of 1. For example the second equation you posted.

3x - y = 4

y = 2x - 2 

Notice how this can EASILY be substituted

Solving:

3x - (2x-2) = 4  (remember there is a -1 multiplying at the parentheses).

(remove parentheses and combine like terms.)

x + 2 = 4

x = 2

Now substitute back into one of your equations and find y.

Elimination. You should use this whenever you can't divide both sides of one of the equations and get integer constants.

4x + 7y = 6

9x - 2y = 49

Notice how the equation will be difficult substituting. If you try to isolate a variable, you will get fraction constants.

So a good strategy is to find the greatest common multiple.

Solving:

4x + 7y = 6            (Mulitply 7y by 2)

9x - 2y = 49           (Multiply 2y by 7)

8x + 14y = 12

63x - 14y = 343

Now you can just add both equations: 71x = 355

Then divide - x=5

Now substitute back into one of the equations and find y

I'm sorry...those answers are incorrect. 

There are 2^10 different subsets in the original set.

Now let us just subtract subsets without a 5

Let us just pretend you have set without a 5

So {1 2 3 4 6 7 8 9 10} Is your new set

So how many subsets can you make from that?

2^9 subsets

2^10 - 2^9 = 512 different subsets

NOTE: 2^10 - 2^9 is NOT 2.

So there you have it! :D:D:D

These are the number of ways Jimmy can choose his shirts:

\(\binom{12}{3} = \frac{12!}{9! \cdot 3!} = \frac{12 \cdot 11 \cdot 10}{6} = \boxed{220}\)

FV=9000; R=0.04/2; N=9*2;PMT = FV*(((1 + R)^N - 1)^-1* R)

PMT =$ 420.32 - This is the semi-annual deposit.

Not sure honestly, thats what the question says.

We need to find this...: (Just watch American Idol)+(Just watch Lost)+(Watch both television series). Thus...we have (21-8)+(39-8)+8=13+31+8=52 people.