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• We try to solve the equation:


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• We insert the solution into one of the initial equations of our system of equations


• We get a system of equations:


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• We insert the solution into one of the initial equations of our system of equations


• For :




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• We get a system of equations:


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Thanks. Makes perfect sense. Didn't think of it this way.

Simplifying
 4x + 3 = 2x + -5
 
 Reorder the terms:
 3 + 4x = 2x + -5
 
 Reorder the terms:
 3 + 4x = -5 + 2x
 
 Solving
 3 + 4x = -5 + 2x
 
 Solving for variable 'x'.
 
 Move all terms containing x to the left, all other terms to the right.
 
 Add '-2x' to each side of the equation.
 3 + 4x + -2x = -5 + 2x + -2x
 
 Combine like terms: 4x + -2x = 2x
 3 + 2x = -5 + 2x + -2x
 
 Combine like terms: 2x + -2x = 0
 3 + 2x = -5 + 0
 3 + 2x = -5
 
 Add '-3' to each side of the equation.
 3 + -3 + 2x = -5 + -3
 
 Combine like terms: 3 + -3 = 0
 0 + 2x = -5 + -3
 2x = -5 + -3
 
 Combine like terms: -5 + -3 = -8
 2x = -8
 
 Divide each side by '2'.
 x = -4
 
 Simplifying
 x = -4

Notice that  [A = 90] + [ B = 38] = 128 degrees.

Since  the three interior angles of a triangle measure 180.....angle C must measure the other 52 remaining degrees.  But an angle bisector would cut this in half.....so 1/2 of 52 = 26 degrees.... 

Note the triangle below - not to scale !!!.....  ACD is the angle formed when ACB is bisected...

"ACD" refers to the angle formed by the points ....well....A,C and D in that order....So ACB would be 52 degrees, ACD = 26 and DCB = 26, since angle C is bisected.....  

Does that make sense ???

(6x-12) / (x-2)   =

6(x-2) / (x -2)  =    [the  x - 2 terms "cancel" ]

6

3x/2y * 8y/3z * 5z/4x  =  [ multiplying numerators and denominators ]

[120 xyz]  / [ 24 yzx]     [ note that x y z and y z x  are the same thing].....so they "cancel"....and we have

120 / 24  =

[12 x 10] / [12 x 4 ]     [ the 12s "cancel]  =

10 / 4  =

5 / 2

"Anchor" one person.......then.....there are 7! arrangements= 5040 ways

I assume you want a committee of 3 Republicans and 2 Democrats......

So, we want to choose 3 of 55 Republicans and 2 of 45 Democrats.

So we  have

C(55, 3) * C (45, 2) = 2,5972,650 ways

This is just the difference in the area of two concentric  circles......we have

pi [ 4^2  - 3^2] =

pi [ 16 - 9] =

[7 pi] m^2  = about = 21.99 m^2

Write the 832 without the decimal

Subtract the non-repeating part from this   =  832 - 8 = 824

Write this over a number having the same number of 9's as the repeating part and the same number of 0's as the non-repeating part = 990

So we have     824 / 990  = 412 / 495

Check for yourself that this is equal to   .8323232....... 

 In(x^6sqrt y/z^3)  =

ln [x^6 * y^(1/2)] - ln z^3  =

lnx^6 + ln y^(1/2) - ln z^3  =

6 ln x  + (1/2) ln y  - 3ln z

This is either  (5/5)^5   = 1^5 = 5 

OR

5 /(5^5)  = 1/5^4  = 1/625

Depending upon what is meant......

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1/3x^3-1/2x^2+1/3x+1/3=0    let's multiply through by the common denominator of 2 and 3  = 6.....so we have

2x^3 - 3x^2 + 2x + 2 = 0

This will not factor......I can see that one solution is between  -1 and 0....let's use the on-site solver to find all of them.....

$${\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{1}}-{\mathtt{1}}{i}\\
{\mathtt{x}} = {i}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\\
{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{1}}{\mathtt{\,-\,}}{i}\\
{\mathtt{x}} = {\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{i}\\
{\mathtt{x}} = -{\mathtt{0.5}}\\
\end{array} \right\}$$

I actually should have used the Rational Zeroes Theorem to find the "real" solution.....the other two could have been found by using the Quadratic formula.

42000cm @ 5 cents per cm =   42000(.05)  = $2100

This makes sense....at 10 cents per cm we would have (1/10) of a dollar

So (1/10)  of 42000 = $4200......but we have half of this  (5 cents per cm....)

In(x-5)-4In(x+1)    we have

ln(x -5) - ln(x +1)^4  =

ln [(x - 5) / (x + 1)^4]

I used the folllowing properties, here

b ln a   = ln a ^b    and    ln a - ln b = ln[a / b]

I'll give this one a shot......but....I'm not overly confident.....!!!

Notice the follwing patterns......

For a 1 x 1 square, we have two possible values fo x and y, 0 and 1. And we have two ways to pick an apex point  and a base of 1. But, we can orient these two triangles in 4 ways......apex points at the "top," apex points to the "right," apeax points to the "left" and apex points on the "bottom"....and each will have a base of 1...so that's eight possible triangles

1x 1 =   (2 apex points)*(1 base) * [4 orientations]  = 8 possible triangles 

For a 2 x 2   square, we have three possible values for x and y  .... 0, 1, 2.  And we have three possible apex points,  three possible bases -  two of one unit and one of two units, and three possible heights - two of one unit and one of two units. And these can be oriented in 4 different ways, as before.

So the possible triangles in a 2 x 2 configuration are....

2 x 2  = [3 apex points]*[3 possible bases]* [3 possible heights]* [4 possible orientations]  = 108 possible triangles

And for our situation, we have a 3 x 3 square.

We have four possible values for x and y - 0,1,2,3. And we have four possible apex points, 6 possible bases and heights- three of one unit, two of two units and three of one unit.... and four possible orientations, as before.

So....the possible triangles are

3 x 3 =  [4 apex points]* [6  bases] *[6 heights]*[4 possible orientations ] =  576 possible triangles

It appears that the pattern for an n x n arrangement is [n + 1]*[C(n+1, 2)]^2 * 4  

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I hope this is correct........can somebody else check my reasoning ????

You can put the first ball in any of 3 boxes, the second ball in any of 3 boxes, etc.

There are 3 choices for each ball, so there are 3x3x3x3x3 = 3⁵ choices = 243 choices.

Depends ...

If you mean:  √(3)/2 x 1/√(3), then:  [√(3) x 1 ] / [2 x √(3) ] = 1/2 (the √(3)s cancel).

If you mean:  √(3/2) x 1/√(3), then:  √(3/2) x √(1/3)  =  √(3/6) = √(1/2)  =  √(2)/2.

If each try is independent from every other try and the probability of success is 1/683 and there are 683 tries, then the probability of failure is (682/683)⁶⁸³ because the probability of failure each time is 682/683 and you have to fail each of the 683 times.

So, the probability of success is 1 - (682/683)⁶⁸³ since the probability of success + the probability of failure is exactly 1 --- which means that the probability of success is 1 - probability of failure.

1/4 = 2.5 (25) (25%)

$${\frac{{\mathtt{4\,000}}}{{\mathtt{25}}}} = {\mathtt{160}}$$

What is this?

Hi Melody,

I was willing to answer that thinking it had to do with Permutations but I didn't want to risk getting it wrong.

I don't understand.

Are you talking about factors or number that literally go in between.

-12 / -4

If I'm correct it should be unless you Times it 0.5.

But that should be correct.