hectictar

I think I got it!!!!!!! 

Adding these three equations together will give us another true equation.

Like this...

ax + a + ay + by + bx + b + c + cy + cx  =  x + 2x + 4x + 6y + y + 7     Then factor it like this...

a(x + 1 + y) + b(y + x + 1) + c(1 + y + x)  =  7x + 7y + 7      Divide through by  (x + y + 1) .

a + b + c  =  \(\frac{7x+7y+7}{x+y+1}\)

a + b + c  =  \(\frac{7(x+y+1)}{x+y+1}\)

a + b + c  =  7

*edit*

Also... to show that it produces a true equation when you add them together...let's say...

A = B  ,   C = D ,  and  E = F                          (And  A, B, C, D, E, and F  can be expressions.)

A + C + E  =  A + C + E      Now we can substitute  B  in for  A ,  D  in for  C , and  F  in for  E .

A + C + E  =  B + D + F

I never could remember the distance formula either.....until I finally understood where it came from!

If     w  =  \(c\cdot\frac{y}{c}-y\)     , then this is how to do it.

c  =  2 · 10⁶

y  =  2 · 10⁵

\(w^2\,=\,c\cdot\frac{y}{c}-y\)                                      Plug in the given values for  c  and  y .

\(w^2\,=\,2\cdot10^6\cdot\frac{2\cdot10^5}{2\cdot10^6}-2\cdot10^5 \\ w^2\,=\, 2\cdot10^6\cdot\frac{10^5}{10^6}-2\cdot10^5 \\ w^2  \,=\,   2\cdot10^6\cdot\frac{1}{10}-2\cdot10^5 \\ w^2  \,=\,   2\cdot10^5-2\cdot10^5 \\ w^2\,=\,0\\w\,\ \,=\,0\)              Reduce the fraction by  2 .

Or....if     \(w^2\,=\,\frac{c\cdot y}{c-y} \)     , then this is how to do it.

\(w^2\,=\,\frac{2\cdot10^6\cdot2\cdot10^5}{2\cdot10^6-2\cdot10^5} \,=\, \frac{4\cdot10^{11}}{20\cdot10^5-2\cdot10^5} \,=\,\frac{4\cdot10^{11}}{18\cdot10^5}\,=\,\frac{2\cdot10^6}{9}\\~\\ w\,=\,\pm\sqrt{\frac{2\cdot10^6}{9}}\,=\,\pm\frac{10^3\sqrt2}{3}\,=\,\pm\frac{1000\sqrt2}{3}\)

The second one is the same answer as Guest's. 

That is a really good way to explain it!! 

What are you trying to do? What do the directions say?

cos 46°  =  b / 120

120 cos 46°  =  b

b  ≈  83.359  km / hr

1)   We want the ratio of width to height to be the same. If we multiply the height by 2, we must

      multiply the width by 2. If we multiply the height by  x , we must multiply the width by  x .

      The number that we multiply the height and width by is the scale factor.

\(\frac{\text{old width}}{\text{old height}}\,=\,\frac{\text{new width}}{\text{new height}} \\~\\ \frac{10}{8}\,=\,\frac{\text{new width}}{3.5} \\~\\ 3.5\,*\,\frac{10}{8}\,=\,\text{new width} \\~\\ 4.375\,=\,\text{new width, in inches}\)          Plug in the information from the problem.

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2)   This is done the same as the last question.

\(\frac{\text{old width}}{\text{old height}}=\frac{\text{new width}}{\text{new height}} \)          Plug in the information from the problem.

\(\frac{14}{10}=\frac{\text{new width}}{2}\)          Can you finish it from here?  

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3)   First let's convert 3 feet into inches.   1 ft  =  12 in   →   3 ft  =  36 in

Now let's call the unknown scale factor  " s " .

4.5s  =  36      Divide both sides by  4.5 .

     s  =  8

So each side of the triangle is multiplied by  8  .

The base of the larger triangle will then   =   (8 * 6) inches   =   48 inches   =   4 feet

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4)   If it takes 0.03 inches to make 1 mile on the map, then it will take 0.03 * 120 inches to make 120

      miles on the map...which is 3.6 inches.

I just noticed something that is kind of interesting!!

You could also have factored it as    (7m)(1 + 2m)(1 – 2m)

So...        ( – 7m)(2m + 1)(2m – 1)  =  (7m)(1 + 2m)(1 - 2m)

which means....                 – (a + b)(a – b)  =  (b + a)(b – a)      

Yeaa!! I got it right!!!!!!!!!!!

Let  a  be the number of species A ants he had to begin with.

Let  b  be the number of species B ants he had to begin with.

On day 0 there are  a + b ants.

On day 1 there are  2a + 3b  ants.

On day 2 there are  2 * 2a + 3 * 3b  ants.

On day 3 there are  2 * 2 * 2 * a + 3 * 3 * 3 * b  ants.

On day 4 there are  2⁴ * a + 3⁴ * b  ants.

On day n there are  2ⁿ * a + 3ⁿ * b

So...

On day 5 there are 2⁵ * a + 3⁵ * b

a + b  =  30     →     b = 30 - a

2⁵ * a  + 3⁵ * b  =  3281

2⁵ * a + 3⁵ * (30 - a)  =  3281

2⁵a + 3⁵ * 30 - 3⁵a  =  3281

2⁵a - 3⁵a  =  3281 - 3⁵ * 30

a(2⁵ - 3⁵)  =  3281 - 3⁵ * 30

a  =  (3281 - 3⁵ * 30) / (2⁵ - 3⁵)

a  =  19

So...he started with  19  ants of species A.

After  5 days, he will have  2⁵ * 19  =  608 ants of species A.