Answer:
C, 7 1/2
Step-by-step explanation:
12 is 2/3 of 18, therefore 5 is 2/3 of something, so you divide 5 by 2/3 which is 15/2 which is 7 1/2
Answer:
<em>Proof in explanation</em>
Step-by-step explanation:
<u>Trigonometric Identities</u>
The basic trigonometric identity is:
![\sin^2\theta+\cos^2\theta=1](https://tex.z-dn.net/?f=%5Csin%5E2%5Ctheta%2B%5Ccos%5E2%5Ctheta%3D1)
We'll use it and some basic algebra to prove that, given:
![a sin^3\theta+b cos^3\theta=sin\theta cos\theta](https://tex.z-dn.net/?f=a%20sin%5E3%5Ctheta%2Bb%20cos%5E3%5Ctheta%3Dsin%5Ctheta%20cos%5Ctheta)
And
![a\sin\theta-b\cos\theta=0](https://tex.z-dn.net/?f=a%5Csin%5Ctheta-b%5Ccos%5Ctheta%3D0)
Then
![a^2+b^2=1](https://tex.z-dn.net/?f=a%5E2%2Bb%5E2%3D1)
From the equation:
![a\sin\theta-b\cos\theta=0](https://tex.z-dn.net/?f=a%5Csin%5Ctheta-b%5Ccos%5Ctheta%3D0)
We have:
![a\sin\theta=b\cos\theta\qquad [1]](https://tex.z-dn.net/?f=a%5Csin%5Ctheta%3Db%5Ccos%5Ctheta%5Cqquad%20%5B1%5D)
The equation
![a sin^3\theta+b cos^3\theta=sin\theta cos\theta](https://tex.z-dn.net/?f=a%20sin%5E3%5Ctheta%2Bb%20cos%5E3%5Ctheta%3Dsin%5Ctheta%20cos%5Ctheta)
Can be rewritten as
![a\sin\theta \sin^2\theta+b \cos^3\theta=\sin\theta \cos\theta](https://tex.z-dn.net/?f=a%5Csin%5Ctheta%20%5Csin%5E2%5Ctheta%2Bb%20%5Ccos%5E3%5Ctheta%3D%5Csin%5Ctheta%20%5Ccos%5Ctheta)
Replacing [1]:
![b\cos\theta \sin^2\theta+b \cos^3\theta=\sin\theta \cos\theta](https://tex.z-dn.net/?f=b%5Ccos%5Ctheta%20%5Csin%5E2%5Ctheta%2Bb%20%5Ccos%5E3%5Ctheta%3D%5Csin%5Ctheta%20%5Ccos%5Ctheta)
Taking the common factor:
![b\cos\theta (\sin^2\theta+ \cos^2\theta)=\sin\theta \cos\theta](https://tex.z-dn.net/?f=b%5Ccos%5Ctheta%20%28%5Csin%5E2%5Ctheta%2B%20%5Ccos%5E2%5Ctheta%29%3D%5Csin%5Ctheta%20%5Ccos%5Ctheta)
The expression in parentheses is 1, thus:
![b\cos\theta =\sin\theta \cos\theta](https://tex.z-dn.net/?f=b%5Ccos%5Ctheta%20%3D%5Csin%5Ctheta%20%5Ccos%5Ctheta)
Dividing by ![\cos\theta](https://tex.z-dn.net/?f=%5Ccos%5Ctheta)
![b=\sin\theta](https://tex.z-dn.net/?f=b%3D%5Csin%5Ctheta)
Replacing in
![a\sin\theta=b\cos\theta](https://tex.z-dn.net/?f=a%5Csin%5Ctheta%3Db%5Ccos%5Ctheta)
We have
![a\sin\theta=\sin\theta\cos\theta](https://tex.z-dn.net/?f=a%5Csin%5Ctheta%3D%5Csin%5Ctheta%5Ccos%5Ctheta)
Dividing by ![\sin\theta](https://tex.z-dn.net/?f=%5Csin%5Ctheta)
![a=\cos\theta](https://tex.z-dn.net/?f=a%3D%5Ccos%5Ctheta)
Now:
![a^2+b^2=(\cos\theta)^2+(\sin\theta)^2](https://tex.z-dn.net/?f=a%5E2%2Bb%5E2%3D%28%5Ccos%5Ctheta%29%5E2%2B%28%5Csin%5Ctheta%29%5E2)
This expression is 1, thus it's proven:
![\boxed{a^2+b^2=1}](https://tex.z-dn.net/?f=%5Cboxed%7Ba%5E2%2Bb%5E2%3D1%7D)
Answer:
![x=-1\\y=-1](https://tex.z-dn.net/?f=x%3D-1%5C%5Cy%3D-1)
Step-by-step explanation:
![5x+5y=-10\\3x-7y=4](https://tex.z-dn.net/?f=5x%2B5y%3D-10%5C%5C3x-7y%3D4)
Let's solve the first equation for either x or y. I'll do it for x.
![5x+5y=-10](https://tex.z-dn.net/?f=5x%2B5y%3D-10)
Begin by subtracting 5y.
![5x=-5y-10](https://tex.z-dn.net/?f=5x%3D-5y-10)
Now divide by 5.
![x=\frac{-5y}{5}-\frac{10}{5}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-5y%7D%7B5%7D-%5Cfrac%7B10%7D%7B5%7D)
Simplify:
![x=-y-2](https://tex.z-dn.net/?f=x%3D-y-2)
Now substitute x in the second equation for this value.
![3x-7y=4\\3(-y-2)-7y=4](https://tex.z-dn.net/?f=3x-7y%3D4%5C%5C3%28-y-2%29-7y%3D4)
Distribute;
![-3y-6-7y=4](https://tex.z-dn.net/?f=-3y-6-7y%3D4)
Add 6
![-3y-7y=4+6](https://tex.z-dn.net/?f=-3y-7y%3D4%2B6)
Combine like terms;
![-10y=10](https://tex.z-dn.net/?f=-10y%3D10)
Divide by -10.
![y=\frac{10}{-10}\\ y=-1](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B10%7D%7B-10%7D%5C%5C%20y%3D-1)
-----------------------------------------------------------------------------------------------------------
Take this value of y and replace it in the first equation to find the value of x.
![5x+5y=-10\\5x+5(-1)=-10\\5x-5=-10\\5x=-10+5\\5x=-5\\x=\frac{-5}{5}\\ x=-1](https://tex.z-dn.net/?f=5x%2B5y%3D-10%5C%5C5x%2B5%28-1%29%3D-10%5C%5C5x-5%3D-10%5C%5C5x%3D-10%2B5%5C%5C5x%3D-5%5C%5Cx%3D%5Cfrac%7B-5%7D%7B5%7D%5C%5C%20x%3D-1)
Answer:
hhhhhhhhhhdbfhrkuvn
Step-by-step explanation:
For every 9 goldfish, there are 4 gallons of water.