Answer:
yyaayayyaay
Step-by-step explanation:
X=hours of plan A
y=hours of plan B
wed: 5x+6y=12
thu: 3x+2y=6
so
5x+6y=12 and
3x+2y=6
we can eliminate y's by multiplying 2nd equation by -3 and adding to first equation
-9x-6y=-18
<u>5x+6y=12 +</u>
-4x+0y=-6
-4x=-6
divid both sides by -4
x=-6/-4
x=3/2
x=1.5
sub back
3x+2y=6
3(1.5)+2y=6
4.5+2y=6
2y=1.5
y=0.75
Plan A=1.5 hours
Plan B=0.75 hours
Use the change-of-basis identity,
![\log_x(y) = \dfrac{\ln(y)}{\ln(x)}](https://tex.z-dn.net/?f=%5Clog_x%28y%29%20%3D%20%5Cdfrac%7B%5Cln%28y%29%7D%7B%5Cln%28x%29%7D)
to write
![xyz = \log_a(bc) \log_b(ac) \log_c(ab) = \dfrac{\ln(bc) \ln(ac) \ln(ab)}{\ln(a) \ln(b) \ln(c)}](https://tex.z-dn.net/?f=xyz%20%3D%20%5Clog_a%28bc%29%20%5Clog_b%28ac%29%20%5Clog_c%28ab%29%20%3D%20%5Cdfrac%7B%5Cln%28bc%29%20%5Cln%28ac%29%20%5Cln%28ab%29%7D%7B%5Cln%28a%29%20%5Cln%28b%29%20%5Cln%28c%29%7D)
Use the product-to-sum identity,
![\log_x(yz) = \log_x(y) + \log_x(z)](https://tex.z-dn.net/?f=%5Clog_x%28yz%29%20%3D%20%5Clog_x%28y%29%20%2B%20%5Clog_x%28z%29)
to write
![xyz = \dfrac{(\ln(b) + \ln(c)) (\ln(a) + \ln(c)) (\ln(a) + \ln(b))}{\ln(a) \ln(b) \ln(c)}](https://tex.z-dn.net/?f=xyz%20%3D%20%5Cdfrac%7B%28%5Cln%28b%29%20%2B%20%5Cln%28c%29%29%20%28%5Cln%28a%29%20%2B%20%5Cln%28c%29%29%20%28%5Cln%28a%29%20%2B%20%5Cln%28b%29%29%7D%7B%5Cln%28a%29%20%5Cln%28b%29%20%5Cln%28c%29%7D)
Redistribute the factors on the left side as
![xyz = \dfrac{\ln(b) + \ln(c)}{\ln(b)} \times \dfrac{\ln(a) + \ln(c)}{\ln(c)} \times \dfrac{\ln(a) + \ln(b)}{\ln(a)}](https://tex.z-dn.net/?f=xyz%20%3D%20%5Cdfrac%7B%5Cln%28b%29%20%2B%20%5Cln%28c%29%7D%7B%5Cln%28b%29%7D%20%5Ctimes%20%5Cdfrac%7B%5Cln%28a%29%20%2B%20%5Cln%28c%29%7D%7B%5Cln%28c%29%7D%20%5Ctimes%20%5Cdfrac%7B%5Cln%28a%29%20%2B%20%5Cln%28b%29%7D%7B%5Cln%28a%29%7D)
and simplify to
![xyz = \left(1 + \dfrac{\ln(c)}{\ln(b)}\right) \left(1 + \dfrac{\ln(a)}{\ln(c)}\right) \left(1 + \dfrac{\ln(b)}{\ln(a)}\right)](https://tex.z-dn.net/?f=xyz%20%3D%20%5Cleft%281%20%2B%20%5Cdfrac%7B%5Cln%28c%29%7D%7B%5Cln%28b%29%7D%5Cright%29%20%5Cleft%281%20%2B%20%5Cdfrac%7B%5Cln%28a%29%7D%7B%5Cln%28c%29%7D%5Cright%29%20%5Cleft%281%20%2B%20%5Cdfrac%7B%5Cln%28b%29%7D%7B%5Cln%28a%29%7D%5Cright%29)
Now expand the right side:
![xyz = 1 + \dfrac{\ln(c)}{\ln(b)} + \dfrac{\ln(a)}{\ln(c)} + \dfrac{\ln(b)}{\ln(a)} \\\\ ~~~~~~~~~~~~+ \dfrac{\ln(c)\ln(a)}{\ln(b)\ln(c)} + \dfrac{\ln(c)\ln(b)}{\ln(b)\ln(a)} + \dfrac{\ln(a)\ln(b)}{\ln(c)\ln(a)} \\\\ ~~~~~~~~~~~~ + \dfrac{\ln(c)\ln(a)\ln(b)}{\ln(b)\ln(c)\ln(a)}](https://tex.z-dn.net/?f=xyz%20%3D%201%20%2B%20%5Cdfrac%7B%5Cln%28c%29%7D%7B%5Cln%28b%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28a%29%7D%7B%5Cln%28c%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28b%29%7D%7B%5Cln%28a%29%7D%20%5C%5C%5C%5C%20~~~~~~~~~~~~%2B%20%5Cdfrac%7B%5Cln%28c%29%5Cln%28a%29%7D%7B%5Cln%28b%29%5Cln%28c%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28c%29%5Cln%28b%29%7D%7B%5Cln%28b%29%5Cln%28a%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28a%29%5Cln%28b%29%7D%7B%5Cln%28c%29%5Cln%28a%29%7D%20%5C%5C%5C%5C%20~~~~~~~~~~~~%20%2B%20%5Cdfrac%7B%5Cln%28c%29%5Cln%28a%29%5Cln%28b%29%7D%7B%5Cln%28b%29%5Cln%28c%29%5Cln%28a%29%7D)
Simplify and rewrite using the logarithm properties mentioned earlier.
![xyz = 1 + \dfrac{\ln(c)}{\ln(b)} + \dfrac{\ln(a)}{\ln(c)} + \dfrac{\ln(b)}{\ln(a)} + \dfrac{\ln(a)}{\ln(b)} + \dfrac{\ln(c)}{\ln(a)} + \dfrac{\ln(b)}{\ln(c)} + 1](https://tex.z-dn.net/?f=xyz%20%3D%201%20%2B%20%5Cdfrac%7B%5Cln%28c%29%7D%7B%5Cln%28b%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28a%29%7D%7B%5Cln%28c%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28b%29%7D%7B%5Cln%28a%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28a%29%7D%7B%5Cln%28b%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28c%29%7D%7B%5Cln%28a%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28b%29%7D%7B%5Cln%28c%29%7D%20%2B%201)
![xyz = 2 + \dfrac{\ln(c)+\ln(a)}{\ln(b)} + \dfrac{\ln(a)+\ln(b)}{\ln(c)} + \dfrac{\ln(b)+\ln(c)}{\ln(a)}](https://tex.z-dn.net/?f=xyz%20%3D%202%20%2B%20%5Cdfrac%7B%5Cln%28c%29%2B%5Cln%28a%29%7D%7B%5Cln%28b%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28a%29%2B%5Cln%28b%29%7D%7B%5Cln%28c%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28b%29%2B%5Cln%28c%29%7D%7B%5Cln%28a%29%7D)
![xyz = 2 + \dfrac{\ln(ac)}{\ln(b)} + \dfrac{\ln(ab)}{\ln(c)} + \dfrac{\ln(bc)}{\ln(a)}](https://tex.z-dn.net/?f=xyz%20%3D%202%20%2B%20%5Cdfrac%7B%5Cln%28ac%29%7D%7B%5Cln%28b%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28ab%29%7D%7B%5Cln%28c%29%7D%20%2B%20%5Cdfrac%7B%5Cln%28bc%29%7D%7B%5Cln%28a%29%7D)
![xyz = 2 + \log_b(ac) + \log_c(ab) + \log_a(bc)](https://tex.z-dn.net/?f=xyz%20%3D%202%20%2B%20%5Clog_b%28ac%29%20%2B%20%5Clog_c%28ab%29%20%2B%20%5Clog_a%28bc%29)
![\implies \boxed{xyz = x + y + z + 2}](https://tex.z-dn.net/?f=%5Cimplies%20%5Cboxed%7Bxyz%20%3D%20x%20%2B%20y%20%2B%20z%20%2B%202%7D)
(C)
Answer:
To write a two-variable equation, I would first need to know how much Maya’s allowance was. Then, I would need the cost of playing the arcade game and of riding the Ferris wheel. I could let the equation be cost of playing the arcade games plus cost of riding the Ferris wheel equals the total allowance. My variables would represent the number of times Maya played the arcade game and the number of times she rode the Ferris wheel. With this equation I could solve for how many times she rode the Ferris wheel given the number of times she played the arcade game.
Step-by-step explanation:
y² = 8y - 15 (alternate angles are equal)
y² - 8y + 15 = 0
(y - 5)(y - 3) = 0
y = 5 or 3
x + 8y - 15 = 180 (angles in a straight line add up to 180)
when y = 5
x + 40 - 15 = 180
x = 155°
when y = 3
x + 24 - 15 = 180
x = 171°