Answer:
![\mathrm{The\:solutions\:to\:the\:system\:of\:equations\:are:}](https://tex.z-dn.net/?f=%5Cmathrm%7BThe%5C%3Asolutions%5C%3Ato%5C%3Athe%5C%3Asystem%5C%3Aof%5C%3Aequations%5C%3Aare%3A%7D)
![x=2,\:y=1](https://tex.z-dn.net/?f=x%3D2%2C%5C%3Ay%3D1)
Step-by-step explanation:
Given the equations
![-5x-18y=-28](https://tex.z-dn.net/?f=-5x-18y%3D-28)
![-10x+9y=-11](https://tex.z-dn.net/?f=-10x%2B9y%3D-11)
solving the system of the equation by elimination method
![\begin{bmatrix}-5x-18y=-28\\ -10x+9y=-11\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D-5x-18y%3D-28%5C%5C%20-10x%2B9y%3D-11%5Cend%7Bbmatrix%7D)
![\mathrm{Multiply\:}-5x-18y=-28\mathrm{\:by\:}2\:\mathrm{:}\:\quad \:-10x-36y=-56](https://tex.z-dn.net/?f=%5Cmathrm%7BMultiply%5C%3A%7D-5x-18y%3D-28%5Cmathrm%7B%5C%3Aby%5C%3A%7D2%5C%3A%5Cmathrm%7B%3A%7D%5C%3A%5Cquad%20%5C%3A-10x-36y%3D-56)
![\begin{bmatrix}-10x-36y=-56\\ -10x+9y=-11\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D-10x-36y%3D-56%5C%5C%20-10x%2B9y%3D-11%5Cend%7Bbmatrix%7D)
![-10x+9y=-11](https://tex.z-dn.net/?f=-10x%2B9y%3D-11)
![-](https://tex.z-dn.net/?f=-)
![\underline{-10x-36y=-56}](https://tex.z-dn.net/?f=%5Cunderline%7B-10x-36y%3D-56%7D)
![45y=45](https://tex.z-dn.net/?f=45y%3D45)
![\begin{bmatrix}-10x-36y=-56\\ 45y=45\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D-10x-36y%3D-56%5C%5C%2045y%3D45%5Cend%7Bbmatrix%7D)
solve 45y=5 for y:
![45y=45](https://tex.z-dn.net/?f=45y%3D45)
![\frac{45y}{45}=\frac{45}{45}](https://tex.z-dn.net/?f=%5Cfrac%7B45y%7D%7B45%7D%3D%5Cfrac%7B45%7D%7B45%7D)
![y=1](https://tex.z-dn.net/?f=y%3D1)
![\mathrm{For\:}-10x-36y=-56\mathrm{\:plug\:in\:}y=1](https://tex.z-dn.net/?f=%5Cmathrm%7BFor%5C%3A%7D-10x-36y%3D-56%5Cmathrm%7B%5C%3Aplug%5C%3Ain%5C%3A%7Dy%3D1)
solve
for x:
![-10x-36\cdot \:1=-56](https://tex.z-dn.net/?f=-10x-36%5Ccdot%20%5C%3A1%3D-56)
![-10x-36=-56](https://tex.z-dn.net/?f=-10x-36%3D-56)
Add 36 to both sides
![-10x-36+36=-56+36](https://tex.z-dn.net/?f=-10x-36%2B36%3D-56%2B36)
![-10x=-20](https://tex.z-dn.net/?f=-10x%3D-20)
![\frac{-10x}{-10}=\frac{-20}{-10}](https://tex.z-dn.net/?f=%5Cfrac%7B-10x%7D%7B-10%7D%3D%5Cfrac%7B-20%7D%7B-10%7D)
![x=2](https://tex.z-dn.net/?f=x%3D2)
![\mathrm{The\:solutions\:to\:the\:system\:of\:equations\:are:}](https://tex.z-dn.net/?f=%5Cmathrm%7BThe%5C%3Asolutions%5C%3Ato%5C%3Athe%5C%3Asystem%5C%3Aof%5C%3Aequations%5C%3Aare%3A%7D)
Answer:
![\frac{d y}{d x} = - (2 x + 3 x^{2} )sin2(x^{2} +x^{3})](https://tex.z-dn.net/?f=%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20%3D%20%20-%20%282%20x%20%2B%203%20x%5E%7B2%7D%20%20%29sin2%28x%5E%7B2%7D%20%2Bx%5E%7B3%7D%29)
Step-by-step explanation:
<u><em>Step(i):-</em></u>
Given y = cos² (x² + x³) ....(i)
By using differentiation formulas
a) ![\frac{d}{dx} (cosx) = -sinx](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%20%28cosx%29%20%3D%20-sinx)
b) ![\frac{d}{dx} (x^{n} ) = n x ^{n-1}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%20%28x%5E%7Bn%7D%20%29%20%3D%20n%20x%20%5E%7Bn-1%7D)
<u><em>Step(ii):-</em></u>
Differentiating equation (i) with respective to 'x'
<em>First apply formula </em>
<em></em>
<em>Now we will apply formula </em>
![\frac{d}{dx} (cosx) = -sinx](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%20%28cosx%29%20%3D%20-sinx)
![\frac{d y}{d x} = 2 cos (x^{2} +x^{3} ) (-sin(x^{2} +x^{3})\frac{d}{dx} (x^{2} +x^{3} )](https://tex.z-dn.net/?f=%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20%3D%20%202%20cos%20%28x%5E%7B2%7D%20%2Bx%5E%7B3%7D%20%29%20%28-sin%28x%5E%7B2%7D%20%2Bx%5E%7B3%7D%29%5Cfrac%7Bd%7D%7Bdx%7D%20%28x%5E%7B2%7D%20%2Bx%5E%7B3%7D%20%29)
<em>Again apply formula </em>
<em></em>
![\frac{d y}{d x} = 2 cos (x^{2} +x^{3} ) (-sin(x^{2} +x^{3}) (2 x + 3 x^{2} )](https://tex.z-dn.net/?f=%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20%3D%20%202%20cos%20%28x%5E%7B2%7D%20%2Bx%5E%7B3%7D%20%29%20%28-sin%28x%5E%7B2%7D%20%2Bx%5E%7B3%7D%29%20%282%20x%20%2B%203%20x%5E%7B2%7D%20%20%29)
![\frac{d y}{d x} = -2 sin (x^{2} +x^{3} ) cos(x^{2} +x^{3}) (2 x + 3 x^{2} )](https://tex.z-dn.net/?f=%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20%3D%20%20-2%20sin%20%28x%5E%7B2%7D%20%2Bx%5E%7B3%7D%20%29%20cos%28x%5E%7B2%7D%20%2Bx%5E%7B3%7D%29%20%20%282%20x%20%2B%203%20x%5E%7B2%7D%20%20%29)
we know that trigonometric formulas
Sin 2θ = 2 sinθ cosθ
![\frac{d y}{d x} = -sin2(x^{2} +x^{3}) (2 x + 3 x^{2} )](https://tex.z-dn.net/?f=%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20%3D%20%20-sin2%28x%5E%7B2%7D%20%2Bx%5E%7B3%7D%29%20%282%20x%20%2B%203%20x%5E%7B2%7D%20%20%29)
<u><em>Final answer:-</em></u>
![\frac{d y}{d x} = - (2 x + 3 x^{2} )sin2(x^{2} +x^{3})](https://tex.z-dn.net/?f=%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20%3D%20%20-%20%282%20x%20%2B%203%20x%5E%7B2%7D%20%20%29sin2%28x%5E%7B2%7D%20%2Bx%5E%7B3%7D%29)
Second choice- multiply by 12
The answer would be 10.27 each person has to pay