Option 1: The system has no solution
Option 2: The system has one solution
Option 3: The system has infinitely many solution.
Explanation:
Option 1: The equations are
and ![-4 x-2 y=30](https://tex.z-dn.net/?f=-4%20x-2%20y%3D30)
Solving using substitution method,
Let us substitute
in ![-4 x-2 y=30](https://tex.z-dn.net/?f=-4%20x-2%20y%3D30)
![\begin{array}{r}{-4 x-2(-12-2 x)=30} \\{-4 x+24+4 x=30} \\{24=30}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Br%7D%7B-4%20x-2%28-12-2%20x%29%3D30%7D%20%5C%5C%7B-4%20x%2B24%2B4%20x%3D30%7D%20%5C%5C%7B24%3D30%7D%5Cend%7Barray%7D)
Since, both sides of the equation are not equal. The system has no solution.
Option 2: The equations are
and ![x-3 y=5](https://tex.z-dn.net/?f=x-3%20y%3D5)
Solving using substitution method,
Let us substitute
in ![\frac{1}{3} y=\frac{7}{3} x+\frac{5}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B3%7D%20y%3D%5Cfrac%7B7%7D%7B3%7D%20x%2B%5Cfrac%7B5%7D%7B3%7D)
![\begin{aligned}\frac{1}{3} y &=\frac{7}{3}(5+3 y)+\frac{5}{3} \\\frac{1}{3} y &=\frac{35}{3}+7 y+\frac{5}{3} \\\frac{1}{3} y-7 y &=\frac{40}{3} \\-20 y &=40 \\y &=-2\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cfrac%7B1%7D%7B3%7D%20y%20%26%3D%5Cfrac%7B7%7D%7B3%7D%285%2B3%20y%29%2B%5Cfrac%7B5%7D%7B3%7D%20%5C%5C%5Cfrac%7B1%7D%7B3%7D%20y%20%26%3D%5Cfrac%7B35%7D%7B3%7D%2B7%20y%2B%5Cfrac%7B5%7D%7B3%7D%20%5C%5C%5Cfrac%7B1%7D%7B3%7D%20y-7%20y%20%26%3D%5Cfrac%7B40%7D%7B3%7D%20%5C%5C-20%20y%20%26%3D40%20%5C%5Cy%20%26%3D-2%5Cend%7Baligned%7D)
substituting y=-2 in
, we get,
![\begin{aligned}x &=5+3(-2) \\&=5-6 \\&=-1\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Dx%20%26%3D5%2B3%28-2%29%20%5C%5C%26%3D5-6%20%5C%5C%26%3D-1%5Cend%7Baligned%7D)
Thus, the system has one solution.
Option 3: The equations are
and ![3 x-3 y=21](https://tex.z-dn.net/?f=3%20x-3%20y%3D21)
Solving using substitution method,
Let us substitute
in ![3 x-3 y=21](https://tex.z-dn.net/?f=3%20x-3%20y%3D21)
![\begin{aligned}3 x-3(x-7) &=21 \\3 x-3 x+21 &=21 \\21 &=21\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D3%20x-3%28x-7%29%20%26%3D21%20%5C%5C3%20x-3%20x%2B21%20%26%3D21%20%5C%5C21%20%26%3D21%5Cend%7Baligned%7D)
Since, both sides of the equation are equal. Thus, the system has infinitely many solution.