Answer:
The solution to the system of equations be:
[tex]y=4,\:x=0[/tex]
As the consistent system of equations has only one solution, it is independent.
Step-by-step explanation:
Given the system of equations
[tex]3y + 2x= 12[/tex]
[tex]36 - 9y = -6x[/tex]
solving the system of equations
[tex]\begin{bmatrix}3y+2x=12\\ 36-9y=-6x\end{bmatrix}[/tex]
Arrange equation variables for elimination
[tex]\begin{bmatrix}3y+2x=12\\ -9y+6x=-36\end{bmatrix}[/tex]
[tex]\mathrm{Multiply\:}3y+2x=12\mathrm{\:by\:}3\:\mathrm{:}\:\quad \:9y+6x=36[/tex]
[tex]\begin{bmatrix}9y+6x=36\\ -9y+6x=-36\end{bmatrix}[/tex]
so
[tex]-9y+6x=-36[/tex]
[tex]+[/tex]
[tex]\underline{9y+6x=36}[/tex]
[tex]12x=0[/tex]
[tex]\begin{bmatrix}9y+6x=36\\ 12x=0\end{bmatrix}[/tex]
solve 12x=0 for x
[tex]12x=0[/tex]
Divide both sides by 12
[tex]\frac{12x}{12}=\frac{0}{12}[/tex]
[tex]x=0[/tex]
[tex]\mathrm{For\:}9y+6x=36\mathrm{\:plug\:in\:}x=0[/tex]
[tex]9y+6\cdot \:0=36[/tex]
[tex]9y=36[/tex]
Divide both sides by 9
[tex]\frac{9y}{9}=\frac{36}{9}[/tex]
[tex]y=4[/tex]
Thus, the solution to the system of equations be:
[tex]y=4,\:x=0[/tex]
As the consistent system of equations has only one solution, it is independent.
