Solve the system of equations: 4x - 3y = 2 8x - 12y = 8

Let's solve the system of equations step by step. The given system of equations is: $$ \begin{cases} 4x - 3y = 2 \\ 8x - 12y = 8 \end{cases} $$ Step 1: Simplify the equations (if possible). We can divide the second equation by 4: $$ \frac{8x - 12y}{4} = \frac{8}{4} $$ $$ 2x - 3y = 2 $$ Now we have: $$ \begin{cases} 4x - 3y = 2 \\ 2x - 3y = 2 \end{cases} $$ Step 2: Eliminate one variable. Let's multiply the second equation by 2: $$ 2 * (2x - 3y) = 2 * 2 $$ $$ 4x - 6y = 4 $$ So now we have: $$ \begin{cases} 4x - 3y = 2 \\ 4x - 6y = 4 \end{cases} $$ Now subtract the second equation from the first equation: $$ (4x - 3y) - (4x - 6y) = 2 - 4 $$ $$ 4x - 3y - 4x + 6y = -2 $$ $$ 3y = -2 $$ $$ y = -\frac{2}{3} $$ Step 3: Substitute the value of y back into one of the original equations to solve for x. Using the first equation: $$ 4x - 3y = 2 $$ $$ 4x - 3(-\frac{2}{3}) = 2 $$ $$ 4x + 2 = 2 $$ $$ 4x = 0 $$ $$ x = 0 $$ Step 4: Check the solution. Substitute $$x = 0$$ and $$y = -\frac{2}{3}$$ into the original equations: First equation: $$ 4(0) - 3(-\frac{2}{3}) = 2 $$ $$ 0 + 2 = 2 $$ $$ 2 = 2 $$ Second equation: $$ 8(0) - 12(-\frac{2}{3}) = 8 $$ $$ 0 + 8 = 8 $$ $$ 8 = 8 $$ Both equations are satisfied. Final Answer: The solution to the system of equations is $$x = 0$$ and $$y = -\frac{2}{3}$$. Answer: x = 0, y = -2/3