Solve for the variable x:
$\left\{\begin{array}{l}2 x-3 y=-14 \\ 6 x+y=8\end{array}\right.$

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Answer

Final Answer: The solution to the system of equations is $\boxed{x = \frac{1}{2}, y = 5}$

Steps

Step 1 ：Given the system of equations: $\begin{array}{l}2 x-3 y=-14 \ 6 x+y=8\end{array}$

Step 2 ：First, multiply the second equation by 3 to get $18x + 3y = 24$

Step 3 ：Add the two equations to eliminate y: $(2x - 3y) + (18x + 3y) = -14 + 24$, which simplifies to $20x = 10$

Step 4 ：Solve for x: $x = \frac{10}{20} = \frac{1}{2}$

Step 5 ：Substitute $x = \frac{1}{2}$ into the second equation: $6(\frac{1}{2}) + y = 8$, which simplifies to $3 + y = 8$

Step 6 ：Solve for y: $y = 8 - 3 = 5$

Step 7 ：Final Answer: The solution to the system of equations is $\boxed{x = \frac{1}{2}, y = 5}$