Use the substitution method to solve the system \[ \left\{\begin{array}{l} -x+y=-6 \\ 4 x-3 y=21 \end{array}\right. \]

Step 1 ：Solve the first equation for y in terms of x: \(y = x + 6\)

Step 2 ：Substitute \(y = x + 6\) into the second equation: \(4x - 3(x + 6) = 21\)

Step 3 ：Solve for x: \(x = 3\)

Step 4 ：Substitute \(x = 3\) back into the first equation to find y: \(-3 + y = -6\)

Step 5 ：Solve for y: \(y = -3\)

Step 6 ：Final Answer: The solution to the system of equations is \(\boxed{x = 3, y = -3}\)