To solve the system of equations
- \( 3x + 6y = 21 \)
- \( -5x + 2y = 13 \)
we can use either the substitution method or the elimination method. Here, I'll use the elimination method for clarity.
First, simplify the first equation:
\[ 3x + 6y = 21 \]
Dividing the entire equation by 3:
\[ x + 2y = 7 \quad \text{(Equation 1)} \]
So, we have two simplified equations:
- \( x + 2y = 7 \)
- \( -5x + 2y = 13 \)
Next, let's eliminate \( y \) by subtracting the first equation from the second equation:
\[ (-5x + 2y) - (x + 2y) = 13 - 7 \]
Simplifying:
\[ -5x + 2y - x - 2y = 6 \]
This reduces to:
\[ -6x = 6 \]
Dividing both sides by -6 gives:
\[ x = -1 \]
Now, we can substitute \( x = -1 \) back into Equation 1 to find \( y \):
\[ -1 + 2y = 7 \]
Adding 1 to both sides:
\[ 2y = 8 \]
Dividing by 2:
\[ y = 4 \]
Thus, the solution to the system of equations is:
\[ (x, y) = (-1, 4) \]
We can summarize the answer as:
\[ \boxed{(-1, 4)} \]