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 substitution method.
First, we can simplify the first equation:
-
\(3x + 6y = 21\) can be simplified by dividing everything by 3:
\[ x + 2y = 7 \]
So we can express \(x\) in terms of \(y\):
\[ x = 7 - 2y \] (Equation 1)
Next, we substitute \(x\) into the second equation:
-
Substitute \(x = 7 - 2y\) into \(-5x + 2y = 13\):
\[ -5(7 - 2y) + 2y = 13 \]
Distributing \(-5\):
\[ -35 + 10y + 2y = 13 \]
Combining like terms:
\[ 12y - 35 = 13 \]
Now, add 35 to both sides:
\[ 12y = 48 \]
Now, divide by 12:
\[ y = 4 \]
Now that we have \(y\), we can find \(x\) using Equation 1:
\[ x = 7 - 2(4) = 7 - 8 = -1 \]
Therefore, the solution to the system of equations is
\[ (x, y) = (-1, 4) \]
So the final answer is
\[ \boxed{(-1, 4)} \]