To solve the system of equations:
- \( 3x + 6y = 21 \)
- \( -5x + 2y = 13 \)
We can use the substitution or elimination method. Here, we'll use the elimination method.
First, let's simplify the first equation:
- From the first equation: \[ 3x + 6y = 21 \] Divide everything by 3: \[ x + 2y = 7 \tag{1} \]
Now we have a simpler first equation:
- \( x + 2y = 7 \)
Next, let's express \( x \) in terms of \( y \) from equation (1): \[ x = 7 - 2y \tag{2} \]
Now, we can substitute equation (2) into the second equation \( -5x + 2y = 13 \): \[ -5(7 - 2y) + 2y = 13 \]
Expanding this, we get: \[ -35 + 10y + 2y = 13 \] Combining like terms: \[ 12y - 35 = 13 \]
Now, add 35 to both sides: \[ 12y = 48 \]
Dividing both sides by 12: \[ y = 4 \]
Now that we have the value of \( y \), we can substitute it back into equation (2) to find \( x \): \[ x = 7 - 2(4) \] \[ x = 7 - 8 \] \[ x = -1 \]
Thus, the solution to the system of equations is: \[ (x, y) = (-1, 4) \]