To solve the system of equations:
- \( 3x + 4y = 48 \)
- \( 6x + 2y = 60 \)
We can use either the substitution method or the elimination method. Here, we'll use the elimination method.
First, let's manipulate the equations to make elimination easier. We notice that the second equation can be simplified by dividing all terms by 2:
\[ 3x + y = 30 \]
Now we have the system:
- \( 3x + 4y = 48 \)
- \( 3x + y = 30 \)
Next, we'll eliminate \(3x\) by subtracting the second equation from the first:
\[ (3x + 4y) - (3x + y) = 48 - 30 \]
This simplifies to:
\[ 4y - y = 18 \] \[ 3y = 18 \] \[ y = 6 \]
Now that we have \(y\), we can substitute it back into the second simplified equation to find \(x\):
\[ 3x + 6 = 30 \]
Subtract 6 from both sides:
\[ 3x = 24 \]
Now divide by 3:
\[ x = 8 \]
So the solution to the system of equations is:
\[ x = 8, \quad y = 6 \]
You can also check the solution by substituting \(x\) and \(y\) back into the original equations to ensure they hold true:
- \( 3(8) + 4(6) = 24 + 24 = 48 \) (True)
- \( 6(8) + 2(6) = 48 + 12 = 60 \) (True)
Both equations are satisfied, confirming that the solution is correct:
\[ (x, y) = (8, 6) \]