To determine how many solutions the given system of equations has, we first need to express each equation in a more standard form.
1. From the first equation, we have:
2y = 24
If we solve for y, we divide both sides by 2:
y = 12
Now we know that for any value of x, y will always be 12.
2. Now let’s scrutinize the second equation:
3x – 6y = 72
If we substitute y = 12 into this second equation:
3x – 6(12) = 72
3x – 72 = 72
Next, we add 72 to both sides:
3x = 144
Dividing both sides by 3 gives us:
x = 48
Now we have a specific solution: x = 48 and y = 12.
In conclusion, the system of equations has exactly one solution because the two variables can be uniquely determined within the context of the given equations.