To determine the number of solutions for the given system of equations, we can manipulate the equations and analyze their relationships. The system of equations is:
- Equation 1: x + 2y = 24
- Equation 2: 3x – 6y = 72
First, let’s simplify Equation 2. We can divide every term by 3:
3x – 6y = 72
=> x – 2y = 24
Now, we can rewrite this as:
- Equation 2 (Simplified): x – 2y = 24
At this point, we have the two equations:
- x + 2y = 24
- x – 2y = 24
Now we can analyze the two equations:
1. The first equation, x + 2y = 24, can be rearranged to find y:
y = (24 – x)/2
2. The second equation, x – 2y = 24, can also be rearranged:
y = (x – 24)/2
As we can observe, both equations can yield different values for y depending on x. However, if we set the values of y from both equations equal to each other, we can see whether there is any overlap:
(24 – x)/2 = (x – 24)/2
Multiplying both sides by 2 gives us:
24 – x = x – 24
Now, let’s solve for x:
24 + 24 = x + x
=> 48 = 2x
=> x = 24
By substituting x = 24 back into either original equation, we will find:
x + 2y = 24
=> 24 + 2y = 24
=> 2y = 0
=> y = 0
Thus, we have found one specific solution: (24, 0).
Since both equations represent straight lines that intersect at exactly one point, we conclude that:
Conclusion:
The given system of equations has exactly one unique solution.