Solve the system of equations:
24 + 4x = 16
44 + 2x = 20
We are given two linear equations in the variable \(x\):
1) \(24 + 4x = 16\)
2) \(44 + 2x = 20\)
Our goal is to determine if there exists a single value of \(x\) that satisfies both equations simultaneously. If such a value exists, we will find it; otherwise, we will conclude that the system of equations is inconsistent and has no solution.
We start by isolating \(x\) in the first equation:
\[24 + 4x = 16\]
Subtract 24 from both sides:
\[4x = 16 - 24\]
\[4x = -8\]
Divide by 4:
\[x = -2\]
So, the first equation is satisfied only when \(x = -2\).
Next, we isolate \(x\) in the second equation:
\[44 + 2x = 20\]
Subtract 44 from both sides:
\[2x = 20 - 44\]
\[2x = -24\]
Divide by 2:
\[x = -12\]
The second equation is satisfied only when \(x = -12\).
We have found that the first equation requires \(x = -2\) and the second equation requires \(x = -12\). Because these two required values are different, there is no value of \(x\) that satisfies both equations at the same time. Therefore, the system is inconsistent and has no solution. The solution set is empty.
In real-world applications, systems of equations are used to model constraints. For example, if two separate manufacturing processes are required to produce the same component, each process might have different requirements for a parameter like temperature or pressure. If the required temperature for process A is \(20^\circ\text{C}\) and process B requires \(10^\circ\text{C}\), you cannot run both processes optimally at the same temperature, implying the constraints are inconsistent for a shared environment setting. Understanding how to identify these conflicts is crucial for engineering and resource management.