- Rearrange the given equations into the standard linear form \(Ax + By = C\).
- Observe that both equations result in the same expression for \(x\) and \(y\), specifically \(9x - 7y\), but equal different constants (\(27\) and \(45\)).
- Deduce that the equations are inconsistent and represent parallel, non-intersecting lines.
- Conclude that the system has \(\boxed{\text{no solution.}}\)
Explanation
- Rewriting the equations in standard form
To determine the nature of the solution set for the given system, we start by expressing both equations in the standard form \(Ax + By = C\). The given system is:
\[ \begin{cases} 27 - 9x = -7y \ 45 + 7y = 9x \end{cases} \]
For the first equation, \(27 - 9x = -7y\), we can add \(9x\) to both sides to get \(27 = 9x - 7y\), which is equivalent to:
\[9x - 7y = 27\]
For the second equation, \(45 + 7y = 9x\), we can subtract \(7y\) from both sides to get \(45 = 9x - 7y\), which is equivalent to:
\[9x - 7y = 45\]
- Analyzing the system for solutions
Now we compare the two equations:
1) \(9x - 7y = 27\)
2) \(9x - 7y = 45\)
We can see that the left-hand sides of both equations are identical: \(9x - 7y\). However, the right-hand sides are different constants, \(27\) and \(45\). This implies that there is no pair \((x, y)\) that can satisfy both equations simultaneously, because the same expression \(9x - 7y\) cannot be equal to both \(27\) and \(45\) at the same time. Since \(27
eq 45\), the system is inconsistent. This means the two lines represented by these equations are parallel and do not intersect. Therefore, the system has no solution.
- Conclusion
Based on our analysis, the system of equations has no solution because the two equations lead to a contradiction. Among the given options, "The system has no solution" is the correct description.
The final answer is \(\boxed{\text{The system has no solution.}}\)
Examples
In a real-life scenario, imagine two trains running on parallel tracks. If you represent the position of the trains as functions of time, you might set up a system of equations to find when they meet. If your equations result in a contradiction, such as '10 = 20', it physically means the trains are on tracks that never cross, so they will never meet. Identifying such inconsistencies is crucial for scheduling and safety in transportation and logistics, ensuring that paths are correctly planned.