To modify the equation \(5x + 12 = x + 9\) so that it has no solution, we can manipulate it so that both sides become equal in a way that cannot occur.
For example, one way to do this is to create a false statement by ensuring the constants or coefficients lead to a contradiction.
A simple alteration could be: \[5x + 12 = x - 5.\]
This can be solved to see it has no solutions:
- Rearranging gives: \[5x - x + 12 + 5 = 0\] \[4x + 17 = 0\] \[4x = -17\] \[x = -\frac{17}{4}.\]
This equation is solvable, so let's try an alternative.
Instead, we can set the equation such that both sides are equal but don't actually exist, for example:
\[5x + 12 = 5.\]
This would imply the solution: \[5x + 12 = 5\] \[5x = 5 - 12\] \[5x = -7\] \[x = -\frac{7}{5}.\]
Final Adjustment
Now changing it to: \[5x + 12 = x + 12.\] In this case, reducing gives \(4x = 0\) so \(x = 0\), and we adjust to \[5x + 12 = 5x + 1.\]
Reduction leads to \(12 ≠ 1\), which is a contradiction thus proving: \[ 5x + 12 = 5x + 1. \] This equation has no solution because it simplifies to \(12 = 1\) which is false.