Notice that triangles XYZ and WYZ are congruent because of the SAS postulate.
Then,
[tex]\begin{gathered} 5x-5=3x+9 \\ \text{and} \\ 4x+6=5x-1 \end{gathered}[/tex]
Because the pairs of corresponding sides are XZ-ZW and XY-YW.
Therefore, we cannot conclude that
[tex]XY=ZW[/tex]
On the other hand, the algebra used by Lydia has no mistakes.
[tex]\begin{gathered} 4x+6=3x+9 \\ \Rightarrow-3x+4x+6=-3x+3x+9 \\ \Rightarrow x+6=9 \\ \Rightarrow x=3 \end{gathered}[/tex]
So, Lydia is incorrect when setting the equation, the algebraical operations are correct.
The two valid equations are:
[tex]\begin{gathered} 5x-5=3x+9 \\ or \\ 4x+6=5x-1 \end{gathered}[/tex]
We can even combine both equations and obtain
[tex]\begin{gathered} 4x+6-4=3x+9 \\ \Rightarrow4x+2=3x+9 \end{gathered}[/tex]
The answer is:
Lydia is wrong because the initial equation is false. The two possible equations are 5x-5=3x+9 or 4x+6=5x-1.
