\[{\color{Magenta} t'}=\frac{x-vt}{\sqrt{c^{2}-v^{2}}}=\frac{(x-vt):c}{(\sqrt{c^{2}-v^{2}}):c}=\frac{t-\frac{vt}{c}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{t-\frac{vtc}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{t-\frac{vx}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\]