Question

Solve the equations and check for the extraneous solutionsy = 15 - In (x-4)y=23

Answer 1

To answer this question we will use the following property:

`\ln a=b\text{ if and only if }e^b=a.`

Substituting the second equation in the first one we get:

`23=15-\ln(x-4).`

Adding ln(x-4) to the above equation we get:

`\begin{gathered} 23+\ln(x-4)=15-\ln(x-4)+\ln(x-4), \\ 23+\ln(x-4)=15. \end{gathered}`

Subtracting 23 from the above equation we get:

`\begin{gathered} 23+\ln(x-4)-23=15-23, \\ \ln(x-4)=-8. \end{gathered}`

Therefore:

`x-4=e^(-8).`

Adding 4 to the above equation we get:

`\begin{gathered} x-4+4=e^(-8)+4, \\ x=e^(-8)+4. \end{gathered}`

Since

`e^(-8)>0`

we get that x>4, therefore

`\ln(x-4)`

is well defined for

`x=e^(-8)+4.`

Answer:

`x=e^(-8)+4,\text{ y=23.}`