Question

How do you solve more difficult simultaneous equations which involve indices?

Solve the simultaneous equations for x and y:

`1/sqrt(x+6) =3/sqrt(y)`

`2y+3x=3`

Thanks!

Answer 1

`color(blue)(x=-5 and y = 9)`

Explanation:

`1/(sqrt(x+6))=3/(sqrt(y)) \ \ \ \ [1]`

`2y+3x=3 \ \ \ \ [2]`

Starting with `[1]`

`sqrt(y)=3/(1/(sqrt(x+6)))=3sqrt(x+6)`

Squaring both sides:

`y=(3sqrt(x+6))^2=9x+54`

Substitute this value of y into `[2]`

`2(9x+54)+3x=3`

Solve for x:

`18x+108+3x=3`

`21x=-105`

`x=-105/21=-5`

Substituting in `[2]`

`2y+3(-5)=3`

`2y-15=3`

`2y=18`

`y=18/2=9`

So we have:

`x=-5` and `y=9`

As a note:

There is not really a general method for simultaneous equations involving indices. It will usually involve substitution as eliminating is usually not possible or practical.