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!

1 Answer
May 11, 2018

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.