Simple solution to system of three equations

Question

I've been given the question;

$$xy = \frac19$$ $$x(y+1) = \frac79$$ $$y(x+1) = \frac5{18}$$ What is the value of $(x+1)(y+1)$?

Of course, you could solve for $x$ and $y$, then substitute in the values. However, my teacher says there is a quick solution that only requires $2$ lines to solve.

How can I solve $(x+1)(y+1)$ without finding $x$ and $y$, given the values above?

Answer 1

If you multiply the second and third equation you will have $xy(y+1)(x+1)$ and you are given a value for $xy$. Simply divide by $xy$ and you are done.

Answer 2

Multiply the second by third then substitute the first.

Answer 3

$$(x+1)=\frac5{18}\cdot\frac1y$$ $$(y+1)=\frac7{9}\cdot\frac1x$$ $$(x+1)(y+1)=\frac5{18}\cdot\frac1y\cdot\frac7{9}\cdot\frac1x=\frac{35}{162}\cdot9=\frac{35}{18}$$

Answer 4

$$2xy+x+y=\frac{19}{18}$$

$$xy+x+y+1=(x+1)(y+1)=\frac{17}{18}+1=\frac{35}{18}$$

And this comes from combining the second and third equations, using the first equation to reduce the $2xy$ term to $xy$ and then adding one to both sides of the equality so that it matches what you are asked to find.

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The other obvious solution:

since $xy=\frac{1}{9}$ you can do this

$$X(y+1)=\frac{7}{9}$$ $$xy+x=\frac{7}{9}$$ $$\frac{1}{9}+x=\frac{7}{9}$$

$$x=\frac{2}{3}$$

use this fact and find y:

$$\frac{2}{3}y=\frac{1}{9}$$

and then solving is trivial

Answer 5

$$ (x+1)(y+1)=x(y+1)+y(x+1)-xy+1 $$

Answer 6

Without using any multiplications or divisions (except to bring the fractions to a common denominator): $$(x+1)(y+1)=x(y+1)+y(x+1)-xy+1 =\frac{14+5-2+18}{18} = \frac{35}{18}. $$