Question 15519
Solve the system of equations:

1) {{{x + y = -1}}}
2) {{{1/x + 1/y = 1/2}}} Simplify this equation.

{{{1/x + 1/y = 1/2}}} Add the fractions of the left side.
{{{(x + y)/xy = 1/2}}} Multiply both sides by xy
{{{x + y = xy/2}}} But {{{x + y = -1}}} so:
{{{xy/2 = -1}}} or {{{xy = -2}}}

Now you can solve this system of two equations:

1) {{{x + y = -1}}} and
2) {{{xy = -2}}}

Rewrite equation 1) as:
1) {{{x = -y-1}}} and substitute into equation 2)

{{{(-y-1)y = -2}}} Now solve for y.
{{{-y^2 - y = -2}}} Add 2 to both sides and simplify.
{{{y^2 + y - 2 = 0}}} Solve by factoring.
{{{(y - 1)(y + 2) = 0}}} Apply the zero products principle.
{{{y - 1 = 0}}} or {{{y + 2 = 0}}}

{{{y = 1}}} and/or {{{y = -2}}}

Now solve for x.

For y = 1, x = -y-1 = -1-1 = -2
For y = -2, x = -(-2)-1 = 2 - 1 = 1

The solutions are:

1) (-2, 1)
2) (1, -2)

Check the graph to see what this looks llike:

{{{graph(300, 200, -6,6,-6,6,-x-1,2x/(x-2))}}}