Answers (2)
x + y = 8
xy = -12
You have two lines. They look like this: www.wolframalpha.com/input/?i=solve+x+%2B+y+%3D+8,+xy+%3D+-12
Where they cross, the values of x and y satisfy both equations simultaneously, so it is called a simultaneous system. There are two ways to find that point. Substitution is when you solve one equation for one variable and substitute that into the other equation.
Elimination is when you add the equations in a way to eliminate one variable. Otherwise the process is the same.
The answer to this example is not simple.
You can write two equations:
x + y = 8
x * y = -12
Isolate y in equation 1
y = 8 - x
and substitute the y in equation 2 by 8 - x
x (8 - x) = -12
Now you have a nice quadratic equation:
-x^2 + 8x + 12 = 0 ;; times (-1)
x^2 - 8x - 12 = 0
or you isolate e.g. y in both equations
y = 8 - x
y = -12/x
and as on the right side of both equations is y, you set the left sides equal:
-12/x = 8 - x
-12 = x (8 - x)
-12 = -x^2 + 8x
x^2 - 8x - 12 = 0
parameterise:
p = -8
q = -12
and solve with the formula
x1,2 = -p/2 +- sqrt( (p/2)^2 - q)
x1 = - (-8)/2 + sqrt( (-8/2)^2 - (-12) )
x1 = 4 + sqrt( -4^2 + 12)
x1 = 4 + sqrt( 16 + 12)
x1 = 4 + sqrt(28)
x1 ≈ 9,2915026221291811810032315072785
x2 = 4 - sqrt(28) ;; the steps are the same
x2 ≈ -1,2915026221291811810032315072785
Plug these values in for x and y and there you are.