Find the coordinates of the point of intersection of the straight line y=x+5 and the hyperbola xy=-6

Answer 1

To find the point of intersection, we need to solve the system of equations given by the straight line and the hyperbola:

1) y = x + 5
2) xy = -6

From equation 1), substitute y in equation 2):

x(x + 5) = -6

Expanding the left side:

x^2 + 5x = -6

Rearranging the equation:

x^2 + 5x + 6 = 0

We can factorize this quadratic equation:

(x + 2)(x + 3) = 0

Setting each factor to 0:

x + 2 = 0 or x + 3 = 0

Which gives us x = -2 or x = -3.

Substituting each value of x back into equation 1) to solve for y:

For x = -2:
y = -2 + 5 = 3

For x = -3:
y = -3 + 5 = 2

Therefore, the points of intersection are (-2, 3) and (-3, 2).