Find the coordinates of the point of intersection of the straight line y=x+5 and the hyperbola xy=-6
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).
To find the coordinates of the point of intersection between the straight line and the hyperbola, we need to solve the system of equations formed by the equations of the line and the hyperbola.
Starting with the equations:
y = x + 5 (Equation 1)
xy = -6 (Equation 2)
Let's substitute the value of y from Equation 1 into Equation 2:
x(x+5) = -6
Expanding:
x^2 + 5x = -6
Rearranging:
x^2 + 5x + 6 = 0
The above equation is a quadratic equation. We can solve it by factoring:
(x + 2)(x + 3) = 0
Setting each factor to zero:
x + 2 = 0 or x + 3 = 0
Solving for x, we have:
x = -2 or x = -3
Now, let's substitute these x-values back into Equation 1 to find the corresponding y-values:
For x = -2:
y = -2 + 5
y = 3
For x = -3:
y = -3 + 5
y = 2
Therefore, the points of intersection are (-2, 3) and (-3, 2).