state,giving reason for your answer,whether the line y=8-x is a tangent to the curve 2xsquared + xy = -16
y = 8 - x
2x^2 + xy = -16
Substitute the 1st equation, y = 8 - x, into the 2nd equation,
2x^2 + x(8 - x) = -16
2x^2 + 8x - x^2 = -16
x^2 + 8x + 16 = 0
(x + 4)^2 = 0
x = -4 (twice)
There is a double root to the quadratic equation, i.e. x = -4. The same root occurs twice meaning that the straight line, y = 8 - x, only touches the curve 2x^2 +xy = -16 in one point only (It touches the same point twice).
Since this is a straight line that touches a curve at one point only, then that straight line is a tangent to the curve (at the point of touching).