Question

Find an equation of the tangent line to (x+y)^3 -5x+y=1 at the point where the curve intersects the line x+y=1

How am I supposed to determine what the point is? Im guessing system of equations but i can only think of one equation..

Answer 1

The system would be
\[\begin{cases}(x+y)^3-5x+y=1\\
x+y=1\end{cases}\]
Solve by substitution: \(x+y=1~~\iff~~x=1-y\)
\[(1-y+y)^3-5(1-y)+y=1\\
1-5(1-y)+y=1\\
y=5-5y\\
6y=5\\
y=\frac{5}{6}~~\Rightarrow~~x=\frac{1}{6}\]
The point of intersection is thus \(\left(\dfrac{1}{6},\dfrac{5}{6}\right)\).

Answer 2

So from here, you would find the derivative of the given curve at this point, then find the equation of the tangent line.

Answer 3

Oh thank you very much! I feel stupid I totally overlooked that. Thankss!