System of equations Find the value of $k$.

Question

for what value of $k \in \mathbb{R}$ does there exist exactly one pair $(x, y) \in \mathbb{R}^2$ that satisfies the simultaneous equations

$$y-14x=k$$ $$x^2 +y = 5$$

Answer 1

$$y-14x=k$$ $$x^2+y=5$$ By the former, $$y=14x+k$$ Substitute it into the latter, $$x^2+14x+(k-5)=0$$ As you want the value of $x$ to be unique, then the discriminant $$\Delta=14^2-4k+20=216-4k$$ Must be $0$.

Thus, $$4k=216$$ $$k=54$$

Answer 2

from the first equation we get $$y=14x+k$$ plugging this in the second equation we get $$x^2+14x+k-5=0$$ solving this we get $$x_{1,2}=-7\pm \sqrt{54-k}$$ we get only one solution if $$k=54$$

Answer 3

You can eliminate $y$ from the second equation giving $$x^2+14x+k=5.$$ Completing the square $$(x+7)^2=54-k.$$ For which $k$ has this quadratic repeated roots?