Simultaneous Radical Equations!

Algebra Level 5

$\large \begin{aligned} x+\sqrt{y}&=11 \\ y+\sqrt{x}&=7 \end{aligned}$

Let $n$ be the number of real pairs $(x, y)$ to the above system of equations. Define these solution pairs as $(x_1, y_1), \ldots, (x_n, y_n)$ . Let

$S=\displaystyle \sum_{i=1}^n \left(x_i + y_i\right).$

Find $n+S$ .

The answer is 14.

2 solutions

Atomsky Jahid
Jun 23, 2016

Sketch or graph the two equations. You'll see they intersect only at one point. Some exploration will show that the point is (9, 4). Hence the answer is 1+9+4=14. [P.S. Also note that $x \leq 11$ and $y \leq 7$ which greatly decreases the checking work you have to do.]

Kartik Wadhwa
Jun 22, 2016

Put x=sec^2z and y=tan^2z and solve the equations. You will get only one solution satisfying the given equations i.e. x=9 and y=4. Hence the answer is 1+9+4 = 14

I don't understand how the substitution is justified. In effect you're assuming that x=y+1. Could you please explain?

Joe Mansley - 3 years ago

Simultaneous Radical Equations!

The answer is 14.

2 solutions

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