Answer: (0,3) in the top right corner
There is only one solution to this system of equations.
The graph is shown below.
============================================================
Explanation:
Because we're given a list of choices, we can plug in each (x,y) coordinates into each equation. The goal is to see if we get true or equations or not. True equations have the same number on both sides after fully simplifying.
Let's try (x,y) = (4,1) and we'll plug this into the first equation.
[tex]y = \sqrt{9-x}\\\\1 = \sqrt{9-4}\\\\1 = \sqrt{5}\\\\[/tex]
The last equation is false, so the first equation must be false for those x,y values. We don't need to check the other equation. We have enough to rule out any answer choice involving (4,1). So that means we can cross off the two answer choices on the left side.
---------------------
Let's try (x,y) = (0,3) in the first equation
[tex]y = \sqrt{9-x}\\\\3 = \sqrt{9-0}\\\\3 = \sqrt{9}\\\\3 = 3\\\\[/tex]
We get a true equation. So far, so good.
Now we need to check the other equation
[tex]x+2y = 6\\\\0+2*3 = 6\\\\6 = 6\\\\[/tex]
That works as well. Therefore, the point (0,3) is confirmed to be a solution to this system. Since the top right corner is the only thing remaining that has (0,3), this must be the final answer. So we can stop here.
---------------------
For the sake of completeness, let's try (8,1)
[tex]y = \sqrt{9-x}\\\\1 = \sqrt{9-8}\\\\1 = \sqrt{1}\\\\1 = 1\\\\[/tex]
Looking good so far. Now the second equation
[tex]x+2y = 6\\\\8+2*1 = 6\\\\10 = 6\\\\[/tex]
which is false. The point (8,1) must make both equations true in order for it to be considered a solution. Therefore, we rule out (8,1). If you didn't check (4,1) earlier, then ruling out (8,1) will let you cross off the top left corner answer choice.
A quick way to find the solution(s) is to graph the two curves and see where they cross. They intersect at (0,3) which is the only solution of this system. See below.
