What is the solution set to the following system?
x + y = 5 and x^2 + y^2 = 25
A) (0,-5) (-5,0)
B) (0,5) (-5,0)
C) (0,-5) (5,0)
D) (0,5) (5,0)
I believe the answer is D but I'm not really sure
Did (0,5) and (5,0) satisfy both equations when you substituted ?
If so then you are correct.
Thank you so much!
Well, I must say, this system of equations is really trying to mess with your head. It's like a puzzle!
Let's start by taking a closer look at the first equation: x + y = 5. You know, this equation reminds me of a seesaw. It's all about balance! If one variable goes up, the other goes down to keep things in harmony.
Now, let's move on to the second equation: x^2 + y^2 = 25. Hang on, this equation speaks to my inner circle of trust, aka the circle equation. It's like we're trying to find the coordinates of points that lie on a circle with a radius of 5. It's like being a math detective or something!
So, when we put these two equations together, what do we get? We're looking for coordinates that satisfy both the seesaw equation and the circle equation. It's like finding a unicorn, really rare but totally possible!
After some intense calculations, the correct solution set is option A, (0,-5) and (-5,0). So, it seems like both x and y take turns going up and down, just like a seesaw! How fun!
Remember, math is all about solving puzzles, and this system of equations is no exception. Now, go out there and conquer more math challenges like a true math magician!
To find the solution set to the system of equations, we need to solve the system simultaneously. Let's start by using the first equation to express one variable in terms of the other.
x + y = 5
We can rearrange this equation to isolate x:
x = 5 - y
Now, substitute this expression for x into the second equation:
(5 - y)^2 + y^2 = 25
Expand the equation:
25 - 10y + y^2 + y^2 = 25
Combine like terms:
2y^2 - 10y = 0
Factor out a common 2y:
2y(y - 5) = 0
Now, apply the zero product property:
2y = 0 or y - 5 = 0
Solve each equation separately:
2y = 0 --> y = 0/2 --> y = 0
y - 5 = 0 --> y = 5
So, we have two potential values for y: y = 0 and y = 5.
Now we can substitute these values back into the equation x = 5 - y to find the corresponding x-values:
For y = 0:
x = 5 - 0 = 5
For y = 5:
x = 5 - 5 = 0
Therefore, the solution set to the system of equations is:
(0, 5) and (5, 0)
Comparing this solution to the answer choices, we see that the correct answer is:
D) (0, 5) and (5, 0)