How would you solve this problem?:

x2 + y2 = 25
x2 - y = 5

You have to solve for one variable either by substitution or elimination method and then use the answer to plug it in one of the equations to get the other variable. I've been at this problem for 30 minutes now and have given up!

Thanks! :)

Well, solving equations can be quite a puzzle! But don't worry, I'm here to make it fun!

Let's start by eliminating one of the variables in these equations. Since both equations have an x² term, we can subtract the second equation from the first one to eliminate that term:

(x² + y²) - (x² - y) = 25 - 5

After simplifying, we have:

x² + y² - x² + y = 20

The x² term cancels out, leaving us with:

y² + y = 20

Now, let's try to solve for y. We have a quadratic equation here, so let's see if we can factor it. Do you know any factorization jokes?

To solve this problem, we can use the elimination method to eliminate one of the variables. Let's start by subtracting the second equation from the first equation:

(x^2 + y^2) - (x^2 - y) = 25 - 5

Simplifying the equation, we get:

2y = 20

Divide both sides of the equation by 2 to solve for y:

y = 10

Now, we can substitute this value of y into one of the original equations to solve for x. Let's use the second equation:

x^2 - y = 5

Substituting y = 10, we get:

x^2 - 10 = 5

Adding 10 to both sides of the equation, we have:

x^2 = 15

Taking the square root of both sides, we find:

x = ±√15

Therefore, the solutions to the system of equations are:

x = √15, y = 10
x = -√15, y = 10

To solve this problem, we can use the elimination method. Here's a step-by-step explanation of the process:

1. Start by writing down the two equations:
Equation 1: x^2 + y^2 = 25
Equation 2: x^2 - y = 5

2. To eliminate one of the variables, we need to manipulate one of the equations so that the coefficients of the variable are the same or differ by a multiple or a factor.

3. In this case, let's eliminate the variable x. We can achieve this by multiplying Equation 2 by -1, which gives us:
-x^2 + y = -5

4. Now, we can add Equation 1 and the modified Equation 2 to eliminate the x terms:
x^2 + y^2 + (-x^2 + y) = 25 + (-5)

Simplifying the equation, we get:
2y = 20
Divide both sides of the equation by 2:
y = 10

5. We have found the value for y. Now, substitute this value into either of the original equations to find the value of x. Let's use Equation 2:
x^2 - y = 5

Substitute y = 10:
x^2 - 10 = 5

Simplify further:
x^2 = 15

Take the square root of both sides:
x = ±√(15)

6. Therefore, we have two solutions for x: x = √(15) or x = -√(15).

By following these steps, you should be able to find the solution to the given problem.