Solve for a and b: Algebra Equations a^2+a^2=200, a+b=30

We are given the following equations:

Let’s solve the first equation for :


Divide both sides by 2:


Take the square root of both sides:

Now we have two possible values for : or .

Case 1:
Substitute into the second equation, :

Subtract 10 from both sides:


Now, calculate :

Case 2:
Substitute into the second equation, :

Add 10 to both sides:


Now, calculate :

Since the problem does not specify whether is positive or negative, there are two possible answers. However, in many contexts where such problems are presented without further constraints, a positive value for variables is often implied, or the simplest positive solution is preferred. If we consider the visual representation, the numbers are written with white chalk on a green board, and there are no explicit constraints that would force us to choose a negative value. Therefore, we will present both possible answers.

The final answer is .