Thomas says that you can take the square root of each side of an equation. Therefore, he feels that (x+5)^2 + (y-7)^2 = 49 and (x+5) + (y-7) = 7 are equivalent. However, Mark says that they aren't equal. Who is right? Explain.

I believe that Mark is correct because if you were to square each value in the equation, you would have: (x+25) + (y - 49) = 49, not (x+5) + (y-7) = 7.

Show Thomas a simplified example of why his idea is wrong

3^2 + 4^2 = 5^2 ---- true
then
3 + 4 = 5 ?????

If you want to take the square root, you have to take it of the whole side, not the individual terms

So, Mark would be correct. Those don't equal each other.

To determine who is right, we need to examine the mathematical reasoning behind both Thomas and Mark's arguments.

Thomas claims that you can take the square root of each side of an equation. This is indeed a valid operation in mathematics. When you take the square root of a number, you are finding the value that, when multiplied by itself, gives you that number. Applying this operation to both sides of an equation can often help solve it.

Let's start with Thomas' equation:
(x+5)^2 + (y-7)^2 = 49

Thomas suggests taking the square root of each side, which would yield:
sqrt((x+5)^2 + (y-7)^2) = sqrt(49)

Taking the square root of 49 gives us 7, so the equation becomes:
sqrt((x+5)^2 + (y-7)^2) = 7

Now, let's consider the equation that Mark claims is equivalent:
(x+5) + (y-7) = 7

Here, Mark is simplifying the equation (x+5) + (y-7) = 7 by removing the squared terms. However, this is not mathematically valid because the square root does not distribute over addition. Mark's simplification overlooks the geometric interpretation of the two equations.

To understand the geometric interpretation, let's visualize the original equation (x+5)^2 + (y-7)^2 = 49. This equation represents a circle with its center at (-5, 7) and a radius of 7 (since the square root of 49 is 7). It describes all the points (x, y) that are exactly 7 units away from the center.

On the other hand, the equation (x+5) + (y-7) = 7 represents a line. By rearranging this equation, we get y = -x + 5. This is the equation of a line with a slope of -1 and a y-intercept of 5.

Comparing the two equations, we can see that they represent different geometric shapes. A circle and a line are not equivalent; they have distinct properties and describe different sets of points.

Therefore, Mark is correct in asserting that the two equations, (x+5)^2 + (y-7)^2 = 49 and (x+5) + (y-7) = 7, are not equivalent. The reasoning provided by Thomas is not valid in this case.