# Freeing up the variables of an equation and assigning signs to them

• graphs
In summary, the equation x^2 - y^2 = 0 has two possible solutions for y, which are y = x and y = -x. This is because factoring the equation results in (x - y)(x + y) = 0, which means one solution is when x = y and the other is when x = -y.
graphs

## Homework Statement

sqrtY^2=sqrtX^2

Solving for Y we get:

Y= X and Y= -X

## The Attempt at a Solution

Since both sides of sqrtY^2=sqrtX^2 are equal I thought the equation would solve simply as Y= X. Turns out it has also the second part Y= -X.

My questions are since both sides of the equation sqrtY^2= sqrtX^2 are EQUAL

1. why don't we just get Y=X

or

2. why Y also doesn't take the negative sign just to keep the both sides of the orginal equation in balance?

Thanks.

Possible solutions of the equation:

y = 2 and x = -2;

y = 2 and x = 2;

y = -2 and x = 2.

graphs said:

## Homework Statement

sqrtY^2=sqrtX^2

Solving for Y we get:

Y= X and Y= -X

## The Attempt at a Solution

Since both sides of sqrtY^2=sqrtX^2 are equal I thought the equation would solve simply as Y= X. Turns out it has also the second part Y= -X.

My questions are since both sides of the equation sqrtY^2= sqrtX^2 are EQUAL

1. why don't we just get Y=X

or

2. why Y also doesn't take the negative sign just to keep the both sides of the orginal equation in balance?

Thanks.
$\sqrt{y^2} = \sqrt{x^2} \Rightarrow y^2 = x^2$
$\Rightarrow y^2 - x^2 = 0 \Rightarrow (y - x)(y + x) = 0$
The solutions of the last equation are y = x and y = -x.

Mark44 said:
$\sqrt{y^2} = \sqrt{x^2} \Rightarrow y^2 = x^2$
$\Rightarrow y^2 - x^2 = 0 \Rightarrow (y - x)(y + x) = 0$
The solutions of the last equation are y = x and y = -x.

Amazing!

Why is that amazing?

Mark44 said:
Why is that amazing?

Well, because I didn't even think about factoring and finding the zeros. Simple and elegant.

X^2- Y^2=0 was given.

What I did was X^2=Y^2, then sqrtX^2=sqrtY^2 to "liberate" the variables Y and X. Then I got stuck with the signs the variable X took, what with X being both positive and negative.

So someone, in another forum, explained it with "Because Y = -X is also a solution. √(2²) = √((-2)²), for example" which was very helpful too. I just never thought about factoring.

Anyway, thank you, people!

If you started with x2 - y2 = 0, then the quickest approach is to factor the left side, and not messing around taking square roots.

## 1. What is the purpose of freeing up variables in an equation?

Freeing up variables in an equation allows for greater flexibility and simplification in solving the equation. It allows us to isolate and manipulate specific variables to find their values.

## 2. How do you free up variables in an equation?

To free up variables in an equation, we use inverse operations to remove any constants or coefficients that are attached to the variable. This allows the variable to be on one side of the equation by itself.

## 3. Why is it important to assign signs to variables in an equation?

Assigning signs to variables in an equation helps to clearly convey the relationship between the variables and constants. It helps to indicate whether the variable is being added, subtracted, multiplied, or divided in the equation.

## 4. Can you assign different signs to the same variable in an equation?

Yes, it is possible to assign different signs to the same variable in an equation. The sign assigned to a variable can change depending on the operation being performed on it. For example, a variable may have a positive sign when it is being added, but a negative sign when it is being subtracted.

## 5. How does assigning signs affect the solution of an equation?

Assigning signs to variables affects the solution of an equation by determining the direction and magnitude of the variable's value. It helps to accurately represent the relationship between the variables and constants in the equation.

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