## why a number does not become negative when it's moved

Complex numbers, rational functions, logarithms, sequences and series, matrix operations, etc.
arya6000
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Joined: Wed Feb 27, 2013 11:24 pm
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### why a number does not become negative when it's moved

Hello

I'm very confused as to why 2 numbers don't become negative when it's moved to the other side of the equal sign. This is from a learning video that I'm watching and I have circles the part that is confusing in red. If you can clarify this I'd highly appreciate it.

Here is the link to the picture

Regards!

FWT
Posts: 153
Joined: Sat Feb 28, 2009 8:53 pm

### Re: why a number does not become negative when it's moved

They gave you this:
$y\, =\, \sqrt{2\, +\, 5x}$

$y^2\, =\, 2\, +\, 5x$
Then you'd probably do like this:
$y^2\, -\, 2\, =\, 2\, +\, 5x\, -\, 2$

$y^2\, -\, 2\, =\, 5x\,$$\,\color{red}\leftarrow$

$5x\, =\, y^2\, -\, 2$
They just skipped the step (with the arrow) where they flipped the sides.

gingersnap0819
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### Re: why a number does not become negative when it's moved

You are also able to reason through this by understanding the properties of equality and the properties of real numbers.

y^2 = 2 + 5x Let's start from this step since this is where you got confused.
The commutative property of addition says that you can add terms in any order, i.e. switch the order of the terms when adding, and your result will be the same. (A numerical example: 3 + 5 is the same as 5 + 3 because it doesn't matter which number comes first and which number comes second when adding.) So by using the commutative property, you can switch the order of the 2 and the 5x on the right side of your equation to get:

y^2 = 5x + 2 A different property you could use now to justify a next step would be the symmetric property of equality. The symmetric property of equality allows you to interchange the left hand side and the right hand side of an equation. (A numerical example: 3 + 5 = 8 is technically the same equation as 8 = 3 + 5 , no matter which side of the equals sign the expressions happen to be on.) So by using the symmetric property of equality, you can switch the left hand side (y^2) with the right hand side (5x + 2) of the equation to get:

5x + 2 = y^2 The last step to get this to look like what you circled in the video is to use the Addition Property of Equality (sometimes seen also as the Subtraction Property of Equality). This property allows you to add (or subtract) the same number from BOTH sides of an equation, and the resulting equation will still be true. We often use this step in equation solving without realizing that there is a property allowing us to perform certain operations in order to complete the steps or method to solving an equation. We tend to take for granted that we are "allowed" to do certain things in math without fully understanding the proof or the reasoning behind it. Anyway, a simple example: x - 5 = 9 To solve for x, you would add 5 to both sides of the equation and get x - 5 + 5 = 9 + 5 Simplifying on both sides of the equation gives you x = 14. So in order to isolate the variable, we had to "undo" what was being done to the variable so that the constant would cancel itself out on the side where the variable was, and move across the equals sign to join its like term. So by using the Addition Property of Equality, you can add (-2) to both sides of the equation to get:

5x + 2 - 2 = y^2 - 2 which simplifies to give you 5x = y^2 - 2

It would be my conjecture that the person who created the example in your photo was assuming that his or her audience did not need each step of equation manipulation to be spelled out. Unfortunately, this is a common misconception of teachers to assume that students will be able to follow the reasoning from one step to the next even if intermediate steps are left out. Hopefully any teacher would be receptive to a question about how to go from Step A to Step B- it's possible the teacher did not even realize the importance of those intermediate steps for learners who are trying to understand a concept for the first time.