Problem with moving variables to other side of an equation

[Pumpel]

Hello everybody.
For the past hour I've been struggling to understand something that at first glance looks super easy to me, yet I just can't solve it.

So, let's say we have and equation [imath]c^2 = a^2 * b^2[/imath] and let's say [imath]a = 4[/imath] and [imath]b = 8[/imath].
Now, [imath]c^2 = a^2 * b^2[/imath] is the same as [imath]c * c = a * a * b * b[/imath].

Now let's say I wanted to move one [imath]a[/imath] from the right-hand side of the equation to the left-hand side of the equation.
Then the equation would look like this: [imath]\frac{c^2}{a} = a * b^2[/imath].

And this would work because in this case, [imath]\frac{c^2}{a}[/imath] is equal to [imath]a * b^2[/imath].


But now, If I changed the multiply symbol to the plus symbol, the equation would no longer work.
If I have an equation [imath]c^2 = a^2 + b^2[/imath], then that is the same as [imath]c * c = a * a + b * b[/imath].

If I tried to move one [imath]a[/imath] from the right-hand side of the equation to the left-hand side of the equation now,
the equation would be [imath]\frac{c^2}{a} = a + b^2[/imath].
But if I try to plug in the numbers, the result is incorrect.

And so my question is: Why the equation [imath]\frac{c^2}{a} = a + b^2[/imath] doesn't work, but equation [imath]\frac{c^2}{a} = a * b^2[/imath] does work?

 

[BigBeachBanana]

Hello everybody.
For the past hour I've been struggling to understand something that at first glance looks super easy to me, yet I just can't solve it.

So, let's say we have and equation [imath]c^2 = a^2 * b^2[/imath] and let's say [imath]a = 4[/imath] and [imath]b = 8[/imath].
Now, [imath]c^2 = a^2 * b^2[/imath] is the same as [imath]c * c = a * a * b * b[/imath].

Now let's say I wanted to move one [imath]a[/imath] from the right-hand side of the equation to the left-hand side of the equation.
Then the equation would look like this: [imath]\frac{c^2}{a} = a * b^2[/imath].

And this would work because in this case, [imath]\frac{c^2}{a}[/imath] is equal to [imath]a * b^2[/imath].


But now, If I changed the multiply symbol to the plus symbol, the equation would no longer work.
If I have an equation [imath]c^2 = a^2 + b^2[/imath], then that is the same as [imath]c * c = a * a + b * b[/imath].

If I tried to move one [imath]a[/imath] from the right-hand side of the equation to the left-hand side of the equation now,
the equation would be [imath]\frac{c^2}{a} = a + b^2[/imath].
But if I try to plug in the numbers, the result is incorrect.

And so my question is: Why the equation [imath]\frac{c^2}{a} = a + b^2[/imath] doesn't work, but equation [imath]\frac{c^2}{a} = a * b^2[/imath] does work?

When you "moved" a to the other side, essentially you are dividing by [imath]a[/imath] on both sides.
For multiplication:
[math]c^2=a^2b^2\\ \frac{c^2}{a}=\frac{a^2b^2}{a}=\frac{\cancel{a}\times a\times b^2}{\cancel{a}}=a\times b^2\\ \implies \frac{c^2}{a}=a\times b^2 [/math]
For addition:
[math]c^2=a^2+b^2\\ \frac{c^2}{a}=\frac{a^2+b^2}{a}=\frac{a^2}{a}+\frac{b^2}{a}=a+\frac{b^2}{a} \\ \implies \frac{c^2}{a}=a+\frac{b^2}{a}[/math]

 

[Pumpel]

When you "moved" a to the other side, essentially you are dividing by [imath]a[/imath] on both sides.
For multiplication:
[math]c^2=a^2b^2\\ \frac{c^2}{a}=\frac{a^2b^2}{a}=\frac{\cancel{a}\times a\times b^2}{\cancel{a}}=a\times b^2\\ \implies \frac{c^2}{a}=a\times b^2 [/math]
For addition:
[math]c^2=a^2+b^2\\ \frac{c^2}{a}=\frac{a^2+b^2}{a}=\frac{a^2}{a}+\frac{b^2}{a}=a+\frac{b^2}{a} \\ \implies \frac{c^2}{a}=a+\frac{b^2}{a}[/math]

Oooh that's why. Now this finally makes sense to me. Thank you so much for helping!

 

[blamocur]

When you "moved" a to the other side, essentially you are dividing by [imath]a[/imath] on both sides.
For multiplication:
[math]c^2=a^2b^2\\ \frac{c^2}{a}=\frac{a^2b^2}{a}=\frac{\cancel{a}\times a\times b^2}{\cancel{a}}=a\times b^2\\ \implies \frac{c^2}{a}=a\times b^2 [/math]
For addition:
[math]c^2=a^2+b^2\\ \frac{c^2}{a}=\frac{a^2+b^2}{a}=\frac{a^2}{a}+\frac{b^2}{a}=a+\frac{b^2}{a} \\ \implies \frac{c^2}{a}=a+\frac{b^2}{a}[/math]

To expand on @BigBeachBanana's post: "moving" is not some magic operation but a relatively simple rule: if you have two equal values and you apply to each of them the same operation (divide by the same non-zero value, add or subtract the same value, etc.) you preserve the equality. But it is important that you apply the same operation to both side. In your second example you are not dividing by [imath]a[/imath] all of the right hand side, but only one part of it ([imath]a^2[/imath]).

 

[Deleted member 4993]

Hello everybody.
For the past hour I've been struggling to understand something that at first glance looks super easy to me, yet I just can't solve it.

So, let's say we have and equation [imath]c^2 = a^2 * b^2[/imath] and let's say [imath]a = 4[/imath] and [imath]b = 8[/imath].
Now, [imath]c^2 = a^2 * b^2[/imath] is the same as [imath]c * c = a * a * b * b[/imath].

Now let's say I wanted to move one [imath]a[/imath] from the right-hand side of the equation to the left-hand side of the equation.
Then the equation would look like this: [imath]\frac{c^2}{a} = a * b^2[/imath].

And this would work because in this case, [imath]\frac{c^2}{a}[/imath] is equal to [imath]a * b^2[/imath].


But now, If I changed the multiply symbol to the plus symbol, the equation would no longer work.
If I have an equation [imath]c^2 = a^2 + b^2[/imath], then that is the same as [imath]c * c = a * a + b * b[/imath].

If I tried to move one [imath]a[/imath] from the right-hand side of the equation to the left-hand side of the equation now,
the equation would be [imath]\frac{c^2}{a} = a + b^2[/imath].
But if I try to plug in the numbers, the result is incorrect.

And so my question is: Why the equation [imath]\frac{c^2}{a} = a + b^2[/imath] doesn't work, but equation [imath]\frac{c^2}{a} = a * b^2[/imath] does work?

We say moving variable for short. Actual operation is:

c*c = a*a + b*b

c*c *1 = 1 * (a*a + b*b)

c*c = a * 1/a * (a*a + b * b)

c*c = a * (1/a * a*a + 1/a * b * b)

c*c = a * (a + 1/a * b * b)

c*c * 1/a = 1/a * a * (a + 1/a * b * b)

c*c * 1/a = 1 * (a + 1/a * b * b)

c*c / a = (a + 1/a * b * b)

I think the key to your confusion is the way 1/a was distributed (c*c = a * 1/a * (a*a + b * b)

This response is very similar to the response by @BigBeachBanana - I did not realize that server went down and mine was not posted.

 

[Steven G]

We know that 12= 4 + 8
The question is does 12/4 = 4/4 + 8?
You can't divide the whole left side by say 4 and only part of the right side by 4 and expect equality.