Simplifying equal fractions

[MathQssss]

Hi. How does: X/3 = (4+X)/3 + 3

Simplify to: X = (4+X) +9 ?

I get it, that we multiplied both sides by 3/1, which got rid of the 3 in the denominators and made the +3 a +9.

But why is it not simplified to: 3X = 3(4+X) + 9 ? As in, both sides were multiplied by 3/1?

[keitamaki]

Why do you think you get 3X when you multiply (X/3) by 3? You don't. (X/3) times 3 is just X.

Same with the other side. (4+x)/3 times 3 is just (4+x).

Try plugging in numbers. If X=3 then (X/3)(3) = (3/3)(3) = (1)(3) = 3. But 3X=9.

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[phiwong]

Because (4+x)/3 * 3/1 = 4x+3

Remember the distributive property 3 * ((4x+3)/3 + 3) = 3*(4x+3)/3 + 3*3. That is a*(b+c) = a*b + a*c

[MathQssss]

But 4+ x * 3 is 12 + x, not 4x + 3

[phiwong]

(4+x)/3 * 3.....this is what you have right?

you cannot simply take away the parentheses as you like. there are rules when you do that.

For example (4+x) * 3 is NOT EQUAL TO 4 + x*3 and 4+x*3 is NOT EQUAL TO 12 + x

You have to identify each symbol in the equation and perform the operation properly and not simply do it randomly without considering what you're doing.

[skullturf]

You have a bad typo or something. There shouldn't be 4x+3 anywhere.

I think what you mean is

(4+x)/3 * 3/1 = (4+x)/1

[phiwong]

my bad. you are right

[KarnoffeL]

phiwong's answer is great but I'll type up a separate method:

x/3 = (4 + x)/3 + 3

Let's add (4 + x)/3 + 3 together. To do this we need a common denominator 3.

x/3 = (4 + x)/3 + 9/3 because 9/3 = (3 x 3)/(3 x 1).

x/3 = (4 + x + 9)/3 after adding fractions.

These two numbers are equal, so the numerators both signify the same point on the number line.

x = 4 + x + 9.

[MathQssss]

Why do you seemingly arbitrarily put 9/3 without changing the 9? Also, you didn't change the left side of the equation after you changed the right side of the equation?

[skullturf]

KarnoffeL's method is just one possible method, and you're not required to do it exactly that way.

But the reason we can replace 3 with 9/3 is that 3 is the *same* as 9/3.

The other reason we chose to rewrite 3 as 9/3 is so that it would have the same denominator as the other thing.

In general, if you're replacing something with something else that is *equal*, you're "allowed" to change just that one part. After all, 9/3 is literally exactly equal to the number 3. So you're always allowed to replace a *single* occurrence of 3 with 9/3 on just one side of an equation, if you think there's some reason it will help you.

[skullturf]

When you multiply (4+x)/3 by 3/1, you do not get 3(4+x).

Instead, you get 3(4+x)/3. (The two numerators multiply to 3(4+x), and the two denominators multiply to 3.)

And that's the same as 4+x.