When subtracting 1 rational expression from another, when I'm multiplying the top & bottom of the first term by the denominator in the second term is that number automatically negative because the whole term is being subtracted?

ex:
(8a-3b)/7a - (2a+5b)/4b

is the 4b negative when I multiply the first term by it?

Thanks a ton! :) ♡

No. However you could do it that way, but that changes the sign of the second part of the numerator, as well as the first term.

What you are fundamentally forgetting, that you are not multiplying the numerator in the first term: you are multiplying the first term by (4x/4x). Doing that makes a common denominator 28ax, and of course, you multiply the second term by (7a/7a)...

no

actually you are multiplying first by 4b/4b
and the second by
7a/7a
so

(4b/4b)[(8a-3b)/7a]-(7a/7a)[(2a+5b)/4b]

={4b[(8a-3b)]-7a[(2a+5b)]} /28ab

Ok, thank you :)

When subtracting one rational expression from another, you need to make sure that both expressions have a common denominator. To achieve this, you must multiply the numerator and denominator of each expression by the missing factors from the other expression's denominator.

In your example, the first rational expression is (8a-3b)/7a, and the second rational expression is (2a+5b)/4b. Let's find a common denominator for these expressions.

The first expression has a denominator of 7a, while the second expression has a denominator of 4b. To find the common denominator, we need to find the least common multiple (LCM) of 7a and 4b, which is 28ab.

To make the first expression have a denominator of 28ab, we multiply the top and bottom of the fraction by (4b):

(8a-3b)/7a = (8a-3b) * (4b) / (7a) * (4b) = (32ab - 12b^2) / (28ab)

Similarly, to make the second expression have a denominator of 28ab, we multiply the top and bottom of the fraction by (7a):

(2a+5b)/4b = (2a+5b) * (7a) / (4b) * (7a) = (14a^2 + 35ab) / (28ab)

Now that both expressions have a common denominator of 28ab, we can subtract them:

[(32ab - 12b^2) / (28ab)] - [(14a^2 + 35ab) / (28ab)]

Since the denominators are the same, we can subtract the numerators directly:

(32ab - 12b^2 - 14a^2 - 35ab) / (28ab)

Now, let's simplify the numerator:

32ab - 12b^2 - 14a^2 - 35ab = 32ab - 35ab - 14a^2 - 12b^2 = (32ab - 35ab) - (14a^2 + 12b^2) = -3ab - (14a^2 + 12b^2)

So, the answer is (-3ab - (14a^2 + 12b^2)) / (28ab).

To answer your specific question about whether the 4b in the first term should be negative when you multiply it, the answer is no. When you multiply the first term by (4b), you should not include a negative sign. The negative sign only appears when subtracting the two expressions.