Question 1

At 5:24, where x = 8 + 5a over a + b, can't we just divide 5a in the numerator by a in the denominator, which would leave us with 5?

Accepted Answer

You cant divide 5a in the numerator with a in the denominator. 
This is because in the numerator and denominator there are no multiplication or division signs.
If u cant understand that, here is another example without variables: 5+5/5 = 10/5 = 2.
You cant cancel the all the 5s in this to be left with 1 + 1/1 = 2/1 = 1 OR two fives to be left with 5+1/1 = 6/1 = 6. Just like that, u can not canel the a's in 8+5a/a+b
If this is still unclear, you should do the multiplication and division course of fractions on KhanAcademy

Question 2

Why did Sal changed from negative ax+bx in to positive ax+bx

Accepted Answer

He did this to avoid having to do this step 
`x(- a - b)/ (- a - b)  = (- 8 - 5a)/ (- a - b) `
 and avoid having to simplify this answer:
`x  = (- 8 - 5a)/ (- a - b)`
Lots of tricky negatives to deal with!  Instead, he multiplied the whole thing, both sides of the equation, by negative 1 and that changes each sign.  In this case, all terms become positive.  That makes it less likely to make a mistake dealing with each negative sign.

Question 3

When will we use this in the real world?

Accepted Answer

My question exactly 🤝

Question 4

In the second example how do you make the (-ax - bx) to (ax + bx) and the (-8 -5a ) to (8 + 5a)?

Accepted Answer

Because  -f(x) = -g(x) is the same as f(x) = g(x) (multiply both sides of the equation by -1).

Question 5

I don't understand how at 4:28 he just randomly decides to multiply by 1. where did that come from and why did he do it it's confusing to me?

Accepted Answer

Sal only multiplied by negative one to turn all of his negative terms to positive ones, to make it easier to work with. If at some point while you are working out your equation, you realize all your terms are negative, then you can multiply all your terms on both sides of the equation by negative 1. It will only change the sign, not the equality as long as you do it to all terms on BOTH sides of the equations. But like I said, all your terms have to be negative in the first place otherwise it defeats the purpose of multiplying by negative one, which is to make it easier to work with the figures.

Question 6

Do you have to multiply all of the numbers on 4:07 by -1? Would it be wrong if I got x= -8-5a/-a-b

Accepted Answer

It is not wrong, it is just incomplete because there is a common factor of -1 in both the numerator and denominator.  These should be factored out and cancelled with each other.
x = [-1(8+5a)] / [-1(a+b)}
The -1's divide out to 1, leaving you with the same answer as in the video.
Sal noticed all the negatives earlier in his process  (at about 4:05 in the video)  and he multiplies both sides of the equation by -1 to clear out all the negatives.

Hope this helps.

Question 7

At 4:56 Sal divides both sides by an Algebraic Expression, (a+b). 
Isn't that problematic? I'm aware there is a rule in math, not to divide by a Variable or Algebraic Expression unless you are sure that said variable/expression is not equal to zero.

In this case, we're not sure that (a+b) is not 0, so if it was 0, we'd end up dividing by zero in this case.

Is my logic wrong?

Accepted Answer

It can't be because a number cannot be divided by zero, not making a+b 0. So lets say you want to divide (a) apples between 0 friends, you can't divide it by 0 as you have 0 friends to share your apples with, so it can never be 0.

PS: On calculators if you divide something by 0 it sometimes comes up as infinity

PS: On calculators if you divide something by 0 it sometimes comes  up as infinity

Question 8

At 4:07, why is he allowed to multiply both sides by -1? Where did the -1 come from? Please Answer

Accepted Answer

You are always allowed to multiply any whole equation (he states this as multiplying both sides) by any number to get an equivalent equation.  The reason he chose -1 is because he noticed that all the terms are negative, and it is easier to work with positive numbers than with negative numbers.

Question 9

At 2:55: Is the order of operations important? Do I first subtract 5a from both sides and then factor out -ax -bx? Or can I first factor out and then subtract like so:

a(5-x) = bx - 8
5a - ax = bx - 8 | -bx
-ax -bx + 5a = -8
x(-a-b) + 5a = -8
x + 5a = (-8 / (-a - b))
x = (-8 / (-a - b)) - 5a

Or is this solution still correct, just not as "elegant" as it could be?

Accepted Answer

The order of operations are not important but you made some errors in your calculations:
Up to your fourth line you're good: x(-a-b) + 5a = -8,
But then if you want to divide by (-a-b) you need to divide all parts of the equation by (-a-b):
x(-a-b)/ (-a-b) +5a/(-a-b) = -8/(-a-b)
this simplifies as: x + 5a/(-a-b) = -8/(-a-b)
Then you can substract 5a/(-a-b) from both sides and you have:
x = -8/(-a-b) -5a/(-a-b)
Which simplify as:
x = (-8 -5a)/(-a-b)
Then to get to Sal's answer you can multiply both sides by (-1)/(-1), which gives:
x = (8+5a)/(a+b)

Hope that helps!

Question 10

if all the variables have an lcm can you simplify it?

Accepted Answer

If you did this you would not be simplifying the problem to lowest terms. Sometimes the lcm will happen to be a common divisor but most likely not.

Course: Algebra 1 > Unit 2

Linear equations with unknown coefficients

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