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## Algebra 1

### Course: Algebra 1>Unit 2

Lesson 4: Linear equations with unknown coefficients

# Linear equations with unknown coefficients

Sal solves the equations ax+3x=bx+5 and a(5-x)=bx-8 for x. Note that these equations include other unknowns (a and b) but we solve them for x.

## Want to join the conversation?

• At , 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? •  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
• Why did Sal changed from negative ax+bx in to positive ax+bx •  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.
• I don't understand how at he just randomly decides to multiply by 1. where did that come from and why did he do it it's confusing to me? •  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.
• In the second example how do you make the (-ax - bx) to (ax + bx) and the (-8 -5a ) to (8 + 5a)? • Do you have to multiply all of the numbers on by -1? Would it be wrong if I got x= -8-5a/-a-b • 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 in the video) and he multiplies both sides of the equation by -1 to clear out all the negatives.

Hope this helps.
• At , why is he allowed to multiply both sides by -1? Where did the -1 come from? Please 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.
• When will we use this in the real world? • At : 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? • 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!
• At 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?  