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### Course: Algebra 1>Unit 6

Lesson 4: Equivalent systems of equations

# Reasoning with systems of equations

When we perform operations on a system of equations, some operations produce an equivalent system, while others don't necessarily produce an equivalent system. When we're solving a system of equations, we need to use operations that guarantee equivalence. Created by Sal Khan.

## Want to join the conversation?

• At couldn't he have multiplied by negative one to cancel out they and then x would be positive, same with the answer at the end?

2x+y=8
x+y=5

2x+y=8
-x-y=-5

x=3

I thought that would be so much easier.
• My sense is, that Sal is trying to teach us to do “mathematical reasoning” (at ) with a little bit tricky stuff--meaning at he says “…not super mathematically rigorous…” which to me, means doing it this particular way shows that you know how to work with algebra and systems that much better. I think the key for this method is to know and be comfortable with substituting x and y for a single number (where 2x + y is substituted for 8). It’s usually totally up to you how “deep” you want to go with the math.
• Can someone please explain this clearer?
• I think the key is at , to watch and listen a few times, as Sal explains what he is going to use instead of the 8 on the left side of the equation in the upper right hand corner. Instead of the 8, he uses the 2x + y. I think this is to show us another way, and also to “play” with the algebra, because I think he is attempting to teach us some “reasoning” skills in the short amount of time this video allows.
• What does 'system' in systems of equations mean? is it like a set??... Please help me!
• A "system" is a <set> or maybe <collection> (usually two or three) of equations that you are dealing with all at once for a particular problem. All the equations in the system are very important and are ALL required to solve for the variables in the system.
• I don't understand anything from this video :D
• Great!! Good for you :)
• Could someone help me with a problem about airplane seats? The following info is given:

"For every 13 seats in economy class there are 5 seats in business class"

my immediate intuition is to write this as 13e = 5b but this is wrong. Instead it was supposed to be 5e = 13b or e/13 = b/5. Somehow I keep making this mistake and was wondering if someone can shed light on why the latter makes more sense! Thank you in advance
(1 vote)
• I like to think about this as a ratio it really helps me so the ratio of business seats to economy seats is 5 to 13 (I used 5 to 13 as the ratio symbol shows the time in the video for example ). So, for every 5 business seats there are 13 economy seats. So we can make a equation where b = business seats and e = economy seats. So we can form the equation (e/13)*5 = b.
(e/13)=(b/5)
So now you can cross multiply and you get 5e = 13b
we can make sure that this equation (e/13)*5 = b works so let's say there are 39 economy seats so (39/13)*5 = b, so now we can say that b = 3*5, and that is b = 15. Now to double check the equation (39/13)*5 = 15
5 to 13, 10 to 26, 15 to 39 are all equal.
• Ok, so this is a bit of a big concept to grasp. Can someone just clarify whether or not I am doing it correctly?

So I think that if you add the equations in any system you will end up with the right answer?

Like this?

2x + y = 8
-2x - 2y = -10

2x + (-2x) = 0
y + (-2y) = -y
8 + (-10) = -2

We are left with -y = -2 or y = 2. Does this apply to all systems, and is it really that simple to find the answer or am I skipping a step?

Any help would be appreciated

• This applies to all systems. You're not skipping anything, you did it great! Personally, I almost always use this method because it's a lot simpler than anything else. I think it is good for you to find this simple.
• In the practice for this video (Resoning with systems of equations) there are sometimes answers like:

A: Replace one equation with the sum/difference of both equations

B: Replace only the left-hand side of one equation with the sum/difference of the left-hand sides of both equations

I'm confused on what's the difference between the two, and what is the "left-side of one equation"

-Thanks (:
• lets say you have 3x + 2y = 12. 3x+2y is left hand side and 12 is right hand side. Then if 4x + y = 15, you have a system of equations. A says to add (7x + 3y = 27 ) or subtract (1st - 2nd gives -x - y = -3 and 2nd-1st gives x+y=3) equations, B says add or subtract left side only to get something like 7x + 3y = 12 or 7x+3y = 15 depending on which equation you start with, hopefully this answer looks incorrect because you could get two different answers.
• that was journey!
• But you made it this far! 🎉🎉🎊🥳