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### Course: Integrated math 1>Unit 6

Lesson 3: 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.
• I don't understand anything from this video :D
• 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.
• 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.
• So basically.. equivalent systems in equations is like simplification? Where two equations have the same product just a different form? (e.x, 2x = 12, x = 6?)

Also, why couldn't Sal just multiple x + y = 5 by -1, and that would get -x - y = -5, wouldn't we still get the same product just less simplification to get the intercepts?
• What if there was 2 systems of equations?
• 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 how when we use the elimination method, we subtract one equation from the other, how we are left off with a single variable, when isolated, gives us the either solutions'x or solution's y of the system of equations.
• Let's say you have a single equation: 2x+y=3+y. Since there is one y on each side of the equation, we can eliminate them both, reducing the equation to 2x=3. Same with 2 equations, except it's like having on equation on the left and the other on the right.
• What is a system?
(1 vote)
• A system of equations just means you have more than 1 equation. Normally, we have the number of equations equal to the number of different variables we use, but this is not an absolute. With two variables, x and y, we would need a system of two equations to solve.
• 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