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# Why can we subtract one equation from the other in a system of equations?

The example of a scale where we try to achieve balance helps to explain why we can subtract one equation from the other in a system of equations. Created by Sal Khan.

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• So the main idea is to get all of the varibles on one side and the numbers on the other so we can simplify the answer? • Suppose you had 3x=9. This is correct, just not simplified. You could multiply both side by 2 and get 6x=18, and the answer would still be correct, just not simplified. You can do whatever you want to the equation, as long as you do it to both sides. Yes, adding 500 to both sides is correct, it is neither simplified nor practical. In essence, you are correct. I hope this helped you understand the subject better as opposed to just memorizing stuff and not understanding why you did it.

I hope this helps!

Bill
• how can you possibly know that x+y is equal to 5?.................. • Sal didn't figure out that x+y=5. I too was confused at first. In actuality, he was just "uncovering" the second part of the expression after making sure we understood there was not enough information on the first scale to answer the question. Had you come across this question on a quiz, it probably would have looked something like this:

If 2x+y=8 and x+y=5, what are the values of x and y?

I hope this helped.
• At Khan adds x and y to the other scale and 5 blocks until they balence. However in algebra in school you don't get a scale and blocks during a test. How would you figure a problem like that out without a scale? •  You are right. You can't solve that problem without real scales and real weights. In a test on that problem through, you will be told that x + y = 5.

When you are doing algebra at school, you can always draw some scales, just like he did. Good teachers like to be able to see how you work things out in a test. Just make sure you always do the same thing to both sides and your "scales" will always balance.
• Well, i know his intentions were to depict two variable equation visual but ........ in the second scale we can put only one x and evaluate its value by placing blocks on the other hand of the scale to know what the value of x is and then find the y value or vice versa! • Sorry. I see other people have asked similar questions but I still feel confused. In the previous videos, Sal has explained that 3x = 6 could be solved by multiplying both sides by 1/3, which would leave us with x = 2. Yes? However in the problem above of 2y+x=8, Sal does not multiply both sides by 1/2. That would have left us with y+x=4. Yet, sal explains with the aid of the second scale that y+x=5. This is confusing. Why is y+x not equal to 4? If the second scale were not part of the problem, would I then be correct in stating that y+x=4? Sorry to bother you guys on this one. • Thank you for the explanation, Sal. Would you please consider demonstrating a formal proof of why, in a system of equations, one equation can be subtracted from another equation. • No proof is needed. There is a property of equality that says we can subtract an equal value from both sides of an equation and we will end up with an equivalent equation. You are already used to using this concept to solve equations like: x+5=2. You would subtract 5 from both sides of the equation to find "x".

When you subtract equations in the elimination method, you are using this same technique. One side of the equation equals the other. So, subtracting 2 equations means your are subtracting equal values from both sides of the other equation. The values just look diffeent.
Hope this helps.
• How do you know it balanced out at 5 blocks that are equal to y and x?
() • Is the scale system how people first learned algebra or is that just taught so kids can understand it easier?   