Do I have this correct, and where would this be used?

Question

I am now working on Solving Systems of Linear Equations. I was shown 3x - 2y = 7 and 6x - 2y = 4.  First I multiply one set by -1 {I'll use the second set for this part.} The second set becomes -6x + 2y = -4. Then I add that to the first set, which gives me -3x = 3. Solving the equation, this tells me that x=-1 and y=-5. [Did I get this right?] My questions are: #1.} Why is one set multiplied by -1 before adding the two sets together? #2.} Where in today's world would I be using this?   

 

[BTW ~ I am not in any school. It has been more than 30 years since I was in school. I have ALWAYS loved math, and usually am running numbers through my mind and playing with them. One time, I was in bed, JUST about ready to fall asleep, and 2 compound fractions came in my mind. My brain told me, "O.k., divide these two fractions." I was thinking, "Oh, COME on! I am just about ready to nod off! But, IN BED, and JUST about ready to fall asleep, I was able to divide those two fractions in my head!.]

Answer 1

Yes, you are correct.

I haven’t been in education for more than 50 years. I know how you feel. Mathematics has interested me since I was about 12. I like solving problems.

Now to answer your questions.

1. You could have subtracted one equation from the other, because it’s the same as multiplying one equation by -1 and adding; but what you’re really doing is eliminating one variable so that you have one equation and one variable. Therefore you are simplifying the problem.
2. In the “real world” this type of problem has many analogies. For example, you could be comparing the costs of two items, and want to know the cost of each item when all you have is the quantity of each item and the differences in costs. The same applies if you were trying to find unit cost. This could be important in managing domestic economy or on a larger scale in manufacturing when deciding on sources of raw materials. Systems of equations can involve many variables and there are a number of methods that can be used to solve them (matrices, for example). Plus the use of computers where manual methods are tedious and prone to error.

Congratulations on your bedtime mathematical gymnastics! Bathtime and bedtime have had their mathematical eureka moments for me on occasions!