Help with "Solving a system of linear equations using elimination with multiplication"

[150mph]

My new method described above has failed. Below is another of this same problem; if you guys explained it, I'm not following....

I need to understand why this time they chose -4 to multiply by (because they decided not to tell us). There isn't even a 4 in the problem...And why does only the bottom equation get multiplied?

 

[lev888]

I need to understand why this time they chose -4 to multiply by (because they decided not to tell us). There isn't even a 4 in the problem...And why does only the bottom equation get multiplied?

Don't pee on my knee and tell me it's raining. There is a perfectly good explanation for -4. You are not making any effort to understand their solution.
And we did cover this case.

 

[lev888]

My new method described above has failed. Below is another of this same problem; if you guys explained it, I'm not following....

We did.
From post 8: "It isn't always necessary to multiply each equation by the coefficient of x in the other; if they have a common factor, you can use smaller numbers to make the coefficients "equal and opposite" so they will cancel. "
From post 16: "Sometimes, we work first on x, other times on y. (or w or q or whatever the variables are.)"

 

[Dr.Peterson]

I'll explain this as if no one had explained it before.

You have the equations

9x + 8y = -15
7x + 2y = 1

They chose to eliminate y. The coefficients are 8 and 2; we want to make them equal and opposite (or as they say, "differ only in sign"). How can you do that? The easiest way is to multiply the 2 by -4, making it -8. So that's what they do.

If you need a rule to follow, the LCM of 8 and 2 is 8 (since 8 is a multiple of 2), so the goal is to make the coefficients 8 and -8. The rest is what we see.

 

[150mph]

Thanks DP, this helps. After re-reading all the responses and working a few more problems, it’s finally sinking in that I wasn't understanding that in his solution to the first problem, the author introduces a solution that requires multiplying both top and bottom equations to get a LCM, which I assumed was required to solve all these types of problems. Until the next problem, I’ve been trying to make all the advice here fit that change-both-top-and-bottom scenario. Confirmation bias perhaps. Phrases like "..it isn't always necessary" and "..you can do this.." can be confusing because they appear to offer alternatives when I haven't grasped even the primary logic yet.

Then in the last problem, he comes up with a number so that changing the coefficient in only one equation seems to be preferable when possible.

It's likely I failed to communicate where I was getting hung up but explaining misunderstandings in writing can be challenging. So yes, I like an omni-rule to follow that works every time I decide to "eliminate x" in the same way as the lesson author.

Thanks all for taking the time to help!

 

[Dr.Peterson]

You can still apply the general method, by saying that you multiplied one equation by -4 and the other by 1!

Or you can say that you multiply either or both equations, as needed, to obtain equal and opposite coefficients. The goal is primary; exactly what you do doesn't matter so much.

By the way, just yesterday I was working with a student face-to-face, and she just couldn't explain to me why my answers didn't deal with her question about a problem. Sometimes all I can do is encourage the student to talk herself through a problem, and listen, hoping to figure out what the hidden issue is. The hard part is then to listen patiently, without interrupting. (In this case, I was already 10 minutes past the time I was supposed to leave, and I hadn't had lunch, so I just had to walk away ...

 

[Steven G]

Don't pee on my knee and tell me it's raining. There is a perfectly good explanation for -4. You are not making any effort to understand their solution.
And we did cover this case.

I remember lev888 explanation this quite well!