Unsure if the answer is correct and if so, why?

[Johnny+Mum]

Hi All,

I'm a homeschooling Mum, helping my 13yo son "Johnny" with his math work. It's been a long time since I did math at school, but mostly I do okay or can work it out. Sometimes we get stuck or occasionally the answer book is wrong, so we figure it out and learn together.

I'm unsure if this should be in the basic instead of intermediate algebra thread, so if I'm in the wrong place, just let me know and I'll move my post. =)

Today, we're unsure if the breakdown down for the problem in the answer book is correct, and if it is, can't quite figure out what we're missing. See attached images of the original problem + the answer book breakdown.

So, the problem we're working on is, "Explain why this works, using algebra" relative to extension/challenge yourself #3 in the first photo.

So far, we've been able to work through and with a little prompting and discussion, Johnny has been able to come up with the expression process through to the third step show as

= 100xsquared + 10xz + 10xy + yz.

We are now stuck though, because, though I understand that because of the "where z + y = 10" condition/paramater, they have separated those terms out and then how the rest of that plays out, I do NOT understand where the extra 10x went as it seems to me (and to Johnny because I ended up showing him their process because I couldn't figure out what I was missing) that you should still be adding 10x to 10x and therefore, the next line should be

100xsquared + 20x(z + y) + yz
which would then become

100xsquared + 20x(10) + yz
and so on from there.

Is someone able to help me to understand why the second 10x (from the 10xy) disappears?

Thanks so much in advance.

 

[Dr.Peterson]

So far, we've been able to work through and with a little prompting and discussion, Johnny has been able to come up with the expression process through to the third step show as

= 100xsquared + 10xz + 10xy + yz.

We are now stuck though, because, though I understand that because of the "where z + y = 10" condition/paramater, they have separated those terms out and then how the rest of that plays out, I do NOT understand where the extra 10x went as it seems to me (and to Johnny because I ended up showing him their process because I couldn't figure out what I was missing) that you should still be adding 10x to 10x and therefore, the next line should be

100xsquared + 20x(z + y) + yz
which would then become

100xsquared + 20x(10) + yz
and so on from there.

Is someone able to help me to understand why the second 10x (from the 10xy) disappears?

I'm sorry about the delay here; there have been technical problems on the site, including access by moderators.



Since you have no trouble with the step from the first line here to the second, you must understand distribution: a(b+c) = ab + ac.

From the second line to the third, they are doing the reverse, which is called factoring (or factorizing): "undistributing" by taking a common factor, in this case 10x, outside of the parentheses.

You can see that this result is right, because if you start with 10x(y+z) and distribute, you will get 10xy + 10xz, which agrees with the previous line. Do you see this? Do you see that your 20 wouldn't work?