Need help with solving this equation: x + 2y = 5, 2x - y = 0

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Savrot

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Sep 28, 2017
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Hi! When i did math back in the day i did not fully understand WHY math did what it did. I just learned the rules and saw them just as that, as rules, rather than something logic that i could understand. So im relearning and i need your help. I'm currently on algebra and have this equation:

x + 2y = 5
2x - y = 0

So the first thing i do is to subtract 2y on both sides: x = 5 -2y

I understand that since this is an equation, you can add and subtract if you do it on both sides. Since equation means that the both sides are the same.

Then i add it into the second equation: 2(5 -2y) - y = 0

Now, i do not understand WHY i put them in parentheses and not just into the equation like ths: 2 * 5 -2y - y = 0

Is it because i need to multiply it with both? If X where a single digit, could i just insert it without the parentheses?


My last question is, if i keep going with the equation i get this: 10 - 5y = 0

So my question is, can i just add the two Y:s together like that and can you always do it?


Here is how i did it for clarification.
2(5 -2y) - y
10 - 5y
 
The principle of Substitution is important. Example:

1) 2x - 5y = 3

2) x = 5 - 6y

We wish to rewrite equation #1, substituting a different expression for 'x'.

It's also important to note that 2x is EXACTLY the same as 2(x). The important thing is the get that 2 multiplied by ALL THAT IS x.

If x = Frog, we have 2(Frog) - 5y = 3. In this case, I added the parentheses only because "xFrog" looks funny and I can't quite tell that it means multiplication. "x(Frog)" is more clear.

If we have equation #2, we have 2(5-6y) - 5y = 3. In this case, I added the parentheses because without them we cause confusion and manage incorrect results.

Without parentheses, we might get: 25-6y - 5y = 3 -- This is no good. We should not be seeing a "25" at all.
Even if we add a multiplication symbol: 2*5-6y - 5y = 3 -- This is no good since that 2 must be multiplied by ALL THAT IS x and we have not multiplied it by the 6.

With the parentheses, 2(5-6y) - 5y = 3, we get 2*5 - 2*(6y) - 5y = 3 or 10 - 12y - 5y = 3

There is no harm in adding additional parentheses. 2*6y is the same as 2*(6y). One or the other may be more clear in some circumstances.
There is much harm in parentheses that must be there. 2*5-6y is NOT the same as 2*(5-6y).
 
Savrot said:
Hi! When i did math back in the day i did not fully understand WHY math did what it did. I just learned the rules and saw them just as that, as rules, rather than something logic that i could understand. So im relearning and i need your help. I'm currently on algebra and have this equation:

x + 2y = 5
2x - y = 0

So the first thing i do is to subtract 2y on both sides: x = 5 -2y

I understand that since this is an equation, you can add and subtract if you do it on both sides. Since equation means that the both sides are the same.
Click to expand...
Yes, that's good.

Then i add it into the second equation: 2(5 -2y) - y = 0

Now, i do not understand WHY i put them in parentheses and not just into the equation like ths: 2 * 5 -2y - y = 0
Click to expand...
Because "2x" means the "2" is multiplied by all of x: 2*5- 2y would just be 10- 2y. 2(5- 2y)= 2*5- 2*2y= 10- 4y.

You can clarify that by taking y to be various number. If y= 1 then x= 5- 2y= 5- 2(1)= 3. 2x= 2(3)= 6 and 2(5- 2y)= 2(5- 2)= 2(3)= 6 while 2*5- 2y would be 2*5- 2= 10- 2= 8.

Again, a(b+ c)= ab+ ac while a*b+ c is just ab+ c.

Is it because i need to multiply it with both? If X where a single digit, could i just insert it without the parentheses?
Click to expand...
What do you mean by "a single digit"? You do not yet know x (not "X", in algebra "x" and "X" are not the same thing) so you do nor know how many "digits" there are in the base 10 expression of x.

My last question is, if i keep going with the equation i get this: 10 - 5y = 0

So my question is, can i just add the two Y:s together like that and can you always do it?
Click to expand...
Yes, 2(5- 2y)- y= 10- 4y- y= 0. Yes, 4 of anything plus one of the same thing is 5 of that thing! 4y+ y= 5y and -4y- y= -(4y+ y)= -5y.


Here is how i did it for clarification.
2(5 -2y) - y
10 - 5y
Click to expand...

I recommend that, until you are really comfortable with algebra (another two or three years, say) you write it in more detail:
2(5- 2y)- y= 10- 4y- y= 10- 5y= 0.

Now what is y? And, knowing that x= 5- 2y, what is x?
 
tkhunny said:
The principle of Substitution is important. Example:

1) 2x - 5y = 3

2) x = 5 - 6y

We wish to rewrite equation #1, substituting a different expression for 'x'.

It's also important to note that 2x is EXACTLY the same as 2(x). The important thing is the get that 2 multiplied by ALL THAT IS x.

If x = Frog, we have 2(Frog) - 5y = 3. In this case, I added the parentheses only because "xFrog" looks funny and I can't quite tell that it means multiplication. "x(Frog)" is more clear.

If we have equation #2, we have 2(5-6y) - 5y = 3. In this case, I added the parentheses because without them we cause confusion and manage incorrect results.

Without parentheses, we might get: 25-6y - 5y = 3 -- This is no good. We should not be seeing a "25" at all.
Even if we add a multiplication symbol: 2*5-6y - 5y = 3 -- This is no good since that 2 must be multiplied by ALL THAT IS x and we have not multiplied it by the 6.

With the parentheses, 2(5-6y) - 5y = 3, we get 2*5 - 2*(6y) - 5y = 3 or 10 - 12y - 5y = 3

There is no harm in adding additional parentheses. 2*6y is the same as 2*(6y). One or the other may be more clear in some circumstances.
There is much harm in parentheses that must be there. 2*5-6y is NOT the same as 2*(5-6y).
Click to expand...

Very good explanation, I understood immediately. Thank you so much!
 
HallsofIvy said:
What do you mean by "a single digit"? You do not yet know x (not "X", in algebra "x" and "X" are not the same thing) so you do nor know how many "digits" there are in the base 10 expression of x.
Click to expand...

I meant what if it's just one number. I understand now that that it would just be for example 2(2). I understand if it didn't make sense.

HallsofIvy said:
Yes, 2(5- 2y)- y= 10- 4y- y= 0. Yes, 4 of anything plus one of the same thing is 5 of that thing! 4y+ y= 5y and -4y- y= -(4y+ y)= -5y.
Click to expand...

Thank you for this answer!


HallsofIvy said:
I recommend that, until you are really comfortable with algebra (another two or three years, say) you write it in more detail:
2(5- 2y)- y= 10- 4y- y= 10- 5y= 0.

Now what is y? And, knowing that x= 5- 2y, what is x?
Click to expand...

Yes that is good advice! I will write in detail from now on. It's a good way to drill these exercises.

2(5-2y) - y = 0, 10 - 5y = 0

So since we know that the sum is 0 y must be 2, 10 - 5(2) = 0

Now that we now that y = 2 we can return to the first equation.

x = 5- 2(2)

Therefore x = 1

Is this correct?
 
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