Knowing when and exactly how to split a simultaneous equation

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Parallel_Platypus

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When working on a problem in Geometry, I came across this simultaneous equation:
xy=24 and 2x+2y=20
After the help of Wolfram and some forum advice, I found the answer, it is either 4 or 6.
This is how:
Make x the subject of the first equation: x=24/y
Substitute into 2:
2(24/y)+2y=20
Expand= 48/y+2y=20
Multiply by y:
48+2y^2=20y
(I used Wolfram after this)
Move to the left side:
48/y+2y-20y=0
Factorise:
2(y-4)(y-6)=0
get rid of 2:
(y-4)(y-6)=0
This was all logical to me, then it said to simply "split" the equation and solve each bracket:
y-4=0
y=4 and
y-6=0
y=6
These answers are correct, but I am curious as to why the equation was split, how it is done, and how to know when it should be done. Thankyou!
 
Parallel_Platypus said:
When working on a problem in Geometry, I came across this simultaneous equation:
xy=24 and 2x+2y=20
After the help of Wolfram and some forum advice, I found the answer, it is either 4 or 6.
This is how:
Make x the subject of the first equation: x=24/y
Substitute into 2:
2(24/y)+2y=20
Expand= 48/y+2y=20
Multiply by y:
48+2y^2=20y
(I used Wolfram after this)
Move to the left side:
48/y+2y-20y=0
Factorise:
2(y-4)(y-6)=0
get rid of 2:
(y-4)(y-6)=0
This was all logical to me, then it said to simply "split" the equation and solve each bracket:
y-4=0
y=4 and
y-6=0
y=6
These answers are correct, but I am curious as to why the equation was split, how it is done, and how to know when it should be done. Thankyou!
Click to expand...
To build a little on what Romesk said.

The values of x such that g(x) = 0 are called the roots (or zeroes) of that function.

Any function that can be expressed as the product of n functions, \(\displaystyle g(x) = f_1(x) *\ ...\ f_n(x)\), has as roots the roots of \(\displaystyle f_i(x) = 0\ for\ 1 \le i \le n.\)

The functions need not be polynomials.
 
well this is interesting :D

I am not sure I fully understand all that Jeff M wrote about the zeroes of a number, but I think I understand the concept, and at the very least I know what it is called so I can do some more research. I am in year 9 at the moment, so I am guessing this will pop up later in school xD So to summarise, when you have the equation to a point where it cannot be simplified any more, and it equals 0. You have to find the possible values that make the equation true, in this case making it equal 0. Could someone give another example in a different equation so I could practise. I feel like I have been looking at the same problem too much :) thank you all
 
Thanks

I have done most of this work on factorisation in class, I just have never encountered an equation which has two solutions other then quadratic ones (Sq root of 16: 4, or -4.)
Thanks everyone
 
If my little explanation can help. If two equations have the same two variables( like X and Y). You will know that the equations are called simultaneous equation. So you need to solve it simultaneously as jeffm did above. Your problen above is more complicated than normal simultaneous equation. It is called simultaneous equation leading to quadratic equation. So, try to look for more problems and solve them as jeffm did above then you will get use to it. Thank you
 
Am sorry for that. I don't believe everybody is used to abbreviation. That's the reason I write everything in full. And again, am in university studying mathematics.
 
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