SOLUTION: 2/x-5/y=5 3/x+10/y=18 find the point where they intersect?

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Question 1049120: 2/x-5/y=5
3/x+10/y=18
find the point where they intersect?
Found 3 solutions by josgarithmetic, MathTherapy, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Try just solving through substitution. Take one equation and solve for one variable. Substitute for this variable in the other equation. Solve for the single variable's value. Now find the value of the other variable, either equation.

Here is a start.
2%2Fx=5%2Fy%2B5
2%2Fx=5%2Fy%2B5y%2Fy
2%2Fx=%285%2B5y%29%2Fy
x%2F2=y%2F%285y%2B5%29
x=%282y%29%2F%285y%2B5%29
You continue the necessary steps for the process..

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

2/x-5/y=5
3/x+10/y=18
find the point where they intersect?
Easiest method: ELIMINATION

2y - 5x = 5xy -------- eq (i)	

3y + 10x = 18xy ------ eq (ii)	

4y - 10x = 10xy	------ Multiplying eq (i ) by 2 ------- eq (iii)

7y = 28xy ---- Adding eqs (iii) & (ii)	

7y%2F28y+=+x	

highlight_green%28x+=+1%2F4%29	

3y+%2B+10%281%2F4%29+=+18y%281%2F4%29 ------- Substituting 1%2F4 for x in eq (ii)

3y+%2B+5%2F2+=+9y%2F2

6y + 5 = 9y ----- Multiplying by LCD, 2	

5 = 9y - 6y 

5 = 3y

highlight_green%28y+=+5%2F3%29

Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.
2/x-5/y=5
3/x+10/y=18
find the point where they intersect?
~~~~~~~~~~~~~~~~~~~~~~~~

The way which "josgarithmetic" offers to you is wrong and without hope to get the result.
It is because this person (I can not call him as a "tutor") is mathematically illiterate and has no any notion on
how to approach the problem correctly.

I will show you the way. 

Introduce new variables u = 1%2Fx  and  v = 1%2Fy.

Then your system takes the form

2u -  5v =  5,    (1)
3u + 10v = 18.    (2)

Let us solve this system by the Elimination method. 
Multiply equation (1) by 2 (both sides) and then add to the equation (2).
You will get

4u + 3u = 10 + 18  --->  7u = 28  --->  u = 28%2F4 = 4.

Next, from (2)  10v = 18 - 3u = 18 - 3*4 = 6  --->  v = 6%2F10 = 3%2F5 = 0.6.

Now recall that u = 1%2Fx.  Hence,  x = 1%2Fu = 1%2F4.

Similarly,  v = 1%2Fy.  Hence,  y = 1%2F%28%283%2F5%29%29 = 5%2F3.

The problem is solved.  The solution to the given/original system is found: x =1%2F4,  y = 5%2F3.

The intersection point is  (x,y) = (1%2F4,5%2F3).
Solved.

It is a standard way solving such problems.
See the lessons
    - Solving systems of non-linear equations in two unknowns using the Cramer's rule
    - Solving systems of non-linear equations in three unknowns using Cramer's rule
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the section "Systems of equations that are not linear".