SOLUTION: please help me solve {{{25x^2-y^2=36}}} 5x+y=2

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Question 1182622: please help me solve
25x%5E2-y%5E2=36
5x+y=2
Found 3 solutions by MathLover1, ikleyn, greenestamps:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

assuming you have a system to solve:
25x%5E2-y%5E2=36....eq.1
5x%2By=2.....eq.2
-------------------------------
5x%2By=2.....eq.2.......solve for y

y=2-5x....eq.2a.substitute in eq.1

25x%5E2-%282-5x%29%5E2=36....eq.1...solve for x
25x%5E2-%284-20x%2B25x%5E2%29=36
25x%5E2-4%2B20x-25x%5E2=36
20x=36%2B4
20x=40
x=2

go to
y=2-5x....eq.2a, substitute x
y=2-5%2A2
y=2-10
y=-8

solutions:
x+=+2, y+=+-8




Answer by ikleyn(52777) About Me  (Show Source):
You can put this solution on YOUR website!
.

Your starting equations are

    25x^2 - y^2 = 36      (1)

    5x    + y   =  2      (2)


This system of two equations (one equation of the degree 2 and the other of the degree 1) is very special.


Although one of the starting equations is of the degree 2, the system can be reduced to the system 
of two LINEAR equations, which can be EASILY solved.


The first equation admits factoring

    (5x - y)*(5x + y) = 36    (3)


In this factored equation, we can replace (5x + y) by the value of 2, based on equation (2).
Doing it, we transform equation (3) to the form

    2*(5x - y) = 36,  

or

    5x - y = 18             (4)

(where 18 = 36/2).


Thus from the original non-linear system of equation, we get the linear system

    5x - y = 18             (4)

    5x + y =  2             (5)


The system of linear equations (4), (5) is EQUIVALENT to the nonlinear system of equations (1), (2).


To solve equations (4), (5), use the ELIMINATION method and add equations (4) and (5).  You will get

    10x = 18 + 2 = 20

      x          = 20/10 = 2.


To find y, substitute the found value x= 2 into equation (5).  You will get

    5*2 + y = 2,  y = 2 - 10 = -8.


Thus the original system of equations (1), (2) has a UNIQUE solution (x,y) = (2,-8).

Solved.

-----------

The fact that the solution is unique  tells  us  that the straight line,  defined by equation  (2),
is tangent to the hyperbola defined by equation  (1).



Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


Tutor @MathLover1 will virtually always solve any system of two equations in two unknowns using substitution. That is a valid method; but often there are much easier ways.

A student with only a little bit of problem solving experience will see the first equation as a difference of squares and will see if that leads to an easy path to the solution.

And it does, as shown in the solution by tutor @ikleyn.

25x%5E2-y%5E2=36 --> %285x%2By%29%285x-y%29=36

But the other equation tells us 5x%2By=2

so

2%285x-y%29=36
5x-y=18

And now we have a system of two linear equation that is easily solved using elimination:

5x%2By=2
5x-y=18

The lesson from this:

Don't be stuck with one method for solving a particular type of problem. Instead, look for clues in the given problem that could possibly lead to a much faster and easier solution to the problem.