# If 5x-11y = 2x+5y, then find the value of 3x² + 2y² : 3x² - 2y²

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ok lets start

5x-11y=2x+5y 3x^2+2y^2:3x^2-2y^2

3x-11y=5y 3x^2+2y^2:3x^2-2y^2

3x=16y 3x^2+2y^2:3x^2-2y^2

x= 5.333...yor 5+1/3y 3x^2+2y^2:3x^2-2y^2

3x^2+2y^2:3x^2-2y^2

3(5+1/3y)^2+2y^2:3(5+1/3y)^2-2y^2

16y^2+2y^2:16y^2-2y^2

256y+4y:256y+4y

260y:260y

y:y or 1:1

hope this helps

First of all, let's focus on the first condition given by the ennunciation, namely 5x-11y = 2x+5y.

We'll group the term in "x" into the left side of the equal and the terms in "y" into the right side and we'll do the math:

5x-2x = 11y+5y

**3x=16y**

For the moment, let's stop in this point of action.

Now,l let's focus on the exression which we have to calculate:

(3x² + 2y²) : (3x² - 2y²)

The expression at numerator, (3x² + 2y²), we could re-write it in this way:

(3x² + 2y²)= (3x + 2y)² - 2*3x*2y

The expression at denominator, (3x² - 2y²), is a difference ofsquares and it could be written as:

(3x² - 2y²)= (3x - 2y)*(3x + 2y)

Now, let's put together the found expressions:

(3x² + 2y²) : (3x² - 2y²)= [(3x + 2y)² - 2*3x*2y]/(3x - 2y)*(3x + 2y)

In the end, let's turn back at the found condition:

**3x=16y**

We'll apply some tricks on this condition, depending on the last form of the expression which we have to calculate:

**3x=16y**

3x + 2y=16y+2y

**3x + 2y=18y**

3x - 2y=16y-2y

**3x - 2y=14y**

Now, all we have to do is to substitute the calculated expressions above, into our expression:

[(3x + 2y)² - 2*3x*2y]/(3x - 2y)*(3x + 2y)=[(18y)²- 2*16y*2y]/(14y)*(18y)

[(18y)²- 2*16y*2y]/(14y)*(18y)=[(18y)²-2²*4²*y²]/14*18*y²=

But, at the numerator we have again a difference of squares:

[(18y)²-2²*4²*y²]=(18y-8y)(18y+8y)=10y*26y

[(18y)²-2²*4²*y²]/14*18*y²=10y*26y/14*18*y²

10y*26y/14*18*y²=2*5*2*13*y²/2*7*2*9*y²

After simplifying:

**3x² + 2y² : 3x² - 2y²=65/63**

By data : 5x-11y=2x+5y

To find the ratio, 3x² + 2y² : 3x² - 2y²

Solution:

5x -11y=2x+5y Or

5x-2x= 5y+11y=16y Or

3x=16y Or

x=(16/3)y. Therefore,

x^2 = (16y/3)^2. Substituting in 3x² + 2y² : 3x² - 2y², we get:

{3(16y^2/3^2)+2y^2} : {3(16y/3)^2-2y^2}. Multiplying both terms by 3^2, we get the ratio,

We can reduce by y^2 both atecedent and precedent terms of the ratio.

(3*16^2+ 2*3^2)y^2 : (3*16^2-2*3^2)Y^2,

Dividing by y^2, we get:

=3*16^2+ 2*3^2 : 3*16^2-2*3^2

= 786 : 750. Both terms of the ratio could be further reduced by 6 and obtain the ratio as:

=131: 125.