if 3x - 4y = 10 and xy = -1, find the value of 9x^2 + 16y^2

Question

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Answer 1

It could be solved as following as well

(3x - 4y)² = (3x)²-2(3x)(4y)+(4x)²

=9x²-24xy+16y²= 100= 10², xy=1

9x²⁺16y²= 100-24xy=100-24=76.

Answer 2

3x - 4y = 10 and xy = -1

 

y = -1/x from 2nd equation.

Substitute into the first.

 

3x - 4(-1/x) = 10

3x + 4/x = 10

multiply through by x

3x² + 4 = 10x

3x² - 10x + 4 = 0

 

Using quadratic equation solve for x:

x = [10 ±√[(-10)²-4(3)(4)]]/(2(3))

x = [10 ±√52]/6

x = (10 ±2√13)/6

x = (5 ±√13)/3

 

y = -1/x   so:

 

For x = (5+√13)/3 

y = -3/(5+√13)

 

For x = (5-√13)/3

y = -3/(5-√13)

*******************

 9x² + 16y²

 

9[(5+√13)/3]² + 16[-3/(5+√13)]²

 

= 9[(38+10√13)/9] + 16[9/(38+10√13)

 

= 38 + 10√13 + 144/(38+10√13)

= 76

 

Now try doing the same with

x = (5-√13)/3
y = -3/(5-√13)