Question 44277
<pre><font size = 5><b>Can you solve x in terms of y and y in terms of x 
for the following equation:

3y² + 4xy - 9x² = -1

Yes with the quadratic formula:

To solve for y in terms of x

3y² + 4xy - 9x² = -1

Get 0 on the right

3y² + 4xy + 1 - 9x² = 0

Now we write it this way

(3)y² + (4x)y + (1 - 9x²) = 0

so you can compare it with

  Ay² + By + C = 0

A = 3, B = 4x and C = (1 - 9x²)

               ___________________
      -(4x) ± <font face = "symbol">Ö</font>(4x)²-4(3)(1 - 9x²)
y = -------------------------------
              2(3) 

             ________________
      -4x ± <font face = "symbol">Ö</font>16x²-12(1 - 9x²)
y = -------------------------------
              6

             _______________
      -4x ± <font face = "symbol">Ö</font>16x²-12 + 108x²
y = --------------------------
              6
 
             ________
      -4x ± <font face = "symbol">Ö</font>124x²-12 
y = ---------------------
              6

             __________
      -4x ± <font face = "symbol">Ö</font>4(31x²-3) 
y = ---------------------
              6
              ______
      -4x ± 2<font face = "symbol">Ö</font>31x²-3 
y = ------------------
              6

               ______
      2(-2x ± <font face = "symbol">Ö</font>31x²-3) 
y = ------------------
            6

      1         ______
      <s>2</s>(-2x ± <font face = "symbol">Ö</font>31x²-3) 
y = ------------------
            <s>6</s>
            3

             ______
      -2x ± <font face = "symbol">Ö</font>31x²-3) 
y = ------------------
           3

That's y solved in terms of x

To solve for x in terms of y:

3y² + 4xy - 9x² = -1

Get 0 on the right

3y² + 4xy + 1 - 9x² = 0

Now we write it this way

-9x² + 4yx + 1 + 3y² = 0

Multiply thru by -1 because it's
easier when the squared term is 
positive:

9x² - 4yx - 1 - 3y² = 0


(9)x² + (-4y)x + (-1 - 3y²) = 0

so you can compare it with

  Ax² + Bx + C = 0

A = 9, B = -4y and C = (-1 - 3y²)

I'll let YOU solve this, and the answer is
            ______
      2y ± <font face = "symbol">Ö</font>31y²+9) 
x = ------------------
           9

That's x solved in terms of y

Edwin</pre>