Solving for the Positive Value of 3x - 2y

We are given two equations:

• Equation 1: \(3x + 2y = 10\)
• Equation 2: \(2xy = 3\)

Our goal is to find the positive value of the expression \(3x - 2y\).

Using Algebraic Identities

We can use the algebraic identity that relates the square of a sum and the square of a difference:

$ (a - b)^2 = (a + b)^2 - 4ab $

Let \(a = 3x\) and \(b = 2y\). Substituting these into the identity, we get:

$ (3x - 2y)^2 = (3x + 2y)^2 - 4(3x)(2y) $

Simplify the term \(4(3x)(2y)\):

$ 4(3x)(2y) = 24xy $

We are given \(2xy = 3\). To find \(24xy\), we can multiply the given value by 12:

$ 24xy = 12 \times (2xy) = 12 \times 3 = 36 $

Now substitute the known values into the expanded identity:

• \( (3x + 2y)^2 = (10)^2 = 100 \)
• \( 24xy = 36 \)

So, the equation becomes:

$ (3x - 2y)^2 = 100 - 36 $

$ (3x - 2y)^2 = 64 $

Determining the Positive Value

To find the value of \(3x - 2y\), we take the square root of both sides:

$ 3x - 2y = \pm \sqrt{64} $

$ 3x - 2y = \pm 8 $

The question specifically asks for the positive value.

Therefore, the positive value of \(3x - 2y\) is 8.