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.
Important Questions from Identities
If \(2x + { {1} \over 3x}= 5, x ≠ 0\) , then what is the value of \(27x^3+{{1} \over 8x^3}\) ?
If x 2 + 4y 2 = 40, xy = 6 and x > 2y then the value of x - 2y is:
- \(\dfrac{(0.73)^3+(0.31)^3}{(0.73)^2-0.73\times0.31+(0.31)^2}\)
- \(\dfrac{(5.17-2.19)^2-(5.17+2.19)^2}{11.3223}\)
If a(a + b + c) 2= 1792; b(a + b + c) 2= 1536; c(a + b + c) 2= 768 then what will be the value of a?
