1/2 (2x+3y )+12/7 (3x -2y )=1/2 :7/(2x +3y )+4/(3x -2y )=2
To solve this system of equations, we will use the method of substitution.
Step 1: Solve the first equation for one variable in terms of the other.
1/2 (2x + 3y) + 12/7 (3x - 2y) = 1/2
Multiply through the equation by 2 to get rid of the fractions:
(2/2)(2x + 3y) + (2/7)(12/7)(3x - 2y) = (2/2)(1/2)
Simplify:
2x + 3y + (24/7)(3x - 2y) = 1/2
Multiply out:
2x + 3y + (72/7)x - (48/7)y = 1/2
Step 2: Solve the second equation for one variable in terms of the other.
7/(2x + 3y) + 4/(3x - 2y) = 2
Multiply through by (2x + 3y)(3x - 2y) to get rid of the denominators:
7(3x - 2y) + 4(2x + 3y) = 2(2x + 3y)(3x - 2y)
Simplify:
21x - 14y + 8x + 12y = 2(6x^2 - 4xy + 9xy - 6y^2)
Combine like terms:
29x - 2y = 12x^2 - 4xy + 18xy - 12y^2
Step 3: Set the two equations equal to each other and solve for one variable.
2x + 3y + (72/7)x - (48/7)y = 29x - 2y = 12x^2 - 4xy + 18xy - 12y^2
Combine like terms:
(74/7)x - (50/7)y = 17x - 2y = 12x^2 + 14xy - 12y^2
Multiply through by 7 to eliminate fractions:
74x - 50y = 7(17x - 2y) = 12(7x^2 + 7xy - 7y^2)
Step 4: Simplify and rearrange the equation.
74x - 50y = 119x - 14y = 84x^2 + 84xy - 84y^2
Combine like terms:
-45x + 36y = 84x^2 + 98xy - 84y^2
Step 5: Rearrange the equation into standard quadratic form.
84x^2 + 98xy - 84y^2 - 45x + 36y = 0
This equation represents a conic section known as a hyperbola. To further solve it, more information is needed, such as additional constraints or conditions.
it would be nice if you asked a question, instead of just posting gibberish. At first I thought you wanted to solve two simultaneous equations:
1/2 (2x+3y )+12/7 (3x -2y )=1/2
7/(2x +3y )+4/(3x -2y )=2
but that became unwieldy fast.
Are you trying to clear fractions?
1/2 (2x+3y )+12/7 (3x -2y )=1/2
multiply by 14:
7(2x+3y) + 24(3x-2y) = 7
14x+21y+72x-48y = 7
84x-27y = 7
or something else entirely?