(2y-3)%3D6xy.jpeg "(3x+2)(2y-3)=6xy")
Solving the Equation (3x+2)(2y-3) = 6xy
This equation involves two variables, x and y, and presents an interesting challenge in terms of solving for specific values. Let's explore how to approach this problem:
Understanding the Equation
The equation (3x+2)(2y-3) = 6xy represents a relationship between x and y. It implies that for any values of x and y that satisfy this equation, the product of the two expressions on the left side will always equal 6 times the product of x and y.
Simplifying the Equation
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Expand the left side: (3x + 2)(2y - 3) = 6xy + 4y - 9x - 6
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Rearrange the terms: 6xy + 4y - 9x - 6 = 6xy 4y - 9x - 6 = 0
Analyzing the Simplified Form
The simplified equation, 4y - 9x - 6 = 0, represents a linear equation. This means it describes a straight line on a graph where any point on that line will satisfy the original equation.
Solving for Specific Values
While we cannot find a single solution for x and y, we can express one variable in terms of the other. Let's solve for y:
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Isolate the y term: 4y = 9x + 6
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Divide both sides by 4: y = (9/4)x + (3/2)
This equation now tells us that for any value of x, we can find a corresponding value for y that will satisfy the original equation.
Example:
Let's say x = 4. Substituting into the equation for y, we get:
y = (9/4)*4 + (3/2) = 9 + 1.5 = 10.5
Therefore, the point (4, 10.5) is a solution to the equation (3x+2)(2y-3) = 6xy.
Conclusion
The equation (3x+2)(2y-3) = 6xy represents a linear relationship between x and y. It does not have a unique solution, but we can express one variable in terms of the other, allowing us to find specific points that satisfy the equation. This understanding is crucial for analyzing and working with such equations in various applications.