When given an equation in terms of x and y, it is often useful to determine the inverse equation in order to understand the relationship between the two variables. In this case, we are tasked with finding the inverse equation of (x-4)^2 – 2/3 = 6y – 12. By following a systematic approach, we can first solve for y in terms of x and then proceed to find the inverse equation.
Solving for y in terms of x
To solve for y in terms of x, we first isolate the term involving y on one side of the equation. Starting with (x-4)^2 – 2/3 = 6y – 12, we can add 12 to both sides to simplify the equation to (x-4)^2 – 2/3 + 12 = 6y. Next, we can combine the constants on the right side to get (x-4)^2 + 34/3 = 6y. Finally, by dividing both sides by 6, we arrive at the expression y = (x-4)^2/6 + 34/18.
This process of solving for y in terms of x allows us to express y solely in terms of x, which is crucial in determining the inverse equation. By manipulating the original equation and isolating y, we have successfully obtained a formula that represents the relationship between x and y in a simplified form. This step sets the stage for the next part of the process, which involves finding the inverse equation.
Finding the inverse equation
The inverse equation is essentially a rearrangement of the equation solved for y in terms of x, but with x and y swapped. In this case, the equation y = (x-4)^2/6 + 34/18 can be rewritten as x = (y-4)^2/6 + 34/18. By interchanging x and y, we have successfully derived the inverse equation of the original equation. This inverse equation now represents the relationship between y and x in a form where y is expressed as a function of x.
By determining the inverse equation of (x-4)^2 – 2/3 = 6y – 12, we have gained insight into the inverse relationship between x and y. Through a systematic process of solving for y in terms of x and finding the inverse equation, we have successfully navigated through the mathematical steps required to uncover this relationship. This approach not only enhances our understanding of the original equation but also provides a new perspective on how x and y are interconnected in the given equation.
In conclusion, the journey to determining the inverse equation of a given equation involves strategic manipulation of variables and a systematic approach to rearranging the terms. By following the steps outlined above, we are able to unlock the inverse relationship between x and y in the equation (x-4)^2 – 2/3 = 6y – 12. This process not only enhances our mathematical skills but also deepens our understanding of the underlying connections between variables in equations. The ability to find the inverse equation is a valuable tool in mathematical analysis and provides a new lens through which to view the relationships between different variables.