Questions: Find two functions defined implicitly by this equation. 6x+2y^2=10 y= ± sqrt(?+ x)

Transcript text: Find two functions defined implicitly by this equation. \[ \begin{array}{c} 6 x+2 y^{2}=10 \\ y= \pm \sqrt{[?]+\quad x} \end{array} \]

Solution

Solution Steps

Step 1: Rearranging the Equation

We start with the implicit equation given by $6x + 2y^2 = 10$ . To isolate $y^2$ , we rearrange the equation as follows: $2y^2 = 10 - 6x$ Dividing both sides by 2 gives: $y^2 = 5 - 3x$

Step 2: Solving for

y

Next, we take the square root of both sides to solve for $y$ : $y = \pm \sqrt{5 - 3x}$ This results in two functions defined implicitly by the original equation.

Final Answer

The two functions defined implicitly by the equation are: $y = \sqrt{5 - 3x} \quad \text{and} \quad y = -\sqrt{5 - 3x}$ Thus, the final answer is: $\boxed{y = \pm \sqrt{5 - 3x}}$

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