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

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. \[ \begin{array}{c} 6 x+2 y^{2}=10 \\ y= \pm \sqrt{[?]+\quad x} \end{array} \]
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Solution

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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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