Questions: Determine whether the equation defines y as a function of x. 2x-3y=5 The equation defines y as a function of x. The equation does not define y as a function of x. Determine whether the equation defines y as a function of x. x^2+(y-5)^2=6 The equation defines y as a function of x. The equation does not define y as a function of x.

Determine whether the equation defines y as a function of x. 2x-3y=5 The equation defines y as a function of x. The equation does not define y as a function of x. Determine whether the equation defines y as a function of x. x^2+(y-5)^2=6 The equation defines y as a function of x. The equation does not define y as a function of x.
Transcript text: Determine whether the equation defines $y$ as a function of $x$. \[ 2 x-3 y=5 \] The equation defines $y$ as a function of $x$. The equation does not define $y$ as a function of $x$. Determine whether the equation defines $y$ as a function of $x$. \[ x^{2}+(y-5)^{2}=6 \] The equation defines $y$ as a function of $x$. The equation does not define $y$ as a function of $x$.
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Solution

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Solution Steps

Step 1: Solve for \( y \) in the first equation

Start with the equation: \[ 2x - 3y = 5 \] Subtract \( 2x \) from both sides: \[ -3y = -2x + 5 \] Divide both sides by \( -3 \): \[ y = \frac{2x - 5}{3} \] Since \( y \) is expressed explicitly as a function of \( x \), the equation defines \( y \) as a function of \( x \).

Step 2: Analyze the second equation for \( y \) as a function of \( x \)

Consider the equation: \[ x^{2} + (y-5)^{2} = 6 \] This is the equation of a circle with center \( (0, 5) \) and radius \( \sqrt{6} \). For a given \( x \), there are two possible values of \( y \) (one above and one below the center). Therefore, \( y \) is not uniquely determined by \( x \), and the equation does not define \( y \) as a function of \( x \).

Step 3: Conclusion for the first equation

The first equation \( 2x - 3y = 5 \) defines \( y \) as a function of \( x \) because \( y \) can be expressed explicitly as \( y = \frac{2x - 5}{3} \).

Step 4: Conclusion for the second equation

The second equation \( x^{2} + (y-5)^{2} = 6 \) does not define \( y \) as a function of \( x \) because \( y \) is not uniquely determined for a given \( x \).

Final Answer

For the first equation: \( \boxed{\text{The equation defines } y \text{ as a function of } x.} \)
For the second equation: \( \boxed{\text{The equation does not define } y \text{ as a function of } x.} \)

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