Which of these lines is Wizard perpendicular to the line $$ y $$ $$ =3 x+2 $$ ?

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UpStudy AI · Accepted Answer

1. The given line is
   $$
   y = 3x + 2,
   $$
   which has a slope $$ m = 3 $$.

2. Two lines are perpendicular if the product of their slopes is $$-1$$. That is, if a line with slope $$ m_1 $$ is perpendicular to a line with slope $$ m_2 $$, then
   $$
   m_1 \cdot m_2 = -1.
   $$

3. Let $$ m_{\perp} $$ be the slope of the line perpendicular to the given line. Then,
   $$
   3 \cdot m_{\perp} = -1.
   $$

4. Solving for $$ m_{\perp} $$:
   $$
   m_{\perp} = -\frac{1}{3}.
   $$

5. Therefore, among the options provided, the line with slope $$-\frac{1}{3}$$ is perpendicular to the line
   $$
   y = 3x + 2.
   $$

Any line with the form
   $$
   y = -\frac{1}{3}x + b,
   $$
where $$ b $$ is any constant, will be perpendicular to the line $$ y = 3x+2 $$.
A line with a slope of $$-\frac{1}{3}$$ is perpendicular to the line $$ y = 3x + 2 $$.

Which of these lines is Wizard perpendicular to the line \( y \) \( =3 x+2 \) ?

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