Which equation represents the line that passes through the point
and is perpendicular to the line whose equation is ?

Ask by Morgan Cox. in the United States

Jan 24,2025

Question

Which equation represents the line that passes through the point $$ (2,-7) $$ and is perpendicular to the line whose equation is $$ y=\frac{3}{4} x+4 $$ ?

UpStudy AI · Accepted Answer

To find the equation of the line that passes through the point $$ (2, -7) $$ and is perpendicular to the line $$ y = \frac{3}{4}x + 4 $$, follow these steps:

1. **Determine the Slope of the Given Line:**
   
   The slope ($$ m $$) of the given line $$ y = \frac{3}{4}x + 4 $$ is $$ \frac{3}{4} $$.

2. **Find the Slope of the Perpendicular Line:**
   
   Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, the slope of the perpendicular line is:
   $$
   m_{\text{perpendicular}} = -\frac{1}{\frac{3}{4}} = -\frac{4}{3}
   $$

3. **Use the Point-Slope Form:**
   
   The point-slope form of a line is:
   $$
   y - y_1 = m(x - x_1)
   $$
   Plugging in the point $$ (2, -7) $$ and the slope $$ -\frac{4}{3} $$:
   $$
   y - (-7) = -\frac{4}{3}(x - 2)
   $$
   Simplifying:
   $$
   y + 7 = -\frac{4}{3}x + \frac{8}{3}
   $$
   $$
   y = -\frac{4}{3}x + \frac{8}{3} - 7
   $$
   $$
   y = -\frac{4}{3}x - \frac{13}{3}
   $$

4. **Final Equation:**
   
   The equation of the desired line in slope-intercept form is:
   $$
   y = -\frac{4}{3}x - \frac{13}{3}
   $$

**Answer:** After simplifying, an equation for the required line is y = –4⁄3 x – 13⁄3.
The equation of the line is $$ y = -\frac{4}{3}x - \frac{13}{3} $$.

Which equation represents the line that passes through the point
and is perpendicular to the line whose equation is ?

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Which equation represents the line that passes through the point and is perpendicular to the line whose equation is ?