Find an equation of the line that satisfies the given conditions.
Through ; perpendicular to the line

Ask by Howell Rogers. in the United States

Jan 23,2025

Question

Find an equation of the line that satisfies the given conditions. Through $$ (-1,-2) $$; perpendicular to the line $$ 2 x+5 y+6=0 $$

UpStudy AI · Accepted Answer

To find the equation of the line that is perpendicular to $$ 2x + 5y + 6 = 0 $$ and passes through the point $$ (-1, -2) $$, follow these steps:

1. **Find the slope of the given line:**
   
   Start by rewriting the equation in slope-intercept form ($$ y = mx + b $$):
   $$
   2x + 5y + 6 = 0 \implies 5y = -2x - 6 \implies y = -\frac{2}{5}x - \frac{6}{5}
   $$
   So, the slope ($$ m_1 $$) of the given line is:
   $$
   m_1 = -\frac{2}{5}
   $$

2. **Determine the slope of the perpendicular line:**
   
   The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore:
   $$
   m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{2}{5}} = \frac{5}{2}
   $$

3. **Use the point-slope form to find the equation:**
   
   The point-slope form of a line is:
   $$
   y - y_1 = m(x - x_1)
   $$
   Plugging in the point $$ (-1, -2) $$ and the slope $$ \frac{5}{2} $$:
   $$
   y - (-2) = \frac{5}{2}(x - (-1)) \implies y + 2 = \frac{5}{2}(x + 1)
   $$

4. **Simplify to slope-intercept form:**
   
   Distribute and solve for $$ y $$:
   $$
   y + 2 = \frac{5}{2}x + \frac{5}{2} \implies y = \frac{5}{2}x + \frac{5}{2} - 2 \implies y = \frac{5}{2}x + \frac{1}{2}
   $$

**Final Equation:**
$$
y = \frac{5}{2}x + \frac{1}{2}
$$
The equation of the line is $$ y = \frac{5}{2}x + \frac{1}{2} $$.

Find an equation of the line that satisfies the given conditions.
Through ; perpendicular to the line

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Find an equation of the line that satisfies the given conditions. Through ; perpendicular to the line