Questions: Use the given conditions to write an equation for the line in point-slope form and general form. Passing through (7,-8) and perpendicular to the line whose equation is x-7 y-9=0 The equation of the line in point-slope form is (Type an equation. Use integers or fractions for any numbers in the equation.)

Use the given conditions to write an equation for the line in point-slope form and general form. Passing through (7,-8) and perpendicular to the line whose equation is x-7 y-9=0 The equation of the line in point-slope form is (Type an equation. Use integers or fractions for any numbers in the equation.)
Transcript text: Use the given conditions to write an equation for the line in point-slope form and general form. Passing through $(7,-8)$ and perpendicular to the line whose equation is $x-7 y-9=0$ The equation of the line in point-slope form is $\square$ (Type an equation. Use integers or fractions for any numbers in the equation.)
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Solution

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Solution Steps

To solve this problem, we need to find the equation of a line that passes through a given point and is perpendicular to another given line.

  1. Identify the slope of the given line: The given line is in the form \(x - 7y - 9 = 0\). We need to convert this to slope-intercept form \(y = mx + b\) to find its slope.
  2. Find the slope of the perpendicular line: The slope of the line perpendicular to the given line is the negative reciprocal of the slope of the given line.
  3. Use the point-slope form: With the slope of the perpendicular line and the given point, use the point-slope form \(y - y_1 = m(x - x_1)\) to write the equation of the line.
  4. Convert to general form: Rearrange the point-slope form equation to the general form \(Ax + By + C = 0\).
Step 1: Identify the Slope of the Given Line

The given line is represented by the equation \( x - 7y - 9 = 0 \). To find its slope, we can rearrange this equation into slope-intercept form \( y = mx + b \):

\[ 7y = x - 9 \implies y = \frac{1}{7}x - \frac{9}{7} \]

From this, we can see that the slope \( m \) of the given line is \( \frac{1}{7} \).

Step 2: Find the Slope of the Perpendicular Line

The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line is:

\[ m_{\text{perpendicular}} = -\frac{1}{\left(\frac{1}{7}\right)} = -7 \]

Step 3: Write the Equation in Point-Slope Form

Using the point-slope form of a line, which is given by \( y - y_1 = m(x - x_1) \), and substituting the point \( (7, -8) \) and the slope \( -7 \):

\[ y + 8 = -7(x - 7) \]

Step 4: Convert to General Form

To convert the point-slope form to general form \( Ax + By + C = 0 \), we expand and rearrange the equation:

\[ y + 8 = -7x + 49 \implies 7x + y - 41 = 0 \]

Final Answer

The equation of the line in point-slope form is:

\[ \boxed{y + 8 = -7(x - 7)} \]

And the equation in general form is:

\[ \boxed{7x + y - 41 = 0} \]

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