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Write an equation of the line satisfying the given conditions.​ (Hint: Recall the relationships among slopes of perpendicular​ lines.)

Perpendicular to xnegative 2yequals9​; ​y-intercept ​(0​,negative 3​)

3 answers

To find the equation of the line that is perpendicular to the line given by \( x - 2y = 9 \) and has a y-intercept of \( (0, -3) \), we will follow these steps:

  1. Convert the equation to slope-intercept form (y = mx + b) to find the slope of the given line.
  2. Determine the slope of the perpendicular line.
  3. Use the slope and the y-intercept point to write the equation of the new line.

Step 1: Convert \( x - 2y = 9 \) to slope-intercept form

Start with the original equation:

\[ x - 2y = 9 \]

Rearranging it to isolate \( y \):

\[ -2y = -x + 9 \]

Now, divide by -2:

\[ y = \frac{1}{2}x - \frac{9}{2} \]

From this, we see that the slope \( m \) of the given line is \( \frac{1}{2} \).

Step 2: Determine the slope of the perpendicular line

The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Thus, for the slope \( \frac{1}{2} \):

\[ m_{\text{perpendicular}} = -\frac{1}{\left(\frac{1}{2}\right)} = -2 \]

Step 3: Write the equation of the new line

Now, we know the y-intercept is \( -3 \) (which means the point is \( (0, -3) \)), and the slope of the line we want to write is \( -2 \).

Using the slope-intercept form \( y = mx + b \):

\[ y = -2x - 3 \]

This is the equation of the line that is perpendicular to \( x - 2y = 9 \) and has a y-intercept of \( (0, -3) \).

Are you sure?

Let me double-check the calculations and steps to ensure everything is correct.

1. Given Line: \( x - 2y = 9 \)

Convert to Slope-Intercept Form:

Start with the equation:

\[ x - 2y = 9 \]

Rearranging it for \( y \):

\[ -2y = -x + 9 \]

Now divide by -2:

\[ y = \frac{1}{2}x - \frac{9}{2} \]

The slope of this line is \( m = \frac{1}{2} \).

2. Find the Perpendicular Slope

To find the slope of the line that is perpendicular to this line, take the negative reciprocal of \( \frac{1}{2} \):

\[ m_{\text{perpendicular}} = -\frac{1}{\frac{1}{2}} = -2 \]

3. Use the Y-Intercept to Write the Equation

Given that the line has a y-intercept of \( (0, -3) \), we can use the slope-intercept form \( y = mx + b \). Here, \( m = -2 \) and \( b = -3 \):

\[ y = -2x - 3 \]

Conclusion

So, the final equation, \( y = -2x - 3 \), is indeed correct. It represents a line that is perpendicular to the original line \( x - 2y = 9 \) and passes through the point \( (0, -3) \).

If you have any specific concerns or additional details, please let me know!