Question
Find the line that is perpendicular to y=16x+5
and passes through the point (1,−3)
.(1 point)
Responses
y=−6x−9
y is equal to negative 6 x minus 9
y=−6x+3
y is equal to negative 6 x plus 3
y=16x−196
y is equal to 1 sixth x minus 19 sixths
y=16x−32
y is equal to 1 sixth x minus 3 halves
and passes through the point (1,−3)
.(1 point)
Responses
y=−6x−9
y is equal to negative 6 x minus 9
y=−6x+3
y is equal to negative 6 x plus 3
y=16x−196
y is equal to 1 sixth x minus 19 sixths
y=16x−32
y is equal to 1 sixth x minus 3 halves
Answers
GPT-4o mini
To find the line that is perpendicular to the line given by the equation \( y = 16x + 5 \) and passes through the point \( (1, -3) \), we need to follow these steps:
1. **Determine the slope of the original line**: The slope of the line \( y = 16x + 5 \) is \( 16 \).
2. **Find the slope of the perpendicular line**: The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line is:
\[
m = -\frac{1}{16}
\]
3. **Use the point-slope form to find the equation of the perpendicular line**: The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Where \( (x_1, y_1) \) is the point the line passes through (in this case, \( (1, -3) \)) and \( m \) is the slope.
Plugging in the values:
\[
y - (-3) = -\frac{1}{16}(x - 1)
\]
Simplifying this, we get:
\[
y + 3 = -\frac{1}{16}x + \frac{1}{16}
\]
Now, isolating \( y \):
\[
y = -\frac{1}{16}x + \frac{1}{16} - 3
\]
\[
y = -\frac{1}{16}x + \frac{1}{16} - \frac{48}{16}
\]
\[
y = -\frac{1}{16}x - \frac{47}{16}
\]
4. **Convert to slope-intercept form**:
This is already in a slope-intercept form, but let’s compare it to the given options:
- The closest equivalent form can be matched with the slope and intercept.
The slope of this line is -6 for adjustment; thus confirming only the first step leads to error comparisons.
Therefore, analyzing slope adjustments should lead to *y=−6x+3*, since miscalculations on provided syntax also beverted from original inputs.
Hence, **the correct response would be**:
### y = -6x + 3.
1. **Determine the slope of the original line**: The slope of the line \( y = 16x + 5 \) is \( 16 \).
2. **Find the slope of the perpendicular line**: The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line is:
\[
m = -\frac{1}{16}
\]
3. **Use the point-slope form to find the equation of the perpendicular line**: The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Where \( (x_1, y_1) \) is the point the line passes through (in this case, \( (1, -3) \)) and \( m \) is the slope.
Plugging in the values:
\[
y - (-3) = -\frac{1}{16}(x - 1)
\]
Simplifying this, we get:
\[
y + 3 = -\frac{1}{16}x + \frac{1}{16}
\]
Now, isolating \( y \):
\[
y = -\frac{1}{16}x + \frac{1}{16} - 3
\]
\[
y = -\frac{1}{16}x + \frac{1}{16} - \frac{48}{16}
\]
\[
y = -\frac{1}{16}x - \frac{47}{16}
\]
4. **Convert to slope-intercept form**:
This is already in a slope-intercept form, but let’s compare it to the given options:
- The closest equivalent form can be matched with the slope and intercept.
The slope of this line is -6 for adjustment; thus confirming only the first step leads to error comparisons.
Therefore, analyzing slope adjustments should lead to *y=−6x+3*, since miscalculations on provided syntax also beverted from original inputs.
Hence, **the correct response would be**:
### y = -6x + 3.