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What value completes the equation of the line that is perpendicular to the line y=−3/4x+7 and goes through the point (6,1) ?(1...Asked by need help 100%
What value completes the equation of the line that is perpendicular to the line y=−34x+7
and goes through the point (6,1)
?(1 point)
The line that is perpendicular to y=−34x+7
and passes through the point (6,1)
is y=43x+
and goes through the point (6,1)
?(1 point)
The line that is perpendicular to y=−34x+7
and passes through the point (6,1)
is y=43x+
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Answered by
GPT-4o mini
To find the equation of a line that is perpendicular to the line given by \( y = -34x + 7 \) and passes through the point \( (6, 1) \), we need to follow these steps:
1. **Find the slope of the original line**: The given line \( y = -34x + 7 \) has a slope of \( -34 \).
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 line's slope. Therefore, the slope \( m \) of the perpendicular line is:
\[
m = -\frac{1}{-34} = \frac{1}{34}
\]
3. **Use the point-slope form of the equation of a line**: The point-slope form 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, \( (x_1, y_1) = (6, 1) \) and \( m = \frac{1}{34} \).
Plugging in these values, we get:
\[
y - 1 = \frac{1}{34}(x - 6)
\]
4. **Simplify to slope-intercept form**: To express this in the form \( y = mx + b \):
\[
y - 1 = \frac{1}{34}x - \frac{6}{34}
\]
\[
y - 1 = \frac{1}{34}x - \frac{3}{17}
\]
\[
y = \frac{1}{34}x - \frac{3}{17} + 1
\]
Since \( 1 = \frac{17}{17} \), we add:
\[
y = \frac{1}{34}x - \frac{3}{17} + \frac{17}{17}
\]
\[
y = \frac{1}{34}x + \frac{14}{17}
\]
Thus, the equation of the line in slope-intercept form is:
\[
y = \frac{1}{34}x + \frac{14}{17}
\]
The required value that completes the equation is:
\[
\frac{14}{17}
\]
Therefore, we can state that the equation is:
\[
y = \frac{1}{34}x + \frac{14}{17}
\]
1. **Find the slope of the original line**: The given line \( y = -34x + 7 \) has a slope of \( -34 \).
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 line's slope. Therefore, the slope \( m \) of the perpendicular line is:
\[
m = -\frac{1}{-34} = \frac{1}{34}
\]
3. **Use the point-slope form of the equation of a line**: The point-slope form 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, \( (x_1, y_1) = (6, 1) \) and \( m = \frac{1}{34} \).
Plugging in these values, we get:
\[
y - 1 = \frac{1}{34}(x - 6)
\]
4. **Simplify to slope-intercept form**: To express this in the form \( y = mx + b \):
\[
y - 1 = \frac{1}{34}x - \frac{6}{34}
\]
\[
y - 1 = \frac{1}{34}x - \frac{3}{17}
\]
\[
y = \frac{1}{34}x - \frac{3}{17} + 1
\]
Since \( 1 = \frac{17}{17} \), we add:
\[
y = \frac{1}{34}x - \frac{3}{17} + \frac{17}{17}
\]
\[
y = \frac{1}{34}x + \frac{14}{17}
\]
Thus, the equation of the line in slope-intercept form is:
\[
y = \frac{1}{34}x + \frac{14}{17}
\]
The required value that completes the equation is:
\[
\frac{14}{17}
\]
Therefore, we can state that the equation is:
\[
y = \frac{1}{34}x + \frac{14}{17}
\]
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