Write an equation in standard form of the line that contains the point (1,−8) and is a. parallel to the line 7x+9y=8 b. perpendicular to the line 7x+9y=8
a. To find the equation of a line that is parallel to the line 7x + 9y = 8, we need to find the slope of the given line. The slope-intercept form of a line is y = mx + b, where m represents the slope. We need to rearrange the equation 7x + 9y = 8 to this form:
9y = -7x + 8
y = (-7/9)x + 8/9
The slope of this line is -7/9. Since the line we want is parallel, it will have the same slope. Now, we can use the point-slope form to find the equation of the line:
y - y1 = m(x - x1)
where (x1, y1) is the given point (1, -8) and m is the slope (-7/9):
y - (-8) = (-7/9)(x - 1)
y + 8 = (-7/9)x + 7/9
7/9x + y = 7/9 - 8
7/9x + y = -65/9
Therefore, the equation of the line that is parallel to 7x + 9y = 8 and passes through the point (1, -8) is:
7x + 9y = -65
b. To find the equation of a line that is perpendicular to the line 7x + 9y = 8, we need to find the negative reciprocal of the slope of the given line. The equation 7x + 9y = 8 is already in the standard form, so we can easily find the slope by rearranging it to the slope-intercept form:
9y = -7x + 8
y = (-7/9)x + 8/9
The slope of this line is -7/9. The negative reciprocal of -7/9 is 9/7. Now, let's use the point-slope form to find the equation of the line:
y - y1 = m(x - x1)
where (x1, y1) is the given point (1, -8) and m is the slope (9/7):
y - (-8) = (9/7)(x - 1)
y + 8 = (9/7)x - 9/7
(9/7)x + y = -65/7
Therefore, the equation of the line that is perpendicular to 7x + 9y = 8 and passes through the point (1, -8) is:
(9/7)x + y = -65/7