Write an equation in standard form of the line that contains the point (3,-8) and is a. parallel to the line 5x+8y=7 and b. perpendicular to the line 5x+8y=7
a. To find the equation of a line parallel to the given line, we use the fact that parallel lines have the same slope. The given line has the equation 5x + 8y = 7. We can rewrite it in slope-intercept form, which is y = mx + b, by isolating y:
8y = -5x + 7
y = (-5/8)x + 7/8
The slope of this line is -5/8. Since the line we want is parallel, it will have the same slope. So, the equation of the line is in the form y = (-5/8)x + b, where b is the y-intercept we need to find.
To find the value of b, we substitute the coordinates (3,-8) into the equation and solve for b:
-8 = (-5/8)(3) + b
-8 = -15/8 + b
b = -8 + 15/8
b = -64/8 + 15/8
b = -49/8
Therefore, the equation of the line parallel to 5x + 8y = 7 that passes through the point (3,-8) is:
y = (-5/8)x - 49/8
b. To find the equation of a line perpendicular to the given line, we use the fact that perpendicular lines have negative reciprocal slopes. The slope of the given line is -5/8, so the slope of the perpendicular line will be the negative reciprocal, which is 8/5.
Using the point-slope form of a line, and substituting the coordinates (3,-8), we can find the equation of the perpendicular line:
y - (-8) = (8/5)(x - 3)
y + 8 = (8/5)x - 24/5
y = (8/5)x - 24/5 - 40/5
y = (8/5)x - 64/5
Therefore, the equation of the line perpendicular to 5x + 8y = 7 that passes through the point (3,-8) is:
y = (8/5)x - 64/5