Write an equation of the line that passes through the point (i) parallel to the given value, and (ii) perpendicular to the given line.
Point(7,1) Line(-5x-5y=90)
The given line is -5x - 5y = 90
or , dividing by -5
x+y = -18
since the new line is parallel to it, its equation differs only in the costant, so let the new equation be
x+y = c , but (7,1) lies on it, so
7+1=c = 8
equation: x+y=8
If perpendicular, then the slop must be negative reciprocal : -1 ----> 1/1 = 1
new equation:
x-y = k
but (7,1) on it, so
7-1 = k = 6
x - y = 6
To find the equation of a line that passes through a given point and is parallel to a given line, we need to find the slope of the given line first.
Given line: -5x - 5y = 90
To get the slope-intercept form of the given line, we rearrange the equation to isolate y:
-5y = 5x + 90
Dividing both sides by -5:
y = -x - 18
Now, we can see that the slope of the given line is -1, since the coefficient of x is -1 in the slope-intercept form.
Since parallel lines have the same slope, the line we want will also have a slope of -1.
So, to find the equation of the line parallel to the given line that passes through the point (7,1), we can use the point-slope form:
y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Substituting the values into the equation:
y - 1 = -1(x - 7)
Expanding the equation:
y - 1 = -x + 7
Finally, rearranging the equation to slope-intercept form:
y = -x + 8
Therefore, the equation of the line that passes through the point (7,1) and is parallel to the given line -5x - 5y = 90 is y = -x + 8.
Now, let's find the equation of the line perpendicular to the given line.
Perpendicular lines have slopes that are negative reciprocals of each other. So, the slope of the perpendicular line will be the negative reciprocal of -1, which is 1.
Using the point-slope form again:
y - 1 = 1(x - 7)
Expanding the equation:
y - 1 = x - 7
Rearranging to slope-intercept form:
y = x - 6
Therefore, the equation of the line that passes through the point (7,1) and is perpendicular to the given line -5x - 5y = 90 is y = x - 6.