write an equation of the line that is parallel to the given line and passes through the given point.
y=-4x-7, (5,-3)
y=-2/3x+4, (-5,5)
First, from a given line, the slope can be identified if the equation of the line is put into the slope-intercept form as:
y = mx + b
For example,
2x + 3y = 5 can be rearranged to
y = (-2/3)x + 5/3
From which we can identify the slope as (-2/3).
One the slope is defined, if the line is to pass through a given point with coordinates (x1,y1), the equation of the line sought is:
y = m(x-x1) + y1
So if a line parallel to the above line is to pass through (3,2), the equation would be:
y = (-2/3)(x - 3) + 2, or
y = (-2/3)x + 4
To find the equation of a line parallel to a given line, we need to know that parallel lines have the same slope.
In the first case, the given line has a slope of -4. So, in order to find a parallel line, we can use the same slope of -4.
Using the point-slope form of a line, the equation can be written as:
y - y1 = m(x - x1)
Where (x1, y1) is the given point and m is the slope.
Substituting the values of the slope (-4) and the point (5, -3) into the equation, we have:
y - (-3) = -4(x - 5)
Simplifying the equation gives us:
y + 3 = -4x + 20
Finally, rearranging the equation to the standard form, we have:
4x + y = 17
So, the equation of the line that is parallel to y = -4x - 7 and passes through the point (5, -3) is 4x + y = 17.
For the second case, the given line has a slope of -2/3. Therefore, a line parallel to it will also have a slope of -2/3.
Using the same approach as before with the point-slope form, we substitute the values of slope (-2/3) and point (-5, 5) into the equation:
y - 5 = (-2/3)(x - (-5))
Simplifying the equation gives us:
y - 5 = (-2/3)(x + 5)
Multiplying through by -3 to eliminate the fraction, we have:
-3y + 15 = 2x + 10
Rearranging the equation to standard form, we have:
2x + 3y = 5
Therefore, the equation of the line that is parallel to y = (-2/3)x + 4 and passes through the point (-5, 5) is 2x + 3y = 5.