Answer:
The distance between parallel lines is [tex]0.6860\ units[/tex]
Step-by-step explanation:
we have
[tex]3x-5y=1[/tex] -----> equation A
[tex]3x-5y=-3[/tex] -----> equation B
Line A and Line B are parallel lines, because their slopes are equal
Step 1
Find the slope of the given lines
isolate the variable y in the equation A
[tex]5y=3x-1[/tex]
[tex]y=(3/5)x-1/5[/tex]
The slope is [tex]m=(3/5)[/tex]
Step 2
Find the slope of the perpendicular line to the given lines
we know that
If two lines are perpendicular, then the product of their slopes is equal to -1
so
[tex]m1*m2=-1[/tex]
we have
[tex]m1=3/5[/tex] -----> given lines
substitute
[tex](3/5)*m2=-1[/tex]
[tex]m2=-5/3[/tex]
Step 3
Find the equation of the perpendicular line that pass though the origin
with the slope [tex]m2=-5/3[/tex] and the point (0,0) Find the equation of the line
[tex]y-y1=m(x-x1)[/tex]
[tex]y-0=(-5/3)(x-0)[/tex]
[tex]y=-(5/3)x[/tex]
Step 4
Using a graphing tool
Graph the two parallel lines and the perpendicular line
The intersection points are
[tex]A(-0.2647,0.4412)[/tex] and [tex]B(0.0882,-0.1471)[/tex]
see the attached figure
we know that
the distance between the points A and B is the distance between parallel lines
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
substitute the values
[tex]d=\sqrt{(-0.1471-0.4412)^{2}+(0.0882+0.2647)^{2}}[/tex]
[tex]d=\sqrt{(-0.5883)^{2}+(0.3529)^{2}}[/tex]
[tex]d=0.6860\ units[/tex]
