SOLUTION: How far from the origin is the line 3x + 4y = 10? (a) 1.5 units (b) 2 units (c) 2.5 units (d) 2.75 units (e) None of the above

Algebra ->  Coordinate-system -> SOLUTION: How far from the origin is the line 3x + 4y = 10? (a) 1.5 units (b) 2 units (c) 2.5 units (d) 2.75 units (e) None of the above      Log On


   

Question 287623: How far from the origin is the line 3x + 4y = 10?
(a) 1.5 units (b) 2 units (c) 2.5 units (d) 2.75 units
(e) None of the above
Found 2 solutions by josmiceli, richwmiller:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Another way to ask this question is:
What is the perpendicular distance from
the origin to this line?
3x+%2B+4y+=+10
4y+=+-3x+%2B+10
y+=+%28-3%2F4%29x+%2B+5%2F2
Any line perpendicular to this line will
have slope = -%281%2F%28-3%2F4%29%29+=+4%2F3
The line through origin with this slope is:
y+=+%284%2F3%29%2Ax
Now the question is: What is the point where this line
intersects the given line?
By substitution:
3x+%2B+4%2A%284%2F3%29%2Ax+=+10
9x+%2B+16x+=+30
25x+=+30
x+=+6%2F5
and, substituting:
y+=+%284%2F3%29%2A%286%2F5%29
y+=+8%2F5
The distance formula is:
d+=+sqrt%28%286%2F5%29%5E2+%2B+%288%2F5%29%5E2%29
d+=+sqrt%2836%2F25+%2B+64%2F25%29
d+=+sqrt%28100%2F25%29
d+=+sqrt%284%29
d+=+2
the distance is (b) 2 units

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
plug in 0 for x and find y 4y=10 y=10/4=5/2=2.5
plug in 0 for y and find x 3x=10 x=10/3=3 1/3
on the y axis it is 2.5
but we need to find the distance from a point to a line.
the slope of our line is -3/4
so the slope of the line from the origin to the line must be perpendicular and so 4/3
y-0=4/3(x-0)
y=4x/3
4x=3y
4x-3y=0
3x+4y=10
They intersect at
x = 6/5, y = 8/5
the distance from the (0,0) to (6/5,8/5) is
d=sqrt(y-y)^2+(x-x)^2
d=sqrt((6/5)^2+(8/5)^2)
d=2
http://www.worsleyschool.net/science/files/linepoint/method1.html