Find the point on the line 6 x + 4 y - 1 =0 which is closest to the point ( 0, -1 ).

rewrite as

y = -(6/4)x + 1/4

line perpendicular has slope +4/6 = +2/3
y = (2/3) x + b
we want it through 0, -1
-1 = (2/3) 0 + b
b = -1
so
y = (2/3) x - 1
where does that hit original line?
(2/3) x - 1 = -(3/2) x + 1/4
4 x - 6 = -9 x + 3/2
13 x = 15/2
x = 15/26
then y = 5/13

now find the distance from
(0,-1) to (15/26 , 5/13) yuuk
check my arithmetic !

the distance is sqrt(x2-x1)^2+(y2 -y1)^2)?

will that be a form of interval?

it sid the distance formula isn't what they're looking for, they're looking for interval

Oh, well I guess I found your point :)

(15/26,5/13)

To find the point on the line 6x + 4y - 1 = 0 that is closest to the point (0, -1), we need to apply the concept of the shortest distance between a point and a line.

Step 1: Write the equation of the line in slope-intercept form (y = mx + b).
Starting with the given equation:
6x + 4y - 1 = 0
Rearrange the equation to isolate y:
4y = -6x + 1
Divide both sides by 4:
y = (-6/4)x + 1/4
Simplify:
y = (-3/2)x + 1/4

Step 2: Determine the slope (m) of the given line.
Comparing the equation to the slope-intercept form, we can see that the slope is -3/2.

Step 3: Find the equation of the line perpendicular to the given line.
For any line perpendicular to the given line, the slope will be the negative reciprocal of the given line's slope.
So, the slope of the perpendicular line will be 2/3.

Step 4: Find the equation of the perpendicular line passing through the point (0, -1).
Using the slope-intercept form (y = mx + b) and substituting the coordinates of the point:
y = (2/3)x + b
Substituting x=0 and y=-1, we can solve for b:
-1 = (2/3)(0) + b
-1 = 0 + b
b = -1

Therefore, the equation of the perpendicular line passing through the point (0, -1) is:
y = (2/3)x - 1

Step 5: Solve the system of equations comprised of the given line and the perpendicular line.
To find the point of intersection between the given line and the perpendicular line, we can solve the system of equations:
y = (-3/2)x + 1/4
y = (2/3)x - 1

By setting the right-side expressions equal to each other, we get:
(-3/2)x + 1/4 = (2/3)x - 1

Multiply both sides by 12 to eliminate the fractions:
12*(-3/2)x + 12*(1/4) = 12*(2/3)x - 12*1
-18x + 3 = 8x - 12

Gather like terms:
-18x - 8x = -12 - 3
-26x = -15

Divide both sides by -26 to solve for x:
x = (-15) / (-26)
x = 15/26

Substitute this value of x back into one of the equations to solve for y:
y = (-3/2) * (15/26) + 1/4
y = -45/52 + 1/4
y = -195/104 + 26/104
y = -169/104

Thus, the point on the line 6x + 4y - 1 = 0 which is closest to the point (0, -1) is approximately (15/26, -169/104).