Find the point on the line -3x+5y-5=0 which is closest to the point (-4,0). Please Help!

the slope of the given line is 3/5.

So, you want a perpendicular line through (-4,0). That line will be

y = -5/3 (x+4)

The two lines intersect at (-115/34,-35/34)

Now just find the distance between those two points

To find the point on the line -3x+5y-5=0 that is closest to the point (-4,0), we can use the concept of perpendicular distance.

1. First, let's rewrite the equation of the line in slope-intercept form (y = mx + c):
-3x + 5y - 5 = 0
5y = 3x + 5
y = (3/5)x + 1

2. The line has a slope of 3/5, which means any line perpendicular to it will have a slope of -5/3 (negative reciprocal).

3. Let's consider the perpendicular line passing through the point (-4,0). The equation of this line can be written as:
y - 0 = (-5/3)(x + 4)

4. Simplifying the equation:
y = (-5/3)x - 20/3

5. Now, we have two lines: y = (3/5)x + 1 and y = (-5/3)x - 20/3. We can solve these two linear equations to find their point of intersection.

6. Equating both equations, we have:
(3/5)x + 1 = (-5/3)x - 20/3

7. Let's multiply both sides by 15 to eliminate the fractions:
9x + 15 = -25x - 100

8. Combining like terms:
34x = -115

9. Dividing by 34:
x = -115/34

10. Substitute the value of x into one of the equations (y = (3/5)x + 1) to find y:
y = (3/5)(-115/34) + 1

11. Simplifying:
y = -69/34 + 34/34
y = -35/34

12. Therefore, the point on the line -3x+5y-5=0 that is closest to the point (-4,0) is (-115/34, -35/34).

To find the point on the line -3x + 5y - 5 = 0 that is closest to the point (-4, 0), we need to find the perpendicular distance between the line and the given point. Here's how you can do it step by step:

1. Start by rewriting the equation of the line in slope-intercept form (y = mx + b):
-3x + 5y - 5 = 0
5y = 3x + 5
y = (3/5)x + 1

2. Determine the slope of the given line. In this case, the slope is 3/5.

3. Find the negative reciprocal of the slope to get the slope of the line perpendicular to the given line. In this case, the negative reciprocal of 3/5 is -5/3.

4. Use the equation of the perpendicular line (y = mx + b) with the negative reciprocal slope and the given point (-4, 0) to find the equation of the perpendicular line:
y = (-5/3)x + b

To find b, substitute the x and y values from the given point:
0 = (-5/3)(-4) + b
0 = 20/3 + b
b = -20/3

So, the equation of the perpendicular line is:
y = (-5/3)x - 20/3

5. Now, we have a system of equations between the original line (y = (3/5)x + 1) and the equation of the perpendicular line (y = (-5/3)x - 20/3):
(3/5)x + 1 = (-5/3)x - 20/3

To solve this system, we can isolate x by moving all the x terms to one side and the constant terms to the other side:
(3/5)x + (5/3)x = -20/3 - 1

Finding the common denominator for the x terms:
(9/15)x + (25/15)x = -20/3 - 3/3
(34/15)x = -23/3

To isolate x, multiply both sides by 15/34:
x = (-23/3)(15/34)

Simplify the expression for x:
x = -345/102

6. Substitute the value of x back into the equation of the original line to find y:
y = (3/5)(-345/102) + 1

Simplify the expression for y:
y = -207/102 + 102/102
y = -105/102

So, the coordinates of the point on the given line that is closest to the point (-4, 0) are (-345/102, -105/102).