Find the point on the line 5x + y = 7 that is closest to the point (−3, 2).

Following are the calculation to the points:

Given:

Line:

point:

To find:

point=?

Solution:

Assuming the point is closest to the distance from

                  using the distance formula

Equation of line:

Substituting the value of

Differentitate D with respect to h

Set

Then,

So,

Therefore, the points are ""

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Step-by-step explanation:

The given equation of line is

        .... (1)

If a line is defined as ax+by=c, then the slope of the line is -a/b.

We need to find the point on the line 5x + y = 7 that is closest to the point (−3, 2).

Draw a perpendicular line from the point (-3,2) on the line 5x + y = 7.

For the given line a=5 and b=1. So, the slope of given line is -5.

Product of slopes of two perpendicular lines is -1. Slope of given line -5, it means the slope of perpendicular line is 1/5.

The slope of perpendicular line is  1/5 and it passes through the point (-3,2). So, the equation of line is

where, m is slope.

Multiply both sides by 5.

    .  .... (2)

The intersection point of line (1) and (2) is the point on the line 5x + y = 7 that is closest to the point (−3, 2).

On solving (1) and (2) we get

Therefore, the point on the line 5x + y = 7 that is closest to the point (−3, 2).

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so, converting firstly the mixed fractions to improper fractions and then "adding".