Find the point on the line 2x – 2y= 9 that is closest to the point (0, 1). Verify by using a graph.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The point on the line 2x – 2y= 9 that is closest to the point (0, 1) is the point where a line perpendicular to 2x – 2y= 9 and passing through the point (0, 1) intersects 2x - 2y = 9.

2x - 2y = 9

=> y = x - 9/2

This line has a slope 1, a line perpendicular to this has a slope of -1.

The equation of the line with slope -1 and passing through (0, 1) is:

(y - 1)/x = -1

=> y - 1 = -x

=> x = 1 - y

To find the point of intersection of x = 1 - y and 2x - 2y = 9 substitute x = 1 - y in 2x - 2y = 9

=> 2(1 - y) - 2y = 9

=> 2 - 2y - 2y = 9

=> -4y = 7

=> y = -7/4

x = 1 - y = 1 + 7/4 = 11/4

From the graph above, the same result is obtained.

The point on the line 2x – 2y= 9 that is closest to the point (0, 1) is (11/4, -7/4).

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