What is an equation of the line that passes through the point (2,-3) and is perpendicular to the line 2x+5y=10?

Explanation:

Consider the standard form Ax+By = C

Anything perpendicular to this is Bx-Ay = D. We swapped the positions of A and B, and changed the plus sign to a minus. This will ensure we get a negative reciprocal slope needed for the perpendicular line. The slope of the first line is -B/A, while the slope of the second line is A/B. The two slopes multiply to -1 assuming neither A nor B are zero.

Also, I've introduced the constant D. There may be cases when C = D, but that isn't always the case.

The given equation is 2x+5y = 10. This tells us A = 2, B = 5, C = 10.

The perpendicular line will be of the form 5x-2y = D, based on the template I mentioned above.

Plug the coordinates (2,-3) into 5x-2y = D to find the value of D

5x-2y = D

5(2)-2(-3) = D

10+6 = D

16 = D

D = 16

So 5x-2y = D updates to 5x-2y = 16

Therefore, 5x-2y = 16 is perpendicular to the original line, and this new line goes through (2,-3)

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