Question

If -5xy=4-2x+5y and it is known that (dy)/(dx)=(-2+5y)/(-5x-5), find all coordinate points on the curve where x=-3 and the line tangent to the curve has slope (3)/(10), or state that no such points exist.

Answer 1

To find the coordinate points on the curve where x = -3 and the tangent line has a slope of 3/10, we substitute x = -3 into the equation and solve for y. Then, we find the derivative (dy/dx) and calculate the slope of the curve at those points. No points satisfy the given conditions.

Step-by-step explanation:

To find the coordinate points on the curve where x = -3 and the line tangent to the curve has slope 3/10, we need to substitute x = -3 into the equation -5xy = 4-2x+5y and find the values of y that satisfy the equation. Then, we calculate the slope of the curve at those points using the given derivative (dy/dx). If the slope matches 3/10, those points are the coordinates we're looking for.

1. Substitute x = -3 into -5xy = 4-2x+5y: -5(-3)y = 4 - 2(-3) + 5y
2. Simplify the equation: 15y = 10 + 6 + 5y
3. Combine like terms: 10y = 16
4. Divide both sides by 10 to solve for y: y = 16/10 = 8/5
5. Now, substitute x = -3 and y = 8/5 into (dy/dx) = (-2+5y)/(-5x-5): (dy/dx) = (-2+5(8/5))/(-5(-3)-5)
6. Simplify the equation: (dy/dx) = (-2+8)/(-5*(-3) - 5) = 6/10 = 3/5

Since the slope of the tangent line does not match 3/10, there are no coordinate points on the curve where x = -3 and the line tangent to the curve has a slope of 3/10.