Find (dy/dx) for the equation (7x+8y/9x-4y)=4.
a. (dy/dx)=(-29/24)
b. (dy/dx)=(43/24)
c. (dy/dx)=(29/24)
d. (dy/dx)=(-43/24)
e. (dy/dx)=4
Correct if you meant
(7x+ 8y)/(9x-4y) = 4 ?
To find (dy/dx) for the equation (7x+8y)/(9x-4y) = 4, we need to differentiate both sides of the equation with respect to x using implicit differentiation.
Step 1: Rewrite the equation in a simpler form.
(7x+8y)/(9x-4y) = 4
Multiply both sides by (9x-4y) to get rid of the denominator:
7x + 8y = 4(9x - 4y)
Simplify the equation:
7x + 8y = 36x - 16y
Step 2: Differentiate both sides with respect to x.
Differentiating the left side with respect to x:
d/dx (7x + 8y) = d/dx (36x - 16y)
7 + 8(dy/dx) = 36 - 16(dy/dx)
Step 3: Solve for (dy/dx).
Rearrange the equation to isolate (dy/dx):
8(dy/dx) + 16(dy/dx) = 36 - 7
24(dy/dx) = 29
(dy/dx) = 29/24
Therefore, the correct answer is c. (dy/dx) = 29/24.