Problem 8

Solve the equation (5*x)/(x+12)-(8-3*x)/(3*x-1)=1.

Solution

a) Multiply both sides by the common denominator (x+12)*(3x-1):

5*x*(3*x-1)-(8-3*x)*(x+12)=(x+12)*(3*x-1)

b) Expand each term. Compute:

5*x*(3*x-1)=15*x2-5*x

and

(8-3*x)*(x+12)=8*x+96-3*x2-36*x=-3*x2-28*x+96.

Since the second term is subtracted, it becomes

+3*x2+28*x-96.

Thus, the left-hand side becomes

15*x2-5*x+3*x2+28*x-96=18*x2+23*x-96.

The right-hand side expands to

(x+12)*(3*x-1)=3*x2+35*x-12.

c) Form the equation: 18*x2+23*x-96=3*x2+35*x-12.

Subtract the right-hand side from the left:

15*x2-12*x-84=0.

d) Divide by 3: 5*x2-4*x-28=0. Apply the quadratic formula:

x=(4±√(,(-4)2-4⋅5⋅(-28)))/(2⋅5)=(4±√(,16+560))/10=(4±√(,576))/10

Since √(,576)=24, we have x=(4±24)/10

Thus,

x=28/10=14/5 or x=(-20)/10=-2.

e) Check that x≠-12 and x≠1/3 from the original denominators; both solutions are acceptable.

Answer: x=14/5 or x=-2.