Rational Expression: Solving the Equation 18x + 1/(2x-3)(3x-1) = p/2x-3
In this article, we will solve the rational expression equation 18x + 1/(2x-3)(3x-1) = p/2x-3. We will start by simplifying the left-hand side of the equation and then solve for p.
Simplifying the Left-Hand Side
The left-hand side of the equation is 18x + 1/(2x-3)(3x-1). To simplify this expression, we need to combine the fractions.
Step 1: Multiply the numerator and denominator of the fraction by the least common multiple (LCM) of 2x-3 and 3x-1, which is 6x-3.
18x + 1 / (2x-3)(3x-1) = 18x + (6x-3) / ((2x-3)(3x-1)(6x-3))
Step 2: Simplify the numerator and denominator.
18x + (6x-3) / (6x^2 - 3x - 6)
Step 3: Combine the fractions.
(18x)(6x^2 - 3x - 6) + (6x-3) / (6x^2 - 3x - 6)
Step 4: Simplify the numerator.
108x^3 - 54x^2 - 108x + 6x - 3 / (6x^2 - 3x - 6)
Solving for p
Now that we have simplified the left-hand side of the equation, we can equate it to p/2x-3.
108x^3 - 54x^2 - 108x + 6x - 3 / (6x^2 - 3x - 6) = p/2x-3
Step 1: Cross-multiply.
p(6x^2 - 3x - 6) = 2x(108x^3 - 54x^2 - 108x + 6x - 3)
Step 2: Simplify both sides of the equation.
p(6x^2 - 3x - 6) = 216x^4 - 108x^3 - 216x^2 + 12x
Step 3: Equate the coefficients of each term.
p = 216x^2 / (6x^2 - 3x - 6)
Thus, we have solved for p.
In conclusion, we have simplified the rational expression 18x + 1/(2x-3)(3x-1) and solved for p in the equation 18x + 1/(2x-3)(3x-1) = p/2x-3.
