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Chapter 0: Problem 40

Solve the equation and check your solution. (If not possible, explain why.) $$ \frac{4}{x-1}+\frac{6}{3 x+1}=\frac{15}{3 x+1} $$

Short Answer

Expert verified
The solution for the equation is \(x = -13/3\). And, it's confirmed that this solution is correct as it does not make any of the original equation's denominators zero.

Step by step solution

01

Simplify the Equation

First, clear the fractions from the equation by finding a common denominator. In this case, the least common denominator would be (x-1)(3x+1). Multiply each ratio by this common denominator \((x-1)(3x+1)\), and do so without altering the invariability.
02

Distribute and Simplify

The equation becomes: \(4*\frac{(x-1)(3x+1)}{(x-1)}+6*\frac{(x-1)(3x+1)}{(3x+1)} = 15*\frac{(x-1)(3x+1)}{(3x+1)}\). Distribute and simplify this to get \(4*(3x+1)+6*(x-1) =15*(x-1)\). Then, expand and simplify that to get \(12x +4+ 6x-6 = 15x-15\).
03

Solve the Resulting Equation

Combine like terms to get \(18x - 2 = 15x -15 \). Move all terms involving x to one side, and constants to the other - this yields \(3x= -13\).
04

Solve for x

Finally, divide by coefficient of x, to get \(x = -13/3\) as the solution for x. The solution to the linear equation is x = -13/3.
05

Validate the Solution

Upon checking, one sees that -13/3 is not a solution that makes the denominators of the original fractions zero. Therefore we can say the solution is valid.

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