Question

Find the real solution(s) of the equation involving fractions. Check your solutions. \(\frac{1}{x}=\frac{4}{x-1}+1\)

Short answer

The exact solution to the equation \(\frac{1}{x}=\frac{4}{x-1}+1\) is \(x = \frac{1}{3}\).

Step-by-step solution

Eliminate the fractions

Since the given equation is \(\frac{1}{x}=\frac{4}{x-1}+1\), we can start by multiplying every term by \(x(x-1)\) which is the least common multiple of \(x\) and \(x-1\), the denominators in the fractions. This will give us: \(x(x-1)\cdot\frac{1}{x}=x(x-1)\cdot\frac{4}{x-1}+x(x-1)\cdot 1\) simplifying this equation we get: \(x-1 = 4x - x^2 + x^2 - x\).

Simplify the equation

After multiplying through by the least common multiple, the equation is then simplified to: \(x-1 = 4x - x^2 + x^2 - x\). This simplifies further to \(x - 1 = 4x\).

Solve for x

Now we solve for x by getting x alone on one side of the equation. Subtracting \(4x\) from both sides we get: \(x - 1 - 4x = 4x -4x\), which simplifies to: \(-3x = -1\). Dividing through by -3, we solve for x: \(x = -1/-3\) which give \(x = \frac{1}{3}\). But don't forget to check this answer in the original equation.

Check the solution

To ensure that \(x = \frac{1}{3}\) is a valid solution, substitute it back into the original equation and check if the equation holds. \(\frac{1}{\frac{1}{3}}=\frac{4}{\frac{1}{3}-1}+1\) simplifies to \(3 = 4(3)-12 + 1\) which is \(3 = 3\), so \(x = \frac{1}{3}\) is indeed a solution to the problem.

Key concepts

Least Common Multiple (LCM)

Understanding the least common multiple (LCM) is essential when solving rational equations with fractions. The LCM of two or more denominators is the smallest number that each of the denominators can divide into without leaving a remainder. To find the LCM, one has to list the multiples of each denominator and then identify the smallest multiple that appears in all lists.

In our exercise, the fractions had denominators of x and x-1. The LCM of these terms is their product, x(x-1), because neither x nor x-1 has common factors other than 1. Using the LCM is a powerful strategy to eliminate fractions, as it simplifies an equation to a form where traditional algebraic methods can be applied more directly.

Simplifying Equations

After finding the LCM, the next step is to simplify the equation. Simplification might involve expanding expressions, combining like terms, or reducing expressions to their lowest terms. In rational equations, we often multiply the entire equation by the LCM to remove fractions, followed by simplifying the resulting expression.

In our example, we multiplied each term by the LCM x(x-1) and then cancelled out the denominators, leaving a simpler, non-fractional equation. This process converted the problem into a standard linear equation, \(x-1 = 4x\), that is much easier to solve. Simplifying equations is like tidying up a messy room; it helps clarify what you're working with and makes it easier to see the solution.

Checking Solutions

Finally, verifying solutions is a crucial step in solving rational equations. Substituting the obtained solution back into the original equation allows you to confirm whether it produces a true statement. This step catches potential mistakes or extraneous solutions—solutions that mathematically appear to be correct but do not actually satisfy the original equation.

In checking the solution \(x = \frac{1}{3}\), we substitute it back into the original equation to ensure the two sides are equal. If the substitution results in a true statement, as it did in our scenario where both sides simplified to 3, then the solution is valid. Skipping this step can lead to undiscovered errors, so it's best to make checking an integral part of the solution process.