Problem 23 - Algebra 1: Concepts and Skills

Question

Solve the equation by multiplying each side by the least common denominator. Check your solutions. \(\frac{3 x}{x-1}=\frac{x}{5}\)

Step-by-Step Solution

Answer

The solution for the given equation \(\frac{3 x}{x-1}=\frac{x}{5}\) is \(x = 16\).

Step 1: Identify the LCD

The first step to solving a rational equation is to identify the lease common denominator (LCD) for all the fractions in the equation. In this case, the LCD is \(5(x - 1)\), because \(x - 1\) appears in the denominator on the left and 5 appears on the right.

Step 2: Multiply both sides by the LCD

Next, multiply both sides of the equation by this LCD. You get \(3x \cdot 5 = x \cdot (x - 1)\), which simplifies to \(15x = x^2 - x\).

Step 3: Arrange the equation

Rearrange the equation to one side, resulting in a quadratic equation: \(0 = x^2 - 16x\). This can further be written as \(0 = x(x - 16)\).

Step 4: Solve for x

The next step is to solve for \(x\). Setting each factor equal to zero gives the solutions \(x = 0\) and \(x = 16\). However, \(x = 0\) isn't a valid solution as it makes the denominator of the original equation to be zero, so we discard it. Therefore, \(x = 16\) is the only solution.

Step 5: Check your answer

The last step is to check the answer. Substitute \(x = 16\) back into the original equation: \(\frac{3 * 16}{16 - 1} = \frac{16}{5}\). Simplifying both sides yields \(\frac{48}{5} = \frac{48}{5}\), confirming that \(x = 16\) is indeed the correct solution.

Key Concepts

Understanding the Least Common Denominator

When solving a rational equation, finding the Least Common Denominator (LCD) helps combine fractions. Imagine you have different parts of a recipe and need a common measuring cup. The denominator is like that measuring cup.
In the equation \( \frac{3x}{x-1} = \frac{x}{5} \), the denominators are \(x - 1\) and \(5\).
To solve the equation, you need a denominator that both \(x - 1\) and \(5\) can fit into.

• The LCD here is \(5(x - 1)\).
• This ensures both terms can be combined and simplified.
  

Multiplying through by the LCD helps clear out the fractions, making the equation easier to work with. You are essentially balancing the scales, removing the fractions entirely.

Solving the Quadratic Equation

Once the fractions are eliminated, you'll often end up with a quadratic equation. In our case, after clearing fractions: \(15x = x^2 - x\).
Quadratic equations usually involve an \(x^2\) term.

• Bring all terms to one side to form \(0 = x^2 - 16x\).
  

Now you've got a standard quadratic form, which allows for solving techniques like factoring, quadratic formula, or completing the square. Getting everything ready on one side simplifies finding rational solutions.

Using Factorization to Find Solutions

Factorization turns a complex expression into simple, solvable pieces. It's like breaking down a puzzle into manageable sections.
For \(0 = x^2 - 16x\), notice that both terms contain \(x\).

• Factor out \(x\), giving \(x(x - 16) = 0\).
  

This is easier since each part can potentially be zero.
Set each factor equal to zero:

• \(x = 0\)
• \(x - 16 = 0\), which simplifies to \(x = 16\).

These solutions arise from the principle that if a product is zero, at least one factor must be zero.

Checking Solutions for Validity

The final step in solving is checking the proposed solutions in the original equation. This ensures they don’t result in undefined mathematical situations.
For \(x = 16\), substitute back in:
\(\frac{3 \times 16}{16 - 1} = \frac{16}{5}\).
Both sides equal \(\frac{48}{5}\), confirming \(x = 16\) is correct.

However, \(x = 0\) was discarded because it makes the denominator \(x - 1\) equal to zero, which is undefined. Always check for these kinds of zero-denominator problems.