Problem 9 - Algebra 2

Question

Solve each equation. Check each solution. $$ \frac{2}{3 x-5}=\frac{4}{x-15} $$

Step-by-Step Solution

Answer

The solution for the given fractional equation is \( x = -1 \) and the check confirms that it is a valid solution.

Step 1: Identify Least Common Multiple

The LCM of \(3x - 5\) and \(x - 15\) can be found by multiplying these two expressions together, so the LCM is \(3x^2 - 60x + 75\).

Step 2: Clear the Fractions

Multiply each side of the equation by the LCM. This results in the equation \( 2(x-15) = 4(3x-5) \). Simplify this expression to get \(2x - 30 = 12x - 20\).

Step 3: Solve the Equation

To solve for \(x\), collect the terms of \(x\) on one side and the constants on the other. This gives \( -10x = 10 \). Solving for \(x\) gives \( x = -1 \).

Step 4: Check the Solution

Substitute \( -1 \) back into the original equation. Therefore, \( \frac{2}{3(-1)-5} = \frac{4}{-1-15} \), which simplifies to \( -\frac{2}{8} = \frac{4}{-16} \), and this simplifies to \( -\frac{1}{4} = -\frac{1}{4} \). As both sides of the equation are equal, this confirms the solution

Key Concepts

Least Common Multiple

When dealing with fractional equations, finding the Least Common Multiple (LCM) of the denominators is crucial. The LCM is the smallest expression that both denominators can divide evenly. This helps to eliminate fractions and simplify the equation. In the given exercise, the denominator terms are \(3x - 5\) and \(x - 15\). To find the LCM, we multiply these expressions: \( (3x - 5)(x - 15) = 3x^2 - 60x + 75 \). This polynomial forms the LCM. Having a single common denominator allows us to clear fractions by multiplying each side of the equation by this LCM.

Solving Equations

Solving equations involves isolating the unknown variable on one side of the equation. Once the equation is cleared of fractions by multiplying with the LCM, it is time to simplify and solve. In our example, multiplying by the LCM gives the new equation:

• \(2(x - 15) = 4(3x - 5)\) transforms into \(2x - 30 = 12x - 20\)

The goal is to collect like terms and solve for \(x\). By rearranging, you'll have the equation \(-10x = 10\). By dividing both sides by \(-10\), you find \(x = -1\). This technique of moving terms and dividing is pivotal in finding where the variable stands alone.

Fractional Equations

Fractional equations involve complicated-looking fractions that can be simplified by proper manipulation. These equations have variables in the denominator and require special steps for solving. You start by clearing fractions using the common denominator, simplifying the structure. After that, tackle it as a regular algebraic problem to isolate the variable. Begin with a clear understanding of each fraction's equivalence in the system:

• Substitute terms thoughtfully and check back into the original equation

Once simplified, ensure your values are correct by substituting back, as verification confirms your solution. Thus, when we solved \(x = -1\) and plugged it back into our original equation, both sides matched, confirming our answer. Remember, managing these fractions initially and testing the result guarantees the equation solution is correct.