Problem 15 - Intermediate Algebra

Question

Complete each solution. $$\frac{6 x-1}{3 x-1}+\frac{3 x-2}{3 x-1}=\frac{6 x-1+\square }{3 x-1} $$ $$=\frac{9 x-\square}{3 x-1}$$ $$=\frac{3(\quad)}{3 x-1}$$ $$=\square$$

Step-by-Step Solution

Answer

The solution to the equation is 3.

Step 1: Identify Common Denominator

Observe that both fractions in the equation \( \frac{6x-1}{3x-1} + \frac{3x-2}{3x-1} \) have the same denominator \(3x-1\). This allows us to combine the numerators directly: \(6x-1 + 3x-2\).

Step 2: Combine Numerators

Add the numerators of the fractions: \(6x - 1 + 3x - 2 = 9x - 3\). Therefore, the combined fraction becomes: \( \frac{9x-3}{3x-1} \).

Step 3: Factor the Numerator

Factor out the greatest common factor from the numerator \(9x-3\). The greatest common factor is 3, giving us: \(3(3x-1)\). Thus, the expression becomes \( \frac{3(3x-1)}{3x-1} \).

Step 4: Simplify the Fraction

Since the numerator \(3(3x-1)\) and the denominator \(3x-1\) are the same except for the factor of 3, they cancel each other out, simplifying to 3. Therefore, \( \frac{3(3x-1)}{3x-1} = 3 \).

Key Concepts

Numerator and Denominator

In a rational expression, understanding the roles of the numerator and denominator is crucial. The numerator is the top part of a fraction, and it represents the number of parts we have. The denominator is the bottom part, and it shows into how many parts the whole is divided. In the expression \( \frac{6x-1}{3x-1} \), \( 6x-1 \) is the numerator and \( 3x-1 \) is the denominator.

The denominator indicates the common base of the values represented by the numerators when adding two fractions. For example, in the exercise given, both fractions had a common denominator of \( 3x-1 \). This made combining their numerators straightforward, as both numerators could be manipulated directly without needing additional adjustments to the denominator.

Factoring

Factoring refers to breaking down a mathematical expression into simpler parts, or factors, that when multiplied together will give us the original expression. In the context of rational expressions, factoring is crucial for simplifying complex fractions.

Consider the expression \( 9x - 3 \) from the solution. The first step in factoring is identifying the greatest common factor (GCF), which is the largest number that divides all terms in the expression. In this example, the GCF is 3 since both 9 and 3 are divisible by 3. Therefore, \( 9x - 3 \) can be rewritten as \( 3(3x-1) \). This simplification is vital before further operations like simplification, as it often reveals hidden opportunities to cancel out parts of the expression.

Simplifying Expressions

Simplifying rational expressions involves reducing them to their simplest form. This process typically includes combining like terms, factoring, and canceling out terms. Once an expression is simplified, it becomes easier to work with in equations or further calculations.

In the problem, reducing \( \frac{9x - 3}{3x - 1} \) to its simplest form involves canceling common factors between the numerator and the denominator. After factoring the numerator to \( 3(3x-1) \), we noticed the \( (3x-1) \) term is identical to the denominator. By canceling out the \( (3x-1) \) from both the numerator and denominator, the expression simplifies to just 3. This elimination step is essential as it reduces complexity and reveals the true value of the expression given the restriction that \( 3x-1 eq 0 \).