Problem 32 - Prealgebra and Introductory Algebra

Question

Solve. $$2+\frac{3}{a-3}=\frac{a}{a-3}$$

Step-by-Step Solution

Answer

The solution to the equation is \(a = 9\)

Step 1: Identify like terms

Looking at the equation, it can be seen that the denominators \((a-3)\) on the right side of the equation and the term \(\frac{3}{a-3}\) on the left side are alike. This presents an opportunity to simplify the equation.

Step 2: Eliminate the denominators

By multiplying every term in the equation by \((a-3)\), the denominators in the equation will be eliminated. This is done in such a way it results in the equation \(2(a-3) + 3 = a\).

Step 3: Simplify the equation

Next the equation is simplified. This involves the expansion of \(2(a-3)\) to get \(2a-6\). Hence the equation becomes \(2a-6+3=a\).

Step 4: Solve for \(a\)

The final step involves solving the above equation for \(a\). This is done by moving the terms involving \(a\) to one side of the equation and the constants to another. The equation becomes \(2a - a = 6 + 3\). Solving for \(a\) gives \(a = 9\).

Key Concepts

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. When faced with an equation like \(2 + \frac{3}{a-3} = \frac{a}{a-3}\), our goal is to determine the value of \(a\).
To solve this equation more effectively, we follow a sequence of steps to transform it into a form where the value of \(a\) can be easily identified. These steps include recognizing the structure of the equation, eliminating fractions, simplifying, and isolating the variable. Each of these steps builds upon the previous one, ultimately leading us to a solution.

Eliminating Denominators

One important strategy in solving rational equations is to eliminate the denominators. In the equation \(2 + \frac{3}{a-3} = \frac{a}{a-3}\), denominators make it challenging to work directly with the terms.
To eliminate these, we multiply every term by the common denominator, which is \((a-3)\) in this case.
By doing so, the equation becomes free of fractions: \(2(a-3) + 3 = a\). This step simplifies the equation significantly and allows us to more easily gather the terms involving \(a\). Eliminating denominators is crucial as it streamlines the equation, setting the stage for further simplification.

Simplifying Equations

After removing the denominators, our next focus is simplifying the equation. The equation \(2(a-3) + 3 = a\) needs to be simplified for clarity.

• First, distribute the \(2\) into \((a-3)\), which gives \(2a - 6\).
• Combining like terms results in \(2a - 6 + 3 = a\). Further simplifying, we have \(2a - 3 = a\).

Each step in the simplification process makes the equation easier to manage. Simplification is about reducing the equation to its simplest form, helping us clearly distinguish the terms that contain variables from those that are constants.

Variable Isolation

The final part of solving an equation is isolating the variable. Once the equation is simplified to \(2a - 3 = a\), our goal is to have \(a\) on one side and constants on the other.

• We subtract \(a\) from both sides to get \(2a - a = 3\) which simplifies to \(a = 3\).

Variable isolation is critical as it helps us find the value of \(a\) without confusion from other terms. This methodical approach not only reveals the solution in a straightforward manner but also underscores the importance of orderly, strategic thinking in solving equations.