Problem 64 - Algebra 2

Question

Solve and check the equation \(\frac{x}{3 x+9}=\frac{x+2}{x+3}\) . Show your work.

Step-by-Step Solution

Answer

There are no solutions to the equation \(\frac{x}{3 x+9}=\frac{x+2}{x+3}\)

Step 1: Simplify the equation

We can see that the denominators of both fractions 3x+9 and x+3 have a common factor of 3. Simplify the equation by dividing each term of the denominator 3x+9 with 3 to have the same denominator in both fractions. Thus, the simplified equation is \(\frac{x}{3(x+3)}=\frac{x+2}{x+3}\).

Step 2: Remove the denominator

To get rid of the fractions, we can multiply both sides of the equation through by (x+3) as it is the common denominator. After multiplying, the fractions will drop out and the equation will simplify to x = x + 2.

Step 3: Find the solution

We then combine like terms and solve for x by either adding or subtracting. Solving x = x + 2, we subtract x from both sides of the equation to get 0 = 2. However, 0 does not equal 2, which suggests that there are no solutions to the given equation.

Step 4: Check the solution

Even though we've found that there are no solutions to the equation, it's always good to double-check. By substituting a solution back into the original equation, one can see if both sides of the equation remain equal. However, as there are no solutions in this case, there's nothing to substitute in.

Key Concepts

Simplification

Simplification is the process of making an equation or expression easier to work with by reducing it to its simplest form. When dealing with rational equations, this often involves factoring and canceling terms. In our example, the equation is \[\frac{x}{3x+9} = \frac{x+2}{x+3}\,\]which has complex denominators.

• The denominator \(3x+9\) can be factored by taking out the common factor of 3, resulting in \(3(x+3)\).
• This makes the equation simpler because both denominators now share \((x + 3)\) as a common term.

Simplifying equations not only makes the equation easier to solve but also provides clearer insight into the structure of the problem.

Common Denominator

The concept of a common denominator is crucial when dealing with rational equations. A common denominator allows us to simplify equations by eliminating the denominators entirely.

• In the given exercise, each side of the equation shares a common denominator of \(x+3\).
• This commonality is used to simplify the equation by multiplying through by \(x+3\).

This process eliminates the fractions, making it easier to deal with the resulting simpler equation. By transforming the equation into a non-fraction form, we can more directly address the solution.

No Solution

In certain mathematical problems, particularly with rational equations, you might encounter a scenario where no solution exists. This happens when solving leads to a mathematical statement that is untrue.

• In the provided exercise, after simplification, we end up with the equation \(x = x + 2\).
• Rearranging terms, we find \(0 = 2\), which is a contradiction.

When a contradiction like this appears, it indicates that there are no values of \(x\) that can satisfy the original equation. It is essential to recognize these situations, as they prevent wasted effort in searching for nonexistent solutions.

Cross Multiplication

Cross multiplication is a technique often used in solving equations involving fractions. It involves multiplying the numerator of one fraction with the denominator of the other, and vice versa, to eliminate the fractions.

• In the exercise, this wasn't strictly necessary because simplifying and using the common denominator was sufficient.
• However, if the equation had different denominators that couldn't be easily simplified, cross multiplication would be the go-to strategy.

Understanding cross multiplication enriches your mathematical toolkit, allowing you to tackle a wider range of rational equations effectively.