Problem 22 - Algebra 1: Concepts and Skills

Question

Solve the equation by multiplying each side by the least common denominator. Check your solutions. \(\frac{x}{x+9}=\frac{9}{x+9}+4\)

Step-by-Step Solution

Answer

After solving the equation, we find that \(x = -9\), however, by substituting this result back into the original equation, it leads to a division by zero, which is undefined. Hence, the equation has no valid solutions.

Step 1: Determine the Least Common Denominator and Multiply

We note that the denominator for all the fractions in the equation is \(x+9\), so that's our least common denominator (LCD). We then multiply each term by \(x+9\) to eliminate the denominators. \[(x+9) \cdot \frac{x}{x+9} = (x+9) \cdot \frac{9}{x+9} + (x+9) \cdot 4\]Which simplifies to: \[x = 9 + 4x + 36\]

Step 2: Solve for x

Rearranging the equation to get all the terms containing \(x\) on one side and the numerical values on the other side gives: \[4x- x = 9 -36\]This simplifies to: \[3x = -27\]Dividing both sides by 3, we find that:\[x = -9\]

Step 3: Check the solution

We substitute the obtained solution into the original equation to verify if it holds true. Substituting \(x = -9\) into the original equation \(\frac{-9}{-9+9} = \frac{9}{-9 + 9} + 4\)gives \(\frac{-9}{0}= \frac{9}{0}+ 4\),which is undefined. Thus, \(x = -9\) is not a solution to the equation. The equation has no valid solutions.

Key Concepts

Least Common Denominator

When solving equations with fractions, finding the Least Common Denominator (LCD) is like finding a common language for all the terms to communicate in. It's the smallest number that each of the denominators can divide into without leaving a remainder.

For instance, consider the equation \(\frac{x}{x+9} = \frac{9}{x+9} + 4\). Here, the denominator is the same for both fractions, namely \(x+9\), so the LCD is simply \(x+9\). By multiplying each term by the LCD, we not only ensure consistency across the equation but also strategically eliminate the fractions, making our equation much simpler to solve.

Mastering the skill of finding the least common denominator is a game-changer in solving rational equations effectively.

Equation Solving

The art of untangling equations is akin to solving a puzzle. Equation solving involves combining like terms, rearranging the equation to isolate the variable, and simplifying the expressions systematically. Look at the simplified version of our earlier equation: \(x = 9 + 4x + 36\). With this, we organize our x's and constants on opposite sides to get \(4x - x = 9 - 36\), simplifying further to \(3x = -27\).

Finally, by dividing by the coefficient of \(x\), which is 3 in this case, we clearly see what x stands for. When you peel back the layers of the equation, you will find that each step is about balance and symmetry—what you do to one side, you must do to the other, ensuring the scale of equality is never tipped.

Checking Solutions

Think of checking solutions as proofreading your own work—this step is crucial. After solving the equation, we must go back to the original equation to ensure our answer is not only correct but also valid. In our case, substituting \(x = -9\) back into our original equation \(\frac{x}{x+9} = \frac{9}{x+9} + 4\) should maintain the balance of the equation. However, when we substitute our \(x\) value, we get an undefined expression. This is a tell-tale sign: just as a sentence with a missing word doesn't make sense, an undefined mathematical expression means our solution doesn't fit. Therefore, checking solutions helps us verify the accuracy of our solutions and reassures us that they are meaningful within the context of the original problem.

Undefined Expressions

While many may fear the unknown, in mathematics, 'undefined' has a specific meaning, especially when dealing with rational expressions. An undefined expression often involves a scenario where we're tempted to divide by zero—as seen when we attempted to substitute \(x = -9\) into the original equation, which would result in \(\frac{-9}{0}\), an impossibility in math.

Division by zero is undefined because it breaks the rules of arithmetic: it's like trying to share zero cookies with your friends, impossible! So, when solving rational equations, always be on the lookout for these undefined expressions. They're red flags indicating that the value you've found cannot be the solution, no matter how neatly it fits algebraically. Understanding why certain expressions are undefined is a vital part of mastering algebraic concepts.