Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation (if possible). $$\frac{2}{x(x-2)}+\frac{5}{x}=\frac{1}{x-2}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2\). However, substituting \(x = 2\) back into the original equation leads to division by zero in the terms. So, this equation has no solution.

Step 1 - Find common denominator

Identify the common denominator, which in this case, is \(x(x-2)\). So, we convert all the fractions to have this common denominator.

Step 2 - Rewrite the equation

Rewrite the original equation by changing each term to the equivalent fraction with the common denominator \(x(x-2)\):\n\( \frac{2}{x(x-2)} + \frac{5(x-2)}{x(x-2)} = \frac{x}{x(x-2)} \).

Step 3 - Simplify equation

To eliminate the fractions, multiply every term with common denominator \(x(x-2)\), which results in \n\(2 + 5(x - 2) = x\). After simplification, this yields the equation \(5x - 8 = x\).

Step 4 - Solve the equation

Move x on the right side over to left side to have equation in standard form\(4x - 8 = 0\). Solve the equation for \(x\), we get \(x = 2\).

Step 5 - Check the solutions

Substitute \(x = 2\) back into the original equation and see if the equation holds, if needed.

Key Concepts

Common Denominator

When solving equations with multiple fractions, it's crucial to find a common denominator, which is a shared multiple of the denominators of these fractions. This commonality allows us to combine the fractions and simplify the equation. Think of it as finding a common ground for all fractions involved.

In the provided exercise, the common denominator is identified as the product of the unique factors from each term's denominator, which is \(x(x-2)\). By ensuring each fraction has this common denominator, we facilitate the addition or subtraction of the fractions, setting the stage for the next steps of solving the equation.

Equation Simplification

Equation simplification is the process of reducing an equation to a simpler or more efficient form. This often involves combining like terms, eliminating fractions, and clearing out parentheses. Simplifying an equation makes it more approachable and easier to solve.

In our exercise, after finding a common denominator, we multiply each term by that denominator to clear the fractions. This leaves us with a simplified equation: \(5x - 8 = x\). This step is essential, as it transforms a complex fractional equation into a simple linear equation, bringing us closer to finding the value of \(x\).

Fractions in Algebra

Dealing with fractions in algebra can be tricky, but understanding how to manipulate them is a valuable skill. When fractions appear in algebraic equations, it's often helpful to eliminate them early on. This can be done by finding a common denominator and multiplying the entire equation by this common denominator.

By doing so in our example, the problematic fractions are removed, leaving behind an algebraic expression that is easier to manage. Remember, the key in algebra is to simplify whenever possible to reduce the chances of errors in computation.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations. In the context of solving equations, algebraic expressions can become complex when they involve fractions. Simplifying these expressions, as we do in the elimination of fractions, can reveal a clearer path to the solution.

In the case of the exercise, once the fractions are eliminated, we are left with a linear algebraic expression. We can then use basic algebraic methods, such as isolating the variable, to solve the equation. Understanding how to manipulate these expressions is fundamental to mastering algebra.