Problem 37 - Intermediate Algebra

Question

Solve each equation. If a solution is extraneous, so indicate. \(4-\frac{3 x}{x-9}=\frac{5 x-72}{x-9}\)

Step-by-Step Solution

Answer

The equation has an extraneous solution: \(x = 9\).

Step 1: Set the Denominators to Be the Same

The denominators on both sides of the equation are already the same: \(x-9\). This allows us to multiply each term by \(x-9\) to remove the denominators.

Step 2: Remove the Denominators

Multiply each term by \(x-9\) to eliminate the fractions: \[ (4 - \frac{3x}{x-9}) \cdot (x-9) = \frac{5x-72}{x-9} \cdot (x-9) \]This simplifies to:\[ 4(x-9) - 3x = 5x - 72 \]

Step 3: Simplify the Equation

Distribute and simplify the terms:\[ 4x - 36 - 3x = 5x - 72 \]Combine like terms:\[ x - 36 = 5x - 72 \]

Step 4: Isolate the Variable

Rearrange the equation to isolate \(x\) on one side:\[ x - 5x = -72 + 36 \]Simplify both sides:\[ -4x = -36 \]

Step 5: Solve for x

Divide both sides by -4:\[ x = \frac{-36}{-4} \]\[ x = 9 \]

Step 6: Check for Extraneous Solutions

Substitute \(x = 9\) back into the original equation to check for validity.The original denominators \(x-9\) become 0 when \(x = 9\), which makes the equation undefined. Thus, \(x = 9\) is an extraneous solution.

Key Concepts

Rational Equations

Rational equations are equations that involve fractions in which the numerator, the denominator, or both contain polynomials. Solving rational equations is all about finding the value of the variable that makes the equation true. In these equations, the key is to eliminate the fractions by finding a common denominator or appropriately multiplying both sides to clear out the fractions.
Let's break it down:

• Identify Rational Equation: Recognize that if the equation has fractions involving variables, it's a rational equation. For example, in our original exercise, the equation \(4-\frac{3x}{x-9}=\frac{5x-72}{x-9}\) is a rational equation.
• Clear the Denominator: To solve such equations, eliminate the fractions by multiplying every term by the least common denominator. This leaves you with a simpler equation to solve.

Rational equations often require checking for extraneous solutions. Why? Because when variables are in the denominator, it means the solution might lead to a zero denominator, making it invalid. Always substitute back possible solutions to verify their validity.

Fractions in Algebra

Fractions in algebra can make solving equations look much more complicated. However, understanding how to manage and simplify fractions is crucial. Here’s how to approach fractions within algebraic equations:

• Find the Common Denominator: It's important to consider the least common denominator when you want to add or subtract fractions. This step is essential in combining fractions into a single expression.
• Elimination of Fractions: In the original exercise, fractions were removed by multiplying the entire equation by the common denominator \(x-9\) simplifying the equation substantially.
• Consider Operations on Fractions: When you multiply or divide fractions, remember to apply the fundamental rules such as multiplying across the numerators and denominators or flipping the fraction and multiplying it when dividing by a fraction.

Working with fractions is like understanding a universal language in algebra. Once you master these concepts, many problems become less intimidating.

Simplifying Equations

Simplifying equations is a crucial step toward finding the solution. It allows you to strip down an equation to its simplest form, making it easier to handle:

• Distribute and Simplify: Once the fractions are removed, distribute any multipliers through the parentheses to clean up the equation. In the exercise, the step from \[4(x-9) - 3x = 5x - 72\] to \[x - 36 = 5x - 72\] involved simple distribution and combining like terms.
• Isolate the Variable: Aim to have the variable on one side of the equation. Rearrange terms by using operations such as addition or subtraction until it's isolated. This involves keeping the balance by doing the same operation to both sides of the equation.
• Combine and Solve: After isolating the variable, solve the simplified form of the equation. Do the arithmetic to find the value of the variable, as in solving \[-4x = -36\] which led to \(x = 9\).

Always remember to double-check your solution, especially in instances involving extraneous solutions. Confirm the solution works in the original equation and doesn't lead to mathematical impossibilities like division by zero.