Problem 34 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{2 x-5}{x-6}=\frac{x+1}{x-6} $$

Step-by-Step Solution

Answer

Answer: Undefined Equation (No valid solution).

Step 1: Cross-multiply the terms.

Since both terms on each side of the equation have the same denominator (x-6), we can cross-multiply the terms, which effectively cancels out the denominators and gives us only the numerators. We are left with: $$ 2x-5=x+1 $$

Step 2: Solve for x.

Now we have a simple linear equation. We will first isolate the x terms on one side and the constants on the other side. $$ 2x-x=1+5 \\ x=6 $$ However, we must confirm that this solution does not cause the denominator to be zero, as this would make the equation undefined.

Step 3: Check for undefined solutions.

We need to make sure the solution does not cause the denominator (x-6) to be zero. If we plug the calculated value of x (6) into the denominator, we get: $$ 6-6 = 0 $$ Since the denominator becomes zero when x is 6, we have an undefined equation and cannot accept this solution.

Step 4: Final Answer: Undefined Equation

As x=6 causes the denominator to become zero, the given rational equation cannot be solved. Therefore, there is no valid solution.

Key Concepts

Denominator

In rational equations, the denominator plays a vital role. The denominator is the number or expression located underneath the fraction bar in a fraction. It gives us information about the size of each part we are considering. For example, in the problem \[\frac{2x-5}{x-6}=\frac{x+1}{x-6}\] the denominator for both sides is \(x-6\). This denominator is essential because if it equals zero, the fraction becomes undefined. Remember: A fraction cannot have a zero denominator because division by zero is impossible.

Whenever working with rational equations, always identify the denominator first. Check if the solution makes this denominator zero. This is crucial because it leads to an undefined equation!

Linear Equation

Linear equations are the simplest forms of algebraic equations. They typically have variables with an exponent of one. When you solved the rational equation by cross-multiplying its terms, you eliminated the denominators and were left with a linear equation \[2x-5=x+1\].

Solving linear equations involves isolating the variable. First, gather all terms involving the variable on one side and constants on the other:

• Subtract \(x\) from both sides, resulting in: \(2x-x=1+5\)
• Combine like terms to find \(x=6\).

Use simple operations like addition, subtraction, multiplication, and division. They help simplify each step. Linear equations are often straightforward once you get the hang of these basic operations.

Undefined Solutions

Sometimes, during the solution of a rational equation, a possible solution might make the equation undefined. This usually happens when the solution causes the denominator of any term to equal zero. For our given equation, \[\frac{2x-5}{x-6}=\frac{x+1}{x-6}\], we found that \(x=6\). However, as we plug \(x=6\) back into the denominator \(x-6\), it becomes \[6-6=0\].

Since a zero in the denominator makes the rational expression undefined, \(x=6\) cannot be accepted as a valid solution.

This check for undefined solutions is important. In rational equations, always validate whether your answers will make any denominator zero to avoid incorrect solutions. It's all about ensuring the problem remains meaningful and mathematically sound!