Problem 81 - Elementary Algebra

Question

Solve each of the following equations. Don't forget that division by zero is undefined. $$\frac{3}{x-2}=\frac{6}{2 x-4}$$

Step-by-Step Solution

Answer

All real numbers except \(x = 2\).

Step 1: Identify the denominator expressions

The given equation is \(\frac{3}{x-2} = \frac{6}{2x-4}\). First, identify the denominators in the equation: one denominator is \(x-2\) and the other is \(2x-4\).

Step 2: Simplify the right side denominator

Notice that \(2x - 4\) can be factored as \(2(x - 2)\). So rewrite the right side of the equation as \(\frac{6}{2(x-2)}\).

Step 3: Cancel common factors

The equation is now \(\frac{3}{x-2} = \frac{6}{2(x-2)}\). Both sides have a common factor of \((x-2)\) in the denominator. Cancel these factors assuming \(x eq 2\) (to avoid division by zero), the equation simplifies to \(3 = \frac{6}{2}\).

Step 4: Simplify the resulting equation

The equation simplifies to \(3 = 3\), which is true for all \(x\) except when \(x = 2\) (since division by zero is undefined at this point).

Step 5: State the solution

Since \(3 = 3\) is an identity, \(x\) can be any real number except \(x = 2\) to avoid division by zero.

Key Concepts

Division by Zero

In mathematics, division by zero is a crucial concept that we always need to be aware of. It happens when a number is divided by zero, such as in an expression like \( \frac{a}{0} \). This operation is undefined, meaning it has no meaningful result. Here's why:

• If we take any number and try to divide it by zero, we're essentially trying to divide something into no parts. This doesn't make sense since division implies distributing an amount into a given number of parts.
• In our original equation, both denominators \( x-2 \) and \( 2(x-2) \) must not be zero because division by zero is not allowed. Thus, \( x \) cannot be 2, otherwise both denominators become zero.
• Whenever solving equations with fractions, always check the denominators first. Ensure that they do not equal zero.

By paying attention to division by zero, we prevent mathematical errors and find valid solutions.

Factoring Expressions

Factoring expressions is a method used to simplify equations by expressing a polynomial expression as a product of its factors. It's a useful tool in algebra to break down complex expressions. In our step-by-step solution, we had the equation \( \frac{3}{x-2} = \frac{6}{2x-4} \).

• The expression \( 2x - 4 \) can be rewritten by factoring out the greatest common factor, which in this case is 2, resulting in \( 2(x-2) \).
• After factoring, the equation becomes easier to solve because we can see common factors that can be canceled out.
• Always look for ways to factor expressions as it can simplify the equation and bring it to a more manageable form.

Factoring is a critical skill in algebra that often leads to finding the solution more quickly and accurately.

Identity Equations

An identity equation is an equation that is true for all values of the variable involved, barring any restrictions such as division by zero. In our solution, after simplifying, we reached the equation \( 3 = 3 \). Here’s what you need to know:

• An identity like \( 3 = 3 \) holds true universally, meaning \( x \) can take any value.
• However, remember to exclude any values that make the original equation undefined. In our case, when \( x=2 \), both fractions have a denominator of zero, which is not allowed.
• With identity equations, focus on the restrictions that might not make the identity hold due to undefined operations like division by zero.

Understanding identity equations help us know when an equation has a solution set covering all possible numbers (except where the operation is undefined).