Solve for x 5/x-3/x=8

Solution:

Step 1: Identify the Least Common Denominator (LCD)

• Step 1.1: The LCD is determined by finding the Least Common Multiple (LCM) of the denominators. The denominators are $x$, $x$, and $1$.

• Step 1.2: To find the LCM, separate the process into two parts: one for the numerical values and one for the variable parts. The numerical values are $1$, $1$, and $1$, while the variable parts are $x^1$ and $x^1$.

• Step 1.3: The LCM is the smallest number that all original numbers can divide into without a remainder. To find it, list out the prime factors and multiply each factor by the highest count it appears in any of the numbers.

• Step 1.4: The number $1$ is unique as it only has one factor, itself, and is not considered prime.

• Step 1.5: The LCM of the numbers $1$, $1$, and $1$ is simply $1$ since it's the product of all unique prime factors taken to their highest power.

• Step 1.6: For the variable part, $x^1$ is just $x$, which appears once.

• Step 1.7: The LCM of the variable parts $x^1$ and $x^1$ is $x$, since it's the product of all unique variable factors taken to their highest power.

Step 2: Eliminate the Fractions

• Step 2.1: Multiply every term in the equation $\frac{5}{x} - \frac{3}{x} = 8$ by the LCD, which is $x$.

• Step 2.2: Simplify the equation by canceling out the common factors.

  • Step 2.2.1: Address each term separately.

    • Step 2.2.1.1: For the term $\frac{5}{x} x$, cancel out the $x$ in the numerator and denominator.

    • Step 2.2.1.1.2: The term simplifies to $5$.

    • Step 2.2.1.2: For the term $-\frac{3}{x} x$, first consider the negative sign in the numerator.

    • Step 2.2.1.2.2: Cancel out the $x$ in the numerator and denominator.

    • Step 2.2.1.2.3: The term simplifies to $-3$.

  • Step 2.2.2: Combine the simplified terms to get $2 = 8x$.

Step 3: Solve for $x$

• Step 3.1: Rearrange the equation to $8x = 2$.

• Step 3.2: Divide both sides of the equation by $8$ to isolate $x$.

  • Step 3.2.1: Perform the division to get $\frac{8x}{8} = \frac{2}{8}$.

  • Step 3.2.2: Simplify the left side by canceling out the $8$.

    • Step 3.2.2.1: After canceling, you get $x = \frac{2}{8}$.

  • Step 3.2.3: Simplify the right side by reducing the fraction.

    • Step 3.2.3.1: Factor out the greatest common divisor, which is $2$.

    • Step 3.2.3.1.2: Cancel out the common factors to get $x = \frac{1}{4}$.

Step 4: Present the Result in Different Forms

• Exact Form: $x = \frac{1}{4}$
• Decimal Form: $x = 0.25$

Knowledge Notes:

• Least Common Multiple (LCM): The smallest positive integer that is divisible by each of the original numbers. When dealing with fractions, the LCM of the denominators is used as the LCD to combine fractions or clear them from an equation.

• Prime Factorization: A method used to find the LCM by breaking down numbers into their prime factors and then taking the highest power of each prime that appears in any of the numbers.

• Simplifying Fractions: Involves canceling out common factors in the numerator and the denominator or reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor.

• Solving Linear Equations: A process that involves isolating the variable on one side of the equation, typically through the use of inverse operations such as addition/subtraction and multiplication/division.

• Equivalent Forms of Numbers: Numbers can be expressed in various forms, such as fractions, decimals, and percentages, which are equivalent in value but may be more suitable for different contexts or calculations.