Problem 30 - College Algebra Essentials

Question

Contain linear equations with constants in denominators. Solve equation. \(\frac{3 x}{5}-\frac{x-3}{2}=\frac{x+2}{3}\)

Step-by-Step Solution

Answer

The solution to the linear equation is \(x = -\frac{25}{8}\)

Step 1: Look for the least common multiple (LCM)

Identify the denominators of the fractions, they are 5, 2, and 3. Their least common multiple is 30.

Step 2: Clear Fractions Using the LCM

Multiply every term of the equation by the LCM of the denominators (which is 30) to eliminate the fractions. It will result in \(30*\frac{3x}{5}-30*\frac{x-3}{2}=30*\frac{x+2}{3}\), which simplifies to \(18x - 15(x - 3) = 10(x+2)\).

Step 3: Simplify the Equation

Simplify the equation obtained from step 2. This will result in: \(18x - 15x + 45 = 10x + 20\)

Step 4: Solve for x

Rearrange the equation to solve for x by moving the x terms to one side and constants to the other side of the equation, i.e., \(18x - 10x = 20 - 45\). Simplify this equation to get \(8x = -25\)

Step 5: Evaluate x

Finally, divide by 8 (the coefficient of x) on both sides of the equation to determine the value of x: \(x = -\frac{25}{8}\)

Key Concepts

Least Common Multiple

The first step in tackling algebraic equations involving fractions is to identify the least common multiple (LCM) of the denominators. When you have denominators like 5, 2, and 3 in the equation, finding their LCM helps you eliminate the fractions. The LCM of a set of numbers is the smallest number that each of the numbers divides into evenly.
To determine the LCM:

• List the multiples of each denominator.
• Find the lowest multiple common to all denominators.

In our example, multiplying these denominators together can also directly give the LCM, which is 30.
Using the LCM to clear out fractions simplifies solving by turning the fractional equation into a cleaner linear equation, free from fractions.

Simplifying Equations

Once the least common multiple clears the fractions from the equation, it's crucial to simplify the resulting equation. Simplification involves distributing any factors and combining like terms. By multiplying every term in the equation by the LCM and applying distribution, you eliminate the fractions.
In the example provided:

• Each numerator is multiplied by the LCM.
• Work through the arithmetic to remove parentheses.
• Combine like terms to streamline the equation further.

This transformation from a complex equation with fractions to a simpler linear form allows you to focus on the core aspects of solving for the variable.

Algebraic Fractions

Algebraic fractions appear daunting but can be easily handled with a few strategic steps. In essence, algebraic fractions are expressions of the form \(\frac{a}{b}\), where both \(a\) and \(b\) can include variables and constants. Solving equations with such fractions demands patience and precision.
When working with algebraic fractions:

• Focus first on understanding the terms within each fraction.
• Identify common denominators to facilitate addition and subtraction.
• Utilize the LCM to convert the equation effectively into a simpler form.

By carefully applying the properties of arithmetic, you can transform an equation with algebraic fractions into a form more easily digestible and solvable.