Problem 78 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{5 x-1}{6}+\frac{3 x+4}{9}=\frac{-8}{9} $$

Step-by-Step Solution

Answer

Answer: The solution for the given rational equation is x = -1.

Step 1: Find the Least Common Denominator

The given equation has denominators 6 and 9, so we'll find the smallest number that both 6 and 9 divide evenly into. The least common denominator for the equation will be 18.

Step 2: Multiply Both Sides of the Equation by the LCD

Now, we'll clear out the denominators by multiplying both sides of the equation by 18: $$ 18\left(\frac{5 x-1}{6}+\frac{3 x+4}{9}\right)=18\left(\frac{-8}{9}\right) $$ Distribute 18 to all the fractions: $$ \frac{18(5 x-1)}{6}+\frac{18(3 x+4)}{9}=\frac{-8(18)}{9} $$

Step 3: Simplify the Equation and Solve for x

Now we will simplify the equation and solve for x: $$ \frac{3(5 x-1)}{1}+\frac{2(3 x+4)}{1}=-16 $$ Now we simplify the equation: $$ 15x-3+6x+8=-16 $$ Combine like terms: $$ 21x+5=-16 $$ Subtract 5 from both sides: $$ 21x=-21 $$ Divide by 21: $$ x=-1 $$ So, the solution of the rational equation is x = -1.

Key Concepts

Understanding Algebra in Rational Equations

Algebra forms the foundation for analyzing and solving rational equations. It involves manipulating symbols and numbers to find unknown values. In the context of algebra, equations like \( \frac{5x-1}{6} + \frac{3x+4}{9} = \frac{-8}{9} \) require identifying unknowns, such as \(x\), and employing algebraic rules to solve them.

• Start by isolating terms and simplifying expressions.
• Use operations like addition, subtraction, multiplication, and division to both sides of the equation.
• Rearranging terms can assist in simplifying and resolving the equation more effectively.

By mastering algebra, students can find unknown variables in complex rational equations with confidence.

Using the Least Common Denominator (LCD)

The least common denominator (LCD) is crucial when working with rational equations. Finding the LCD allows us to eliminate denominators, which simplifies the equation. In the equation \( \frac{5x-1}{6} + \frac{3x+4}{9} = \frac{-8}{9} \), the denominators are 6 and 9. Here's a quick guide on calculating the LCD:

• List the multiples of each denominator.
• Identify the smallest multiple that is common to both lists, which is 18 in this case.

Multiplying the entire equation by the LCD results in eliminating fractions, thus easing the process of solving the equation. This method is essential when dealing with equations involving multiple denominators.

Solving Equations Step by Step

Solving equations, particularly rational ones, requires a clear methodical approach. Each step builds on the previous, ensuring accuracy throughout the solving process. Here's a logical flow for solving equations like the example given:

• Multiply each term by the LCD to remove the fractions.
• Distribute and simplify each term thoroughly to keep track of operations.
• Combine like terms to simplify the equation further. For instance, \(15x-3+6x+8=-16\) simplifies to \(21x+5=-16\).
• Isolate the variable by performing inverse operations, such as subtraction and division, giving us \(x=-1\).

This systematic approach facilitates effective and precise solutions, especially in complex scenarios.

Exploring Algebraic Fractions

Algebraic fractions are expressions where the numerator and/or the denominator contains algebraic expressions. They appear throughout rational equations and are pivotal to understanding and managing these equations efficiently.

• These fractions are similar to simple fractions but include variables.
• Managing algebraic fractions involves operations like factorization, which simplifies expressions.
• Knowing how to manipulate algebraic fractions allows for simplification, aiding in solving equations where these fractions are present, as in the example \(\frac{5x-1}{6}\).

Understanding and working with algebraic fractions expands one's algebraic toolbox, enabling better tackling of rational equations.