Problem 19 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{b+11}{3}+8=6 $$

Step-by-Step Solution

Answer

Answer: \(b=-17\)

Step 1: Combine constant terms

Subtract 8 from both sides of the equation to combine constant terms. The equation becomes: $$ \frac{b+11}{3}=-2 $$

Step 2: Get rid of the fraction

To get rid of the fraction, multiply both sides of the equation by 3. The equation becomes: $$ (b+11)=-6 $$

Step 3: Solve for b

Subtract 11 from both sides to isolate b. The equation becomes: $$ b=-6-11 $$ Now solve for b: $$ b=-17 $$ The solution to the given rational equation is \(b=-17\).

Key Concepts

Rational Equations

Rational equations are an essential part of algebra. They are equations in which variables are found in the denominators of fractions. A typical rational equation looks like this: \(\frac{b+11}{3} + 8 = 6\). Solving these equations involves a series of strategic steps to ultimately isolate the variable. Understanding how to clear fractions is crucial when dealing with rational equations. This involves multiplying every term by the denominator, thus eliminating the fractions and simplifying the equation. This technique allows us to deal with a simpler, linear equation. Moreover, knowing how to manage these equations will help in numerous areas of math, like algebraic fractions and calculus, making it a fundamental skill to master.

Step-by-Step Solutions

Mastering rational equations requires following specific steps methodically. Let's break down the process using our given rational equation:

• Step 1: Simplify the equation. Begin by adjusting constants. For instance, subtracting 8 from both sides in this exercise to simplify the equation to \(\frac{b+11}{3} = -2\).
• Step 2: Eliminate fractions. Multiplying every term by the denominator, which is 3 in our case, results in \(b + 11 = -6\). Now, the equation is more straightforward because it no longer has fractions.
• Step 3: Solve for the variable. Isolate \(b\) by subtracting 11, resulting in \(b = -6 - 11\), which simplifies to \(b = -17\).

By systematically applying these steps, rational equations become easier to manage and resolve. Each step builds on the previous one, ensuring clarity and leading us to the solution efficiently.

Algebra Problems

Algebra problems can sometimes be daunting, especially when they involve rational equations. However, breaking the problems down into smaller, manageable parts can simplify the process. When tackling algebra, key strategies include

• Identifying all parts of the equation and how they relate to one another.
• Using inverse operations like addition or subtraction to isolate variables.
• Carefully managing operations to maintain the balance of the equation.

Approaching algebra problems with these strategies helps ensure errors are minimized. Practice is crucial; working through similar problems strengthens your comprehension and skill. As you gain fluency with algebraic operations and methods, such challenges will become more accessible and less intimidating.

Educational Mathematics

Educational mathematics covers a wide array of topics, and rational equations are just one part of this extensive field. Studying math encourages critical thinking and problem-solving skills, which are applicable in various aspects of life. By engaging with exercises like solving rational equations, students learn how to:

• Develop logical reasoning skills.
• Apply mathematical concepts to real-world problems.
• Explore various solution strategies and adapt to different types of problems.

Educational mathematics aims to build a solid foundation that supports advanced studies in science, technology, engineering, and other math-related fields. Beyond academia, these skills foster analytical thinking and decision-making, which are valuable in everyday life. Embracing such challenges in educational settings cultivates a mindset ready to tackle any problem, large or small.