Problem 66 - Elementary Algebra Within Reach

Question

In Exercises 65-68, solve the rational equation. $$ \frac{3 x}{5}+\frac{4 x}{8}=\frac{11}{10} $$

Step-by-Step Solution

Answer

The solution to the rational equation is \( x = 1 \)

Step 1: Multiply the equation by 40 (the least common multiple of 5 and 8)

Multiplication will simplify the fractions in the equation. \[ 40 (\frac{3 x}{5})+ 40 (\frac{4 x}{8}) = 40 (\frac{11}{10}). \] This simplifies to \( 24x + 20x = 44.\)

Step 2: Combine like terms

Combine the terms with the variable x on the left side, resulting in: \[ 44x = 44. \]

Step 3: Solve for x

Final step is to isolate x by dividing each side of the equation by 44, giving us \( x = 1 \) as the solution.

Key Concepts

Least Common Multiple

When solving equations involving fractions, finding the least common multiple (LCM) of the denominators is crucial. The LCM is the smallest positive integer that is divisible by each denominator in an equation. In our example, the denominators are 5 and 8.

• The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, ...
• The multiples of 8 are: 8, 16, 24, 32, 40, ...

As you can see, the smallest multiple common to both sets is 40. We use this LCM to eliminate the fractions, making it easier to work with the equation by turning all terms into whole numbers. This process simplifies further operations, as seen when we multiply the entire equation by 40 in the first step of our solution.

Combining Like Terms

After multiplying fractions by the LCM, the next step often involves combining like terms. Like terms have identical variable parts. For instance, in our problem, terms such as \(24x\) and \(20x\) on the left side of the equation are both coefficients of the variable \(x\).

• Write all like terms together: \(24x + 20x\).
• Add their coefficients together: \(24 + 20\).

This results in the term \(44x\). Combining like terms simplifies the equation by reducing the number of terms, allowing us to proceed to the next steps of solving the rational equation more easily.

Isolating Variables

Isolating the variable is a fundamental step in solving equations. This process involves manipulating the equation to get the unknown variable on one side by itself. In our example, once we've combined like terms and obtained \(44x = 44\), we need to solve for \(x\). This is done by dividing both sides of the equation by the coefficient of \(x\), which is 44.

• Divide both sides by 44: \(\frac{44x}{44} = \frac{44}{44}\).
• Simplify to find \(x = 1\).

By isolating the variable, we determine the value of \(x\) that satisfies the original equation, completing the solution efficiently.

Algebraic Expressions

Understanding algebraic expressions is key to manipulating and solving equations. These expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division. In our example, the expression includes terms like \(\frac{3x}{5}\) and \(\frac{4x}{8}\). These are rational expressions because they have variables in the numerator or could form part of a fraction.

• Recognize and rewrite expressions when needed.
• Use operations to simplify and rearrange them.

Mastering how to handle algebraic expressions is essential in simplifying problems, combining like terms, and ultimately arriving at a solution for equations. Understanding the components and operators in expressions helps navigate through complex algebra problems efficiently.