Problem 37 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation and check your proposed solution. Begin your work by rewriting each equation without fractions. \(\frac{x}{3}+\frac{x}{2}=\frac{5}{6}\)

Step-by-Step Solution

Answer

Solving the given equation results in \(x = 1\), which has been verified by back substitution into the initial equation.

Step 1: Remove fractions from the equation

Multiply each term by 6, which is the least common multiple of the denominators 3, 2, and 6. This yields \(2x + 3x = 5\).

Step 2: Simplify the equation

Combine like terms on the left side to get \(5x = 5\).

Step 3: Solve for the variable \(x\)

Divide both sides by 5 to isolate \(x\). This gives us \(x = 1\).

Step 4: Check the solution

Substitute \(x = 1\) back into the original equation and check if the left hand side equals to the right hand side. \(\frac{1}{3}+\frac{1}{2}=\frac{2}{3}+\frac{3}{6}=\frac{5}{6}\). So, \(x = 1\) is indeed a solution.

Key Concepts

Algebraic Fractions

Algebraic fractions are fractions that contain variables in their numerator, denominator, or both. They operate under the same principles as numerical fractions, but they often require additional steps to simplify or solve due to the presence of variables.

When solving equations with algebraic fractions, like the exercise \(\frac{x}{3} + \frac{x}{2} = \frac{5}{6}\), the first step usually involves finding a common denominator to combine terms or to eliminate the fractions altogether. This is helpful because working with whole numbers can make the equation more manageable and less intimidating for many students.

In the given problem, the algebraic fractions all had different denominators (3, 2, and 6). By finding a least common multiple, we could rewrite the equation without these fractions, which then made it easier to proceed to simplify and solve for the variable.

Least Common Multiple

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers. More simply, it's the smallest number that all the numbers can go into evenly. It's a concept that often comes into play when dealing with algebraic fractions, especially when you need to combine them or eliminate them from an equation.

For instance, during the solution of our textbook exercise, the LCM of 3, 2, and 6 is 6. Finding the LCM allowed us to multiply each term of the equation by 6, which removed the fractions and gave us a simpler equation with integers. This process is crucial because it serves as a foundational step in many algebra problems, and being proficient at finding the LCM can greatly simplify challenging algebraic tasks.

Simplify Equations

To simplify an equation means to make it easier to solve by combining like terms, reducing fractions, or performing other algebraic operations that do not change the solutions of the original equation. Simplifying can also involve removing parentheses and clearing fractions, as we did in our exercise.

After finding the LCM and eliminating the fractions, we were left with the equation \(2x + 3x = 5\). The next step was to combine the like terms on the left side, which are the terms with the variable \(x\). This gave us \(5x = 5\), a much simplified equation. By doing so, we not only make the equation look 'cleaner', but we also set it up in a way that makes the variable \(x\) easier to isolate and solve for.

Equation Solutions

The solution to an equation is the value of the variable that makes the equation true. In other words, when you substitute the solution into the equation, the left-hand side should equal the right-hand side, verifying that your solution is correct.

Once we simplified our equation to \(5x = 5\), we solved for \(x\) by dividing both sides by 5, which gave us \(x = 1\). This is the proposed solution. However, it is equally important to always check this solution by substituting it back into the original equation. In our example, substituting \(x = 1\) into \(\frac{x}{3} + \frac{x}{2}\) and verifying that the sum is equal to \(\frac{5}{6}\) confirms that the solution is indeed correct.

This step of verification not only ensures that our answer is accurate but also reinforces the understanding of how equations work. By checking your work, you're practicing good mathematical habits and building confidence in your problem-solving abilities.