Problem 26 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \frac{x}{2}+\frac{x}{3}=10 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 12 \).

Step 1: Identify a Common Denominator

To combine the fractions \( \frac{x}{2} \) and \( \frac{x}{3} \), we need a common denominator. The least common multiple of 2 and 3 is 6. Therefore, we adjust each term to have this denominator.

Step 2: Rewrite the Equation with a Common Denominator

Multiply \( \frac{x}{2} \) by \( \frac{3}{3} \) to get \( \frac{3x}{6} \), and \( \frac{x}{3} \) by \( \frac{2}{2} \) to get \( \frac{2x}{6} \). So the equation becomes \( \frac{3x}{6} + \frac{2x}{6} = 10 \).

Step 3: Combine the Fractions

Combine the fractions since they have the same denominator: \( \frac{3x + 2x}{6} = 10 \). This simplifies to \( \frac{5x}{6} = 10 \).

Step 4: Solve for x by Eliminating the Fraction

To solve \( \frac{5x}{6} = 10 \), multiply both sides of the equation by 6 to get rid of the fraction: \( 5x = 60 \).

Step 5: Isolate x

Divide both sides by 5 to solve for x: \( x = \frac{60}{5} \). This simplifies to \( x = 12 \).

Key Concepts

Fractions in Equations

When dealing with algebraic equations, fractions may appear at times. Working with fractions in such equations can seem daunting but they follow the same arithmetic rules as normal fractions.

• The main challenge is to handle them carefully to carry out operations like addition, subtraction, and simplification.
• This can involve finding common denominators and rewriting fractions to make the equation easier to solve.
• It's crucial to remember that whatever operation is applied to one side of the equation must also be applied to the other to maintain balance.

Understanding fractions in equations is all about seeing how they form a part of the equation's terms. Once you manipulate them correctly, finding the variable becomes much more manageable.

Step-by-Step Solving

Solving equations step-by-step is a method that reveals the thought process in a logical and orderly manner. It ensures each part of the problem is tackled in stages, which reduces errors and builds confidence.

• Start by identifying what needs to be done at each stage of the problem.
• In our example, begin by looking for a way to combine the fractions on one side.
• Then move on to eliminate fractions by finding a common denominator. This helps in simplifying the equation.
• Ultimately, progress to isolating the variable using algebraic operations until it's by itself on one side of the equation.

Each step should be approached patiently and with clarity. Mark progress by simplifying before fully solving the equation. This way, complexities are gradually stripped away, making the solution process clearer.

Common Denominators

Finding common denominators is a key concept when dealing with fractions in equations. It is essential for adding or subtracting fractions since it ensures each fraction has the same base.

• To find the common denominator, identify the least common multiple (LCM) of the denominators in the fractions.
• In the example equation, the LCM of 2 and 3 is 6.
• With this LCM, rewrite each fraction so they can be combined into a single term.
• This harmonizes the equation, allowing for straightforward simplification.

Once fractions share a common denominator, you can add them by simply adding their numerators. This simplification leads to a more manageable equation, facilitating an easier path to the solution.