Problem 12 - Precalculus

Question

Solve each equation. $$\frac{x}{3}+\frac{2 x}{5}=\frac{-11}{5}$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -3 \).

Step 1: Identify the goal and the equation

We aim to solve the equation \( \frac{x}{3} + \frac{2x}{5} = \frac{-11}{5} \). This means finding the value of \( x \) that makes the equation true.

Step 2: Find a common denominator

Identify a common denominator for the fractions. The common denominator between 3 and 5 is 15. Rewrite each term with this common denominator.

Step 3: Rewrite the fractions with the common denominator

Convert \( \frac{x}{3} \) to \( \frac{5x}{15} \) and \( \frac{2x}{5} \) to \( \frac{6x}{15} \). The equation now becomes \( \frac{5x}{15} + \frac{6x}{15} = \frac{-33}{15} \).

Step 4: Combine like terms

Add the fractions on the left-hand side since they have the same denominator: \( \frac{5x + 6x}{15} = \frac{11x}{15} \). The equation now is \( \frac{11x}{15} = \frac{-33}{15} \).

Step 5: Clear the fractions

Multiply both sides of the equation by 15 to eliminate the denominators: \( 11x = -33 \).

Step 6: Solve for x

Divide both sides by 11 to isolate \( x \): \( x = -3 \).

Key Concepts

Common Denominator

When dealing with fractions in an equation, finding a common denominator is crucial. This is because adding or subtracting fractions with different denominators is not straightforward. To add fractions readily:

• Identify the denominators of the fractions involved.
• Find the smallest multiple both denominators share, which is known as the least common denominator (LCD).

In this problem, the denominators are 3 and 5. The smallest number that both 3 and 5 divide into evenly is 15, which is our common denominator. Using a common denominator allows us to rewrite fractions so they are compatible for addition or subtraction. This step prepares us to combine fractions seamlessly.

Fractions

Fractions consist of a numerator (the top number) and a denominator (the bottom number). They represent parts of a whole. In equations, they often require special attention to be manipulated correctly.When rewriting fractions with a new common denominator, you must:

• Multiply both the numerator and the denominator by the same number so that the new denominator matches the common denominator.

For instance, to modify \( \frac{x}{3} \) with the common denominator of 15:

• Multiply numerator (x) and denominator (3) by 5, getting \( \frac{5x}{15} \).

This process ensures the value of the fraction remains the same while the denominator matches others in the equation. Proper handling of fractions allows for easier manipulation and solves the equation effectively.

Combining Like Terms

Combining like terms is a fundamental step in simplifying equations. It involves merging terms with the same variables and powers.In this context:

• After standardizing the fractions' denominators, fractions like \( \frac{5x}{15} \) and \( \frac{6x}{15} \) become simpler to add.

You can then combine them by adding the numerators, as they share a denominator:

• The sum is \( \frac{5x + 6x}{15} = \frac{11x}{15} \).

Combining like terms makes the equation less complex and further sets the stage for solving the variable. This process highlights the importance of simplification in algebraic operations.

Isolation of Variable

The ultimate goal in solving a linear equation is to isolate the variable, making it the sole entity on one side of the equation.Here's how you achieve this:

• First, eliminate fractions by multiplying both sides of the equation by the common denominator.

This transforms \( \frac{11x}{15} = \frac{-33}{15} \) to \( 11x = -33 \).

• Next, divide each side by the coefficient of the variable to isolate it. For this equation, divide by 11.

This yields:

• \( x = -3 \)

The isolation of the variable simplifies the equation to its essential form, providing the solution directly. This approach ensures clarity and precision, enabling precise solutions in algebra.