Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 3x/5 = 2x/3 + 1

Verified step by step guidance

1. Multiply every term by the least common denominator (LCD) of the fractions to eliminate the denominators. The LCD of 5 and 3 is 15.
2. Distribute the LCD to each term: \(15 \cdot \frac{3x}{5} = 15 \cdot \frac{2x}{3} + 15 \cdot 1\).
3. Simplify each term: \(3x \cdot 3 = 2x \cdot 5 + 15\).
4. This simplifies to: \(9x = 10x + 15\).
5. Rearrange the equation to isolate \(x\) on one side: Subtract \(10x\) from both sides to get \(9x - 10x = 15\).

Key Concepts

Linear Equations

Linear equations are mathematical statements that express the equality of two linear expressions. They typically take the form ax + b = cx + d, where a, b, c, and d are constants. Understanding how to manipulate these equations is essential for finding the value of the variable, often involving operations like addition, subtraction, multiplication, and division.

Solving for a Variable

Solving for a variable involves isolating the variable on one side of the equation to determine its value. This process often requires combining like terms, eliminating fractions, and applying inverse operations. In the context of the given equation, it is crucial to find a common denominator to simplify the fractions before isolating x.

Fractions and Common Denominators

Fractions represent parts of a whole and can complicate equations. To solve equations involving fractions, it is often necessary to find a common denominator, which allows for the simplification of the equation by eliminating the fractions. This step is vital in the provided equation to facilitate easier manipulation and ultimately solve for the variable.