Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. x/5 - 1/2 = x/6

Verified step by step guidance

1. Identify the least common denominator (LCD) for the fractions involved. In this case, the denominators are 5, 2, and 6. The LCD is 30.
2. Multiply every term in the equation by the LCD (30) to eliminate the fractions: \(30 \cdot \frac{x}{5} - 30 \cdot \frac{1}{2} = 30 \cdot \frac{x}{6}\).
3. Simplify each term: \(6x - 15 = 5x\).
4. Rearrange the equation to isolate the variable \(x\) on one side: \(6x - 5x = 15\).
5. Simplify the equation to solve for \(x\): \(x = 15\).

Key Concepts

Linear Equations

Linear equations are mathematical statements that express the equality of two linear expressions. They typically take the form ax + b = c, where a, b, and c are constants, and x is the variable. Understanding how to manipulate these equations to isolate the variable is crucial for solving them.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When solving equations involving fractions, finding a common denominator allows you to eliminate the fractions by multiplying through the equation, simplifying the process of isolating the variable.

Isolating the Variable

Isolating the variable involves rearranging an equation to get the variable on one side and all other terms on the opposite side. This process often includes adding, subtracting, multiplying, or dividing both sides of the equation by the same number, ultimately leading to a solution for the variable.