Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 3x/5 - x = x/10 - 5/2

Verified step by step guidance

1. Identify the least common denominator (LCD) for all the fractions in the equation. The denominators are 5, 1, 10, and 2. The LCD is 10.
2. Multiply every term in the equation by the LCD (10) to eliminate the fractions: \(10 \times \frac{3x}{5} - 10 \times x = 10 \times \frac{x}{10} - 10 \times \frac{5}{2}\).
3. Simplify each term: \(6x - 10x = x - 25\).
4. Combine like terms on each side of the equation: \(-4x = x - 25\).
5. Isolate \(x\) by adding \(4x\) to both sides: \(0 = 5x - 25\).

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. Solving these equations involves isolating the variable, often requiring the application of algebraic operations such as addition, subtraction, multiplication, and division.

Fractions and Denominators

Fractions represent a part of a whole and consist of a numerator and a denominator. In equations involving fractions, it is crucial to understand how to manipulate these fractions, especially when they contain variables. Common techniques include finding a common denominator to combine fractions or multiplying through by the least common multiple to eliminate denominators.

Isolating the Variable

Isolating the variable is a fundamental technique in solving equations, where the goal is to express the variable on one side of the equation. This often involves rearranging the equation by performing inverse operations, such as adding or subtracting terms from both sides. Mastery of this concept is essential for finding the value of the variable in linear equations.