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

Verified step by step guidance

1. Identify the least common denominator (LCD) of the fractions involved. Here, the denominators are 3 and 2, so the LCD is 6.
2. Multiply every term in the equation by the LCD (6) to eliminate the fractions: \(6 \cdot \frac{x}{3} = 6 \cdot \frac{x}{2} - 6 \cdot 2\).
3. Simplify each term: \(2x = 3x - 12\).
4. Rearrange the equation to isolate terms involving \(x\) on one side: \(2x - 3x = -12\).
5. Combine like terms to solve for \(x\): \(-x = -12\).

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. The goal is to find the value of the variable (x) that makes the equation true. Understanding how to manipulate these equations is essential for solving them.

Solving for a Variable

Solving for a variable involves isolating the variable on one side of the equation to determine its value. This often requires performing inverse operations, such as addition, subtraction, multiplication, or division, to both sides of the equation. In the context of the given equation, this means eliminating the fractions and simplifying to find the value of x.

Fractions and Common Denominators

Fractions represent parts of a whole and can complicate linear equations. To solve equations with fractions, it is often necessary to find a common denominator, which allows for the elimination of the fractions by multiplying through by that denominator. This step simplifies the equation, making it easier to isolate the variable and solve.