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

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1. Combine all terms involving x on one side of the equation. Start by subtracting \( \frac{x}{10} \) from both sides: \( \frac{3x}{5} - x - \frac{x}{10} = -\frac{5}{2} \).
2. Find a common denominator for the fractions involving x on the left-hand side. The least common denominator (LCD) of 5, 1, and 10 is 10. Rewrite each term with this denominator: \( \frac{6x}{10} - \frac{10x}{10} - \frac{x}{10} = -\frac{5}{2} \).
3. Simplify the terms on the left-hand side by combining the fractions: \( \frac{6x - 10x - x}{10} = -\frac{5}{2} \), which simplifies further to \( \frac{-5x}{10} = -\frac{5}{2} \).
4. Eliminate the fractions by multiplying through by the least common denominator of the denominators on both sides. The LCD of 10 and 2 is 10. Multiply through by 10: \( 10 \cdot \frac{-5x}{10} = 10 \cdot -\frac{5}{2} \), resulting in \( -5x = -25 \).
5. Solve for x by dividing both sides of the equation by -5: \( x = \frac{-25}{-5} \). Simplify the result to find the value of x.

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 is essential for solving them, as it involves isolating the variable on one side of the equation.

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 find a common denominator to simplify the equation. This process often involves multiplying through by the least common multiple of the denominators to eliminate fractions, making the equation easier to solve.

Isolating the Variable

Isolating the variable is a fundamental technique in solving equations, where the goal is to get the variable (e.g., x) alone on one side of the equation. This often involves performing inverse operations, such as addition, subtraction, multiplication, or division, to both sides of the equation. Mastery of this concept allows for the effective simplification and solution of linear equations.