Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. (x + 3)/6 = 3/8 + (x - 5)/4

Verified step by step guidance

1. Multiply every term by the least common denominator (LCD) to eliminate the fractions. The LCD of 6, 8, and 4 is 24.
2. Multiply each term by 24: 24 * ((x + 3)/6) = 24 * (3/8) + 24 * ((x - 5)/4).
3. Simplify each term: 4(x + 3) = 9 + 6(x - 5).
4. Distribute the numbers outside the parentheses: 4x + 12 = 9 + 6x - 30.
5. Combine like terms and solve for 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 finding the value of x that satisfies the equation.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When solving equations involving fractions, it is often necessary to find a common denominator to combine or eliminate the fractions effectively. This simplifies the equation and makes it easier to isolate the variable.

Cross-Multiplication

Cross-multiplication is a technique used to eliminate fractions in equations. When two fractions are set equal to each other, you can multiply the numerator of one fraction by the denominator of the other and vice versa. This method helps to simplify the equation into a linear form that can be solved for the variable.