Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 5 + (x - 2)/3 = (x + 3)/8

Verified step by step guidance

1. Step 1: Identify the least common denominator (LCD) of the fractions in the equation. The denominators are 3 and 8, so the LCD is 24.
2. Step 2: Multiply every term in the equation by the LCD (24) to eliminate the fractions. This gives: 24 * 5 + 24 * (x - 2)/3 = 24 * (x + 3)/8.
3. Step 3: Simplify each term after multiplying by the LCD. For example, 24 * (x - 2)/3 becomes 8 * (x - 2), and 24 * (x + 3)/8 becomes 3 * (x + 3). Rewrite the equation as: 120 + 8(x - 2) = 3(x + 3).
4. Step 4: Distribute the constants across the parentheses. For example, 8(x - 2) becomes 8x - 16, and 3(x + 3) becomes 3x + 9. Rewrite the equation as: 120 + 8x - 16 = 3x + 9.
5. Step 5: Combine like terms and isolate the variable x. Combine constants on one side and x terms on the other side to 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 = cx + d, where a, b, c, and d are constants. Understanding how to manipulate these equations is crucial for finding the value of the variable, x, that satisfies the equation.

Solving for x

Solving for x involves isolating the variable on one side of the equation. This process often includes combining like terms, eliminating fractions, and applying inverse operations. Mastery of these techniques allows students to find the specific value of x that makes the equation true.

Fractions and Common Denominators

When dealing with equations that contain fractions, finding a common denominator is essential for simplifying the equation. This involves identifying a number that can be used to eliminate the denominators, making it easier to solve for the variable. Understanding how to manipulate fractions is key to successfully solving these types of linear equations.