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

Verified step by step guidance

1. Step 1: Identify the least common denominator (LCD) of all the denominators in the equation. The denominators are 5, 2, and 3. The LCD is 30.
2. Step 2: Multiply every term in the equation by the LCD (30) to eliminate the fractions. This means multiplying each term by 30 and simplifying.
3. Step 3: Distribute the multiplication across each term. For example, 30 * (3x/5) becomes (30 * 3x) / 5, which simplifies to 18x. Repeat this process for the other terms.
4. Step 4: After clearing the fractions, simplify the resulting equation by combining like terms and isolating the variable (x). This may involve distributing, combining terms, and moving terms across the equals sign.
5. Step 5: Solve for x by dividing or performing any necessary operations to isolate x completely. Check your solution by substituting it back into the original equation to ensure it satisfies the equation.

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.

Common Denominator

When dealing with fractions in linear equations, finding a common denominator is crucial for simplifying the equation. The common denominator allows you to eliminate the fractions by multiplying each term by this value, making it easier to solve for the variable. This step is particularly important when the equation contains multiple fractions with different denominators.

Isolating the Variable

Isolating the variable is a key step in solving linear 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 is essential for finding the solution to the equation.