Solve: 3x-1/5 + x+2/2 = -3/10. (Section 1.4, Example 4)

Verified step by step guidance

1. Identify the least common denominator (LCD) for the fractions involved. The denominators are 5, 2, and 10. The LCD is 10.
2. Multiply every term in the equation by the LCD (10) to eliminate the fractions: 10(3x - 1)/5 + 10(x + 2)/2 = 10(-3)/10.
3. Simplify each term: (10/5)(3x - 1) + (10/2)(x + 2) = -3.
4. Distribute the numbers in the simplified terms: 2(3x - 1) + 5(x + 2) = -3.
5. Expand and combine like terms: 6x - 2 + 5x + 10 = -3.

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 and solve these equations is fundamental in algebra, as it allows for finding the value of the variable that satisfies the equation.

Fraction Operations

Operations with fractions involve addition, subtraction, multiplication, and division of fractional numbers. When solving equations that include fractions, it is essential to find a common denominator to combine terms effectively. This concept is crucial for simplifying expressions and solving equations accurately, especially when fractions are involved.

Isolating the Variable

Isolating the variable is a key technique in solving equations, where the goal is to get the variable (e.g., x) on one side of the equation and all other terms on the opposite side. This often involves performing inverse operations, such as adding, subtracting, multiplying, or dividing both sides of the equation. Mastery of this concept is vital for finding the solution to the equation.