Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. x/5 - 1/2 = x/6

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Step 1: Identify the least common denominator (LCD) of all the denominators in the equation. The denominators are 5, 2, and 6. The LCD is 30.

Step 2: Multiply every term in the equation by the LCD (30) to eliminate the fractions. This gives: 30 * (x/5) - 30 * (1/2) = 30 * (x/6).

Step 3: Simplify each term after multiplying by the LCD. For example, 30 * (x/5) becomes 6x, 30 * (1/2) becomes 15, and 30 * (x/6) becomes 5x. The equation now reads: 6x - 15 = 5x.

Step 4: Isolate the variable x by subtracting 5x from both sides of the equation. This simplifies to: 6x - 5x - 15 = 0, which further simplifies to: x - 15 = 0.

Step 5: Solve for x by adding 15 to both sides of the equation. This gives: x = 15.

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주요 개념

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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 with fractions, it is often necessary to find a common denominator to eliminate the fractions, making the equation easier to solve. This involves multiplying each term by the least common multiple of the denominators.

Isolating the Variable

Isolating the variable involves rearranging the equation to get the variable (in this case, x) on one side and the constants on the other. This process typically includes adding, subtracting, multiplying, or dividing both sides of the equation by the same number. Mastering this technique is crucial for solving linear equations effectively.