91Ó°ÊÓ

Fractions are mathematical expressions representing a part of a whole. They consist of a numerator—the top number—and a denominator—the bottom number. In the context of linear equations, fractions often need to be simplified or eliminated to solve for a variable.

For example, in the exercise equation \( \frac{3a}{7} - 1 = \frac{a}{3} \), you see fractions that need to be dealt with. When solving linear equations involving fractions, one common method is to find a way to remove the fractions by converting them to whole numbers temporarily.

This method calls for identifying a common denominator, which is crucial for clearing the equation of fractions, allowing simplification and solution.

A common denominator is a shared multiple of the denominators in a set of fractions. Finding the common denominator is key to eliminating fractions within an equation effectively. In the equation from our exercise, \( \frac{3a}{7} - 1 = \frac{a}{3} \), the denominators are 7 and 3.

• To eliminate these fractions, find the least common multiple (LCM) of the denominators.
• The LCM of 7 and 3 is 21.
• Multiply each term of the equation by this common denominator.
• This results in the new equation: \( 9a - 21 = 7a \).

From this step, you can see how we converted the fractions into whole numbers, making it easier to proceed with solving for the variable.

Solving for a variable means isolating the variable on one side of the equation to find its value. After the fractions are eliminated in the exercise, you are left with the equation \( 9a - 21 = 7a \).

• First, simplify the equation by getting all terms containing \( a \) on one side and constant numbers on the other.
• Subtract \( 7a \) from both sides, resulting in \( 2a - 21 = 0 \).
• Next, isolate \( a \) by adding 21 to both sides, giving you \( 2a = 21 \).
• Finally, divide each side by 2 to solve for \( a \): \( a = \frac{21}{2} \).

By following these steps, you can determine that \( a \) equals \( \frac{21}{2} \) or 10.5, which completes the solution to the original equation.