Problem 19 - Algebra and Trigonometry

Question

Contain linear equations with constants in denominators. Solve each equation. $$20-\frac{x}{3}=\frac{x}{2}$$

Step-by-Step Solution

Answer

The solution for 'x' is 24.

Step 1: Clear the denominators

To clear the denominators, multiply each term of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 3 and 2 is 6. Our equation becomes: \(6*(20) - 6*(\frac{x}{3}) = 6*(\frac{x}{2})\), which simplifies to \(120 - 2x = 3x\).

Step 2: Collect like terms

Move all terms containing 'x' to one side of the equation and the constants to the other: \(2x + 3x = 120\).

Step 3: Simplify and solve for 'x'

Combine like terms and solve for 'x': \(5x = 120\). Dividing both sides by 5, we get \(x = \frac{120}{5}\).

Step 4: Evaluate

Performing the division, the solution for 'x' is 24.

Key Concepts

Clearing Denominators in Equations

When solving linear equations containing fractions, it's often useful to eliminate the fractions as a first step. This is known as 'clearing the denominators.' To do this, find a number which each denominator can divide into, otherwise known as the 'least common multiple' (LCM). By multiplying every term in the equation by this LCM, the denominators can effectively be removed.

For instance, in the equation \(20-\frac{x}{3}=\frac{x}{2}\), we have denominators of 3 and 2. The LCM of 3 and 2 is 6. Multiplying each term by 6 yields \(6\cdot20 - 6\cdot\frac{x}{3} = 6\cdot\frac{x}{2}\), which simplifies to \(120 - 2x = 3x\). Now, we have an equation without fractions, which is generally easier to work with.

Finding Least Common Multiple

The least common multiple of two or more numbers is the smallest number that is a multiple of all the numbers in the set. Finding the LCM is crucial in the process of clearing denominators as it ensures that each term in the equation can be multiplied without introducing new fractions.

The LCM can be found using various methods, including listing multiples, prime factorization, or using the greatest common divisor (GCD). However, for small numbers, listing multiples is usually the quickest method. For our example, where the denominators are 3 and 2, the multiples of 3 (\(3, 6, 9, 12, ...\)) and the multiples of 2 (\(2, 4, 6, 8, ...\)) both include the number 6. Therefore, 6 is our LCM.

Combining Like Terms

Once you've cleared the denominators, the equation may have terms that can be combined. 'Combining like terms' refers to the process of simplifying an equation by adding or subtracting terms that have the same variable raised to the same power.

In the example \(120 - 2x = 3x\), the like terms \( -2x\) and \(3x\) each contain the variable 'x' to the first power. By moving them to the same side of the equation, we get \(2x + 3x = 120\), which simplifies to \(5x = 120\). Combining like terms is an essential step in solving linear equations, as it reduces the equation to its simplest form and makes it easier to solve.