Problem 23 - Algebra and Trigonometry

Question

Contain linear equations with constants in denominators. Solve each equation. $$\frac{3 x}{5}-x=\frac{x}{10}-\frac{5}{2}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Simplify the equation

Start by simplifying the equation. This involves multiplying each term by the least common multiple (LCM) of the denominators to eliminate the fractions. Assuming we have:\[\frac{3 x}{5}-x=\frac{x}{10}-\frac{5}{2}\] The least common multiple for 5, 5 (from -x which can be assumed to be over 1, thus 5x/5 = x), 10 and 2 is 10. So we multiply each term by 10 to get:\[ 10 * \frac{3 x}{5} - 10 * x = 10 * \frac{x}{10} - 10 * \frac{5}{2}\]. This simplifies to:\[6x - 10x = x - 25\].

Step 2: Combine like terms

Next, combine like terms on both sides of the equation to simplify further:\[-4x = x - 25\].

Step 3: Isolate the variable

The main goal is to isolate x on one side of the equation. In order to accomplish that, move x from the right side of the equation to the left side by subtracting x from both sides:\[-4x - x = -25\].Finally, this gives:\[-5x = -25\].

Step 4: Solve for x

Lastly, solve for x by dividing both sides of the equation by -5:\[x = 5\].

Key Concepts

Understanding Constants in Denominators

When dealing with linear equations, constants in denominators can make things complex. This is because they introduce fractions, which can be cumbersome to handle directly. In our equation, we have terms like \( \frac{3x}{5} \) and \( \frac{x}{10} \). These fractions result from the constants (5 and 10) in the denominators. The key is to remove these fractions to simplify the equation. That way, calculations can be much more straightforward. We'll explore this further when we learn about eliminating fractions.

Finding the Least Common Multiple (LCM)

The least common multiple (LCM) is essential when you want to clear out fractions. Here, we look at the denominators in the equation: 5, 1 (from the term \(-x\)), 10, and 2. To find the LCM, determine the smallest number that each denominator can divide evenly into. For 5, 10, and 2, the LCM is 10. When we multiply each term by this LCM, the fractions disappear, making the equation simpler to solve. By multiplying through by 10, every term aligns to whole numbers, easing our path to solution.

Eliminating Fractions from Equations

Eliminating fractions involves multiplying each term of the equation by the LCM, which you've determined in the previous step. This process ensures that all fractions have been cleared from the equation. So, when our equation \( \frac{3x}{5} - x = \frac{x}{10} - \frac{5}{2} \) is multiplied by 10, the fractions become whole numbers: \( 6x - 10x = x - 25 \). Now, we have an equation with only whole numbers, making the steps towards the solution much more straightforward.

Combining Like Terms

After eliminating fractions, the next step is to combine like terms, which are the terms that have the same variables raised to the same power. This process simplifies the equation even further. In our simplified equation, \( 6x - 10x = x - 25 \), the like terms \( 6x \) and \( -10x \) are combined to give \( -4x \). Similarly, the \( x \) on the right side stands alone, and we further simplify by manipulating these like terms to isolate the variable, solving the equation bit by bit.