Problem 22 - College Algebra

Question

Exercises \(17-30\) contain equations with constants in denominators. Solve each equation. $$ \frac{x}{2}=\frac{3 x}{4}+5 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -20\).

Step 1: Identify the LCM and Multiply Every Term

The LCM of 2 and 4 is 4. So, multiply every term by 4 to get rid of the fractions. The equation will then be \(4*\frac{x}{2} = 4*\frac{3x}{4} + 4*5\). This simplifies to \(2x = 3x + 20\).

Step 2: Move Variable to One Side of the Equation

Create a new equation with \(x\) on one side by subtracting \(3x\) from both sides, resulting in \(-x = 20\).

Step 3: Solve for x

Finally, multiply both sides of the equation by -1 to get x on its own, resulting in \(x = -20\).

Key Concepts

Least Common Multiple

When dealing with equations containing fractions, it becomes crucial to simplify them as much as possible. One effective technique is using the Least Common Multiple (LCM). The LCM of two numbers is the smallest number that both can divide into without leaving a remainder. For instance, in the equation \( \frac{x}{2} = \frac{3x}{4} + 5 \), we identify the denominators, 2 and 4. Finding the LCM involves listing the multiples of both numbers until we find the smallest common one.

• The multiples of 2 are: 2, 4, 6, 8, etc.
• The multiples of 4 are: 4, 8, 12, 16, etc.

The LCM of 2 and 4 is 4 because it is the smallest number in both lists. Multiplying each term of the equation by this LCM will help us eliminate the fractions and simplify our calculations.

Variable Isolation

After clearing fractions in an equation, the next step is variable isolation. This process involves getting the variable on one side of the equation by itself. From the equation \(2x = 3x + 20\), we want all terms with the variable \(x\) on just one side. To do this, we perform operations that move terms around:

• Subtract \(3x\) from both sides, which gives us \(-x = 20\).

By moving the variable terms to one side, we isolate \(x\). The key is maintaining balance. Whatever operation you do on one side, you must do the same on the other. This ensures that the equation remains valid as you step towards solving for \(x\).

Fraction Elimination

Eliminating fractions from equations can make solving them much simpler. This process converts the equations into more comfortable forms to handle. Using our example, the original equation \(\frac{x}{2} = \frac{3x}{4} + 5\) contained fractions. By multiplying each term by the LCM of the denominators (which we've decided was 4), the equation becomes:

• \(4 \cdot \frac{x}{2} = 4 \cdot \frac{3x}{4} + 4 \cdot 5\)

This simplifies to \(2x = 3x + 20\).
By clearing the fractions early on, we execute "Fraction Elimination," making the equations simpler to manipulate, paving the way for straightforward algebraic operations that lead to an answer. Hence, always address the fractions first for smooth sailing through the rest of the solution!