Problem 13 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{x-7}{8}=\frac{x+5}{6} $$

Step-by-Step Solution

Answer

Answer: x = -1.

Step 1: Determine the Least Common Denominator (LCD)

First, find the Least Common Denominator (LCD) of the two fractions. For the fractions \(\frac{x-7}{8}\) and \(\frac{x+5}{6}\), the denominators are 8 and 6. The LCD is the smallest multiple that these two numbers share. The LCD of 8 and 6 is 24.

Step 2: Multiply both sides by the LCD

To eliminate the fractions, multiply both sides of the equation by the LCD, which is 24: $$ 24\left(\frac{x-7}{8}\right)=24 \left(\frac{x+5}{6}\right) $$

Step 3: Simplify and Solve

Now, simplify by multiplying out the 24 on each side: $$ 24 \cdot \frac{x-7}{8} = 24 \cdot \frac{x+5}{6} $$ $$ 3(x-7) = 4(x+5) $$ Next, distribute the 3 and 4 on both sides: $$ 3x - 21 = 4x + 20 $$ Now, move all the x terms to one side and the constants to the other side to solve for x: $$ -21 + 20 = 4x - 3x $$ $$ -1 = x $$ So, x = -1.

Key Concepts

Least Common Denominator

When dealing with rational equations, one of the first steps is to find the Least Common Denominator (LCD). The LCD is essentially the smallest number that is a multiple of the denominators in the equations. This step is crucial because it allows you to eliminate the fractions, making the equation easier to solve.

For example, in the equation \(\frac{x-7}{8} = \frac{x+5}{6}\), the denominators are 8 and 6. We need to find the smallest multiple common to both 8 and 6, which is 24. This LCD will be used to multiply both sides of the equation, thereby eliminating the fractions. By considering the LCD, you set the stage for simpler algebraic manipulation. Always remember that the LCD is specific to the denominators in question, and finding it correctly is key to simplifying the problem efficiently.

Distributive Property

After determining the LCD and multiplying it across both sides of the equation, the next step involves using the Distributive Property. This property is a fundamental concept in algebra that allows you to remove parentheses by distributing a multiplication over addition or subtraction inside the parentheses.

In our exercise, after multiplying by the LCD, we have the equation \(3(x-7) = 4(x+5)\). Here, we apply the Distributive Property to open up the expressions:

• \(3(x-7)\) becomes \(3x - 21\)
• \(4(x+5)\) becomes \(4x + 20\)

By spreading the multiplication over the terms inside the parentheses, you simplify complex expressions into linear ones, making calculations straightforward and easier to follow.

Algebraic Fractions

In algebra, dealing with fractions can often complicate solving equations. Algebraic fractions are fractions that contain an algebraic expression in the numerator or the denominator (or both).

For the rational equation \(\frac{x-7}{8} = \frac{x+5}{6}\), each side consists of an algebraic fraction. The approach to solve these equations is to eliminate the fractions first. This is typically done by multiplying each term by the LCD, as mentioned before.

Converting algebraic fractions into equivalent whole-number terms ensures all terms are easier to work with. This strategy reduces potential errors and helps in achieving the solution quickly. Understanding how to handle these fractions is a core skill in algebra.

Equation Solving Steps

Solving equations systematically involves a series of logical steps. This structured approach not only helps in finding the correct solution but also ensures clarity of understanding.

For the given problem, here's a recap of the solving process:

• Step 1: Identify the denominators and find the Least Common Denominator (LCD). Here, the number is 24.
• Step 2: Multiply each term by the LCD to eliminate fractions, simplifying the equation.
• Step 3: Use the Distributive Property to expand any expressions. Then, align like terms (variables on one side, constants on the other).
• Step 4: Solve for the variable, ensuring to balance both sides till the variable is isolated.

These steps, if followed diligently, provide a clear path to arrive at the solution, allowing you to handle similar problems with confidence.