Problem 23 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{2 k+7}{3 k}=\frac{5}{4} $$

Step-by-Step Solution

Answer

Answer: The value of k in the given rational equation is 4.

Step 1: Find the LCM of the denominators

The denominators are 3k and 4. To find the LCM, we need to find the smallest multiple that is divisible by both 3k and 4. The LCM is 12k.

Step 2: Multiply both sides of the equation by the LCM

Now, we will multiply both sides of the equation by 12k to eliminate the denominators: $$ 12k \cdot \frac{2k + 7}{3k} = 12k \cdot \frac{5}{4} $$

Step 3: Simplify the equation

The multiplication by 12k will cancel out the denominators on both sides. We will now simplify the equation: $$ (12k \cdot \frac{2k + 7}{3k}) = \frac{(12k)(2k + 7)}{3k} =\frac{(4)(2k+7)}{1} = 8k+28 $$ $$ (12k \cdot \frac{5}{4}) = \frac{(12k)(5)}{4} = \frac{(3k)(5)}{1} = 15k $$ So, our simplified equation is: $$ 8k + 28 = 15k $$

Step 4: Solve for k

Now, we will isolate k to find its value: $$ 28 = 15k - 8k $$ $$ 28 = 7k $$ $$ k = \frac{28}{7} $$ $$ k = 4 $$ Thus, the solution for the rational equation is k = 4.

Key Concepts

Least Common Multiple

When working with rational equations, finding the Least Common Multiple (LCM) of the denominators is essential. It allows us to eliminate fractions and simplify the problem. A rational equation can have complex fractional expressions with different denominators, which complicates calculations. To determine the LCM, follow these steps:

• Identify each denominator. In the equation \( \frac{2k+7}{3k} = \frac{5}{4} \), the denominators are 3k and 4.
• Find the smallest number that each denominator can divide. For 3k and 4, the smallest common multiple is 12k because:
  • 3 divides into 12, and k divides into k.
  • 4 divides into 12.
  Therefore, 12k is the smallest number satisfying both conditions.

Finding the LCM simplifies the equation by allowing us to clear fractional expressions through multiplication. This step sets the equation up for straightforward operations, bringing clarity and ease to the solving process.

Solving for Variables

Once the LCM has cleared the rational expressions in an equation, the next step is to solve for the variable. To solve the equation \(8k + 28 = 15k\), you need to perform operations that isolate the variable \(k\) on one side of the equation.Here's how to solve for the variable:

• Start with the equation in its simplified form, \(8k + 28 = 15k\).
• Subtract \(8k\) from both sides, resulting in \(28 = 15k - 8k\).
• Simplify the right side: \(28 = 7k\).
• Divide both sides by 7 to isolate \(k\):
• \(k = \frac{28}{7}\).
• Simplify the fraction to get \(k = 4\).

By practicing these steps, students can become adept at isolating variables, which is crucial for solving complex equations. Understanding this concept also helps in algebraic manipulation, an essential tool in advanced mathematics.

Simplifying Equations

Simplifying equations is a critical aspect in solving rational equations. It involves reducing expressions to their simplest form, thereby making calculations easier and faster.Consider the equation \(12k \cdot \frac{2k + 7}{3k} = 12k \cdot \frac{5}{4}\):

• Multiply both sides by the LCM, 12k, to eliminate denominators.
• The process simplifies the left side:
  • Multiply: \( (12k \cdot \frac{2k + 7}{3k}) = \frac{24k^2 + 84k}{3k} \).
  • Reduce: \( \frac{24k^2 + 84k}{3k} = 8k + 28 \).
  Notice how dividing by 3k removes the fractional complexity.
• For the right side, simplify:
  • Multiply: \( (12k \cdot \frac{5}{4}) = \frac{60k}{4} \).
  • Reduce: \( \frac{60k}{4} = 15k \).
  This results in no fractions remaining.

Simplification is not just about arithmetic; it involves understanding how operations affect equations and how to maintain equality. Mastery of this skill is beneficial for both simple and complex problems, enhancing overall mathematical fluency.