Problem 1 - Elementary Algebra

Question

Solve the following rational equations. \(\frac{2 x}{5}=\frac{x-14}{6}\)

Step-by-Step Solution

Answer

Answer: The value of x that satisfies the given rational equation is \(x = -10\).

Step 1: Identify the equation

We are given the rational equation: \(\frac{2x}{5} = \frac{x-14}{6}\)

Step 2: Clear the denominators

In order to clear the denominators, we need to find the least common multiple (LCM) of the two denominators 5 and 6. In this case, the LCM of 5 and 6 is 30. Now, we'll multiply both sides of the equation by the LCM: \(30 \cdot \frac{2x}{5} = 30 \cdot \frac{x-14}{6}\)

Step 3: Simplify the equation

After multiplying both sides by 30, we can simplify the equation: \((30/5) \cdot 2x = (30/6) \cdot (x-14)\) \(12x = 5(x-14)\)

Step 4: Distribute the values

Now we will distribute the 5 to both terms inside the parenthesis: \(12x = 5x - 70\)

Step 5: Isolate x

To isolate x, we will subtract the 5x from both sides: \(7x = -70\) Now, divide both sides by 7 to get the final x-value: \(x = -10\) So, the solution to the rational equation \(\frac{2x}{5} = \frac{x-14}{6}\) is \(x = -10\).

Key Concepts

Understanding Least Common Multiple (LCM)

When solving rational equations, one key step is to clear the denominators. To do this, we find the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide without a remainder. For example, in the equation \(\frac{2x}{5} = \frac{x-14}{6}\), the denominators are 5 and 6.
To find the LCM of 5 and 6:

• List the multiples of each number:
  
  • Multiples of 5: 5, 10, 15, 20, 25, 30, ...
  • Multiples of 6: 6, 12, 18, 24, 30, ...
• The smallest common multiple is 30.

By multiplying both sides of the equation by 30, we eliminate the fractions and make the equation easier to solve.

Isolating the Variable

Once the equation is free of fractions, the next step is to isolate the variable, which means getting the variable by itself on one side of the equation. In the example \(12x = 5(x-14)\), we first distribute the 5 to the terms inside the parenthesis.
This gives us:

• \(12x = 5x - 70\)

To isolate \(x\), we subtract \(5x\) from both sides:

• \(12x - 5x = -70\)
• \(7x = -70\)

Finally, divide both sides by 7 to solve for \(x\):

• \(x = -10\)

This step-by-step approach allows us to find the value of the variable.

Simplifying Equations

Simplifying equations involves making them as straightforward as possible, often by combining like terms and eliminating complex fractions. After using the LCM to clear fractions, we'll apply simplification techniques.
In our example, we simplified the equation:

• After multiplying both sides by 30, we ended up with \(12x = 5(x-14)\).
• Distribution led to \(12x = 5x - 70\).

Next, combine like terms:

• Subtract \(5x\) from \(12x\) to simplify the left side.
• Combine to get \(7x = -70\).

Divide by 7 to find \(x = -10\).
Simplification helps to solve equations more easily by reducing clutter and focusing only on the necessary steps.