Problem 18 - College Algebra

Question

Exercises \(17-30\) contain equations with constants in denominators. Solve each equation. $$ \frac{x}{5}=\frac{x}{6}+1 $$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{x}{5}=\frac{x}{6}+1\) is \(x=30\).

Step 1: Clear the equation from fractions

To remove the fraction, it's ideal to find a common multiple of the denominators \(5\) and \(6\), which is \(30\). Then, both sides of the equation are multiplied by \(30\), to clear out the fractions. This gives the equation \(6x=5x+30\).

Step 2: Simplify the equation

Now, subtract \(5x\) from both sides of the equation to keep the \(x\) terms on one side of the equation. This will result in \(6x-5x=30\). Simplifying it gives \(x=30\).

Step 3: Verify the solution

Finally, verify this solution by substituting \(x=30\) into the original equation. If both sides of the equation are equal, then the solution is correct. After substituting, we get \(\frac{30}{5}=\frac{30}{6}+1\), simplifying it we get \(6=6\) which shows the solution is correct.

Key Concepts

Fractions

Fractions are essential in algebra, especially when dealing with equations that involve divided quantities. In this exercise, the fractions have constants as their denominators. For instance, with the equation \( \frac{x}{5} = \frac{x}{6} + 1 \), the numerators involve the variable \(x\), while 5 and 6 serve as the denominators.

Fractions indicate parts of a whole. In algebra, they can often make equations a bit challenging to solve directly. This challenge arises because each fraction might represent a different proportional relationship in the equation.

• Understanding Fractions: The numerators of the fractions contain the variable \(x\), indicating parts of certain wholes divided by the denominators.
• Denominator Role: Here, the denominators 5 and 6 indicate how many equal parts the whole is divided into.

Understanding these components helps us to manipulate and simplify the equations for solving.

Common Multiple

Finding a common multiple helps in eliminating fractions from an equation, simplifying it to a more solvable form. In this specific solution, the denominators are 5 and 6, and the least common multiple (LCM) of these two integers is 30.

By multiplying every term in the equation by 30, we effectively remove the fraction component, converting it into a linear equation that's easier to handle.

• Why Multiply by LCM: Multiplying by the LCM clears the fractions because you're essentially 'unwrapping' the parts of each fraction until they become whole numbers.
• Simplification: After multiplying, the equation \(6x = 5x + 30\) is the result, which is now much simpler to work with than dealing with the original fractional form.

Using a common multiple is a powerful tool in solving equations involving fractions, making the problem more straightforward.

Verifying Solutions

Once an algebraic equation has been solved, it's crucial to verify that the proposed solution actually satisfies the original equation. Thus, after finding \(x = 30\), we substitute it back into the initial equation to confirm its validity.

Verification is a step to ensure no errors were made during simplification and solving. If both sides of the equation are equal after substituting the solution, it means the solution holds true. In our exercise:

• Substitution Method: Replace \(x\) with 30, turning the equation into \(\frac{30}{5} = \frac{30}{6} + 1\).
• Simplifying Both Sides: Evaluate each side to get \(6 = 6\), confirming the solution.

Verification acts as the proof step, reassuring that the method and solution are accurate, thus affirming the result is indeed correct.