Problem 60 - Algebra for College Students

Question

Solve each equation, and check the solution. \(\frac{x}{5}-\frac{x}{4}=1\)

Step-by-Step Solution

Answer

The solution is x = -20.

Step 1: Find a Common Denominator

Identify a common denominator for the fractions on the left side of the equation. The fractions are \(\frac{x}{5}\) and \(\frac{x}{4}\). The least common denominator (LCD) of 5 and 4 is 20.

Step 2: Rewrite Fractions with the Common Denominator

Rewrite each fraction using the common denominator of 20. This gives us: \(\frac{4x}{20} - \frac{5x}{20} = 1\).

Step 3: Simplify the Equation

Combine the fractions on the left side: \(\frac{4x - 5x}{20} = 1\). Simplifying this gives us: \(\frac{-x}{20} = 1\).

Step 4: Solve for x

To isolate \x\, multiply both sides of the equation by -20: \(-x = 20\). Hence, \x = -20\.

Step 5: Check the Solution

Substitute \x = -20\ back into the original equation to verify: \(\frac{-20}{5} - \frac{-20}{4} = 1\). This simplifies to \(-4 + 5 = 1\), which is true. Hence, the solution is verified.

Key Concepts

Common Denominator

When solving equations involving fractions, finding a common denominator is crucial. A common denominator is a shared multiple of the denominators of the fractions involved in the equation. This step simplifies the process by giving us a single denominator across all fractions, allowing us to easily combine them.
For example, in the equation \(\frac{x}{5} - \frac{x}{4} = 1\), the denominators are 5 and 4. To find a common denominator, we need the least common multiple (LCM) of these numbers, which is 20. This common denominator will let us rewrite each fraction so they have the same base, making it straightforward to combine them later on.

Simplifying Fractions

Once a common denominator is found, the next step is to rewrite each fraction to have this common denominator. This makes it easier to perform operations like addition or subtraction. To adjust the fractions correctly, multiply both the numerator and the denominator of each fraction by the necessary number to achieve the common denominator.
In our example \(\frac{x}{5} - \frac{x}{4}\), we rewrite these fractions as \(\frac{4x}{20} - \frac{5x}{20}\). Now, they share the common denominator of 20. This rewrite is essential as it helps us combine the fractions easily by simply subtracting the numerators.

Isolating Variables

Isolating the variable means getting the variable on one side of the equation by itself. This often involves combining like terms and performing operations such as addition, subtraction, multiplication, or division to both sides of the equation.
In our example, after combining the fractions to get \(\frac{4x}{20} - \frac{5x}{20} = 1\), we obtain \(\frac{-x}{20} = 1\). To isolate \(x\), we multiply both sides of the equation by -20. This gives us \(-x = 20\), meaning \(x = -20\). Now, the variable is isolated and solved, providing us with our potential solution.

Solution Verification

After solving for the variable, it's important to verify the solution. This step checks if the obtained value satisfies the original equation. Substitution is the common method used for verification.
By substituting \(x = -20\) back into the original equation \(\frac{x}{5} - \frac{x}{4} = 1\), we get \(\frac{-20}{5} - \frac{-20}{4}\). This simplifies to \(-4 + 5 = 1\), which holds true. Therefore, the solution \(x = -20\) is correct and verified. Verification is a critical step to ensure the accuracy of our solution.