Problem 2 - Technical Mathematics with Calculus

Question

Solve for \(x\). Assume the integers in these equations to be exact numbers, and leave your answers in fractional form. \(4 x+\frac{x}{5}=42\)

Step-by-Step Solution

Answer

x = 10

Step 1: Find the common denominator

Identify the least common denominator for the fractions in the equation. Here, the least common denominator between 1 (implicit for the term 4x) and 5 is 5. Multiply each term by 5 to eliminate the fraction.

Step 2: Eliminate the fraction

Multiply each term by 5 to eliminate the fractional part of the equation: 5(4x) + 5(\(\frac{x}{5}\)) = 5(42). This results in 20x + x = 210.

Step 3: Combine like terms

Combine the x terms on the left side of the equation: 20x + x = 21x. The equation now reads 21x = 210.

Step 4: Solve for x

Divide both sides of the equation by 21 to solve for x: x = \(\frac{210}{21}\).

Step 5: Simplify the fraction

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 21: x = \(\frac{210 \/ 21}{21 \/ 21}\) = \(\frac{10}{1}\).

Key Concepts

Understanding Common Denominator

When solving linear equations involving fractions, finding a common denominator is essential. It's the common ground that allows us to combine fractions or eliminate them altogether. Think of it as the least common multiple of all the denominators involved.

In our exercise, we had fractions with denominators 1 (implied for the term 4x) and 5. To find the least common denominator (LCD), we look for the smallest number that both denominators can divide into evenly. In this case, the LCD is 5. Multiplying every term by the LCD transforms the equation into a simpler form without fractions, making it easier to solve.

Eliminate Fractions for a Simpler Equation

Fractions can complicate equations, so it's often helpful to eliminate fractions early on in the solving process. After finding a common denominator, we multiply every term in the equation by that number. This step is a game-changer—it effectively removes fractions and turns the equation into a straightforward linear one.

In our exercise, multiplying each term by 5, the common denominator, magically erased the division by 5 and left us with whole numbers to work with. This simplification step prevents common mistakes that occur when dealing with fractions directly.

Combining Like Terms to Streamline the Equation

To solve an equation efficiently, combining like terms is a must. It means adding or subtracting terms that have the same variable to a power. This consolidation not only simplifies the equation but also brings us one step closer to isolating the variable.

In the textbook problem, once we eliminated the fractions, we combined the terms with the variable 'x' (20x + x). It's a bit like gathering apples; if you have 20 apples plus another apple, you now have 21 apples, or in our case, 21x. This simplicity helps prevent errors in the later stages of problem-solving.

Simplify Fractions for the Final Solution

After working through the previous steps, we often end up with an equation where the variable equals a fraction. To present the answer in the simplest form, we should simplify the fraction. This means dividing the numerator and denominator by their greatest common divisor until you can't simplify further.

For example, we had x = 210/21 in our exercise. Both the numerator (210) and the denominator (21) share a greatest common divisor of 21. Dividing them by 21, we get the simplified fraction x = 10/1, which is the same as x = 10. Simplification makes the final answer more understandable and easier to check for correctness.