Problem 93 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the given number is a solution of the equation. $$w-\frac{2}{3}=\frac{3}{4} ; 1 \frac{5}{12}$$

Step-by-Step Solution

Answer

Yes, the number \(1 \frac{5}{12}\) is a solution of the equation \(w-\frac{2}{3}=\frac{3}{4}\)

Step 1: Substitute the given number

Substitute \(w = 1 \frac{5}{12}\) into the equation \(w-\frac{2}{3}=\frac{3}{4}\). This will give you \(1 \frac{5}{12} - \frac{2}{3}\)

Step 2: Simplify the expression

Simplify the expression on the left side of the equation. To simplify, convert both numbers to improper fractions to make computation easier. \(1 \frac{5}{12} = \frac{17}{12}\). Now compute \(\frac{17}{12} - \frac{2}{3} = \frac{17}{12} - \frac{8}{12} = \frac{9}{12}\).

Step 3: Simplify to lowest terms

Reduce the fraction to its lowest terms. \(\frac{9}{12} = \frac{3}{4}\)

Step 4: Compare the results

Finally, check to see if the left hand side simplifies to equal the right hand side of the equation. As both simplify to \(\frac{3}{4}\), the answer is yes, the number \(1 \frac{5}{12}\) is a solution to the equation.

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator is larger than or equal to the denominator. This means the fraction represents a value greater than or equal to one. For example, instead of writing mixed numbers like \(1 \frac{5}{12}\), we can convert them into improper fractions like \(\frac{17}{12}\).

• Identify the whole number and fractional part of the mixed number.
• Multiply the whole number by the denominator of the fractional part.
• Add this result to the numerator of the fractional part.
• Place the result over the original denominator.

This conversion is useful as it simplifies calculations and is essential in solving equations involving fractions. Once converted, you can perform arithmetic operations such as addition and subtraction more straightforwardly.

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. A fraction is in simplest form when the numerator and the denominator have no common factors other than 1.
To simplify \(\frac{9}{12}\), you identify the greatest common divisor (GCD) of 9 and 12. Here, both numbers are divisible by 3, which is the GCD.

• Divide the numerator and the denominator by the GCD.
• The fraction \(\frac{9}{12}\) becomes \(\frac{3}{4}\) after division.

Simplifying fractions is crucial for clean results and makes comparison and computation just a bit easier when solving equations. It's also a step that ensures your answers are presented neatly and accurately.

Checking Solutions to Equations

Checking solutions to equations involves substituting a proposed solution back into the original equation to verify its correctness. This step is critical in validating whether the given number actually satisfies the equation.
In our example, substituting \(w = 1 \frac{5}{12}\) into the equation \(w-\frac{2}{3}=\frac{3}{4}\) involves these steps:

• Convert \(1 \frac{5}{12}\) to \(\frac{17}{12}\) to ease calculations.
• Perform the subtraction: \(\frac{17}{12} - \frac{8}{12} = \frac{9}{12}\).
• Simplify \(\frac{9}{12}\) to half, resulting in \(\frac{3}{4}\).
• Compare this with the right side of the equation \(\frac{3}{4}\).

Since both sides of the equation match, the proposed number is indeed a solution. This step is essential to ensure accuracy, and prevent errors, confirming that the solution holds true across the entire equation.