Problem 102 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the given number is a solution of the equation. $$(y \div 6)+\frac{1}{3}=(y \div 2)-\frac{5}{9} ; 2 \frac{2}{3}$$

Step-by-Step Solution

Answer

The given number \(2\frac{2}{3}\) is not a solution of the equation \( (y \div 6)+\frac{1}{3}=(y \div 2)-\frac{5}{9}\).

Step 1: Understand the given equation

The given equation is \( (y \div 6)+\frac{1}{3}=(y \div 2)-\frac{5}{9} \). Here, operations of division and addition/subtraction with fractions are involved. The term (y) here is an unknown number.

Step 2: Substitute the given number in the place of variable y

Substitute given number \(2\frac{2}{3}\) in place of \(y\). Express the mixed number in the form of an improper fraction which is \( \frac{8}{3}\). So the equation becomes \( \left(\frac{8}{3} \div 6\right)+\frac{1}{3} = \left(\frac{8}{3} \div 2\right)-\frac{5}{9} \). Remember that division of a number by another is equivalent to multiplication of the first number by the reciprocal of the second.

Step 3: Simplify each side of the equation

By simplifying, we get \( \frac{4}{9} + \frac{3}{9} = \frac{4}{3} - \frac{5}{9}\). Further simplifying gives us \( \frac{7}{9} = \frac{4}{3} - \frac{5}{9}\) and then \( \frac{7}{9} = \frac{31}{27}\). Convert the right side of the equation into nine as the denominator, we get \( \frac{7}{9} = \frac{31}{27}\). Now both sides of the equation are expressed with nine as the denominator.

Step 4: Check if both sides of the equation are equal

Here, the left side \( \frac{7}{9}\) is not equal to the right side \( \frac{31}{27} \). Thus, both sides of the equation are not equal.

Key Concepts

Fractions

Fractions are numbers that represent a part of a whole. They consist of a numerator (top number) and a denominator (bottom number). In the equation, we deal with fractions like \( \frac{1}{3} \) and \( \frac{5}{9} \). Understanding fractions is crucial when solving algebraic equations, especially when they involve operations like addition and subtraction.

• The numerator tells us how many parts we are considering.
• The denominator tells us into how many equal parts the whole is divided.

Fractions can be simplified by finding the greatest common factor of the numerator and the denominator. Additionally, combining fractions requires a common denominator, which allows us to add or subtract them more easily.

Division

Division is one of the fundamental operations in mathematics and involves splitting a number into equal parts. In the given equation, we see division in terms like \( y \div 6 \) and \( y \div 2 \). To handle division in equations involving fractions, we need to use the idea of the reciprocal.

• The reciprocal of a number is just flipping the numerator and denominator.
• Dividing by a number is the same as multiplying by its reciprocal.

For example, dividing by \( 6 \) is equivalent to multiplying by \( \frac{1}{6} \). This knowledge helps simplify complex expressions and make equation solving straightforward.

Equation Solving

Equation solving is the process of finding an unknown number represented by a variable. In algebra, you often solve equations to find the value of this unknown that makes the equation true. The given equation is \( (y \div 6) + \frac{1}{3} = (y \div 2) - \frac{5}{9} \). Let's break down the solving steps.

• Substitute the given number for the variable \( y \).
• Simplify each side separately to see if both sides are equal.

We start by replacing the variable \( y \) with \( 2 \frac{2}{3} \), then converting this mixed number to an improper fraction. After substitution and simplification, we compare both sides. If they are equal, the number is a solution; otherwise, it is not.

Mixed Numbers

Mixed numbers are numbers that contain both a whole part and a fractional part. In the exercise, the number given is \( 2 \frac{2}{3} \). Knowing how to work with mixed numbers is important in many algebra problems.

• A mixed number is always greater than its whole number component.
• To compute easily, convert mixed numbers to improper fractions, where the numerator is greater than the denominator.

To do this, multiply the whole number by the denominator and add the numerator. For \( 2 \frac{2}{3} \), multiply \( 2 \times 3 \) and add \( 2 \), resulting in \( \frac{8}{3} \). This conversion simplifies the operation with fractions and division in algebra.