Problem 97 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the given number is a solution of the equation. $$\frac{2}{9} y+\frac{1}{3} y=\frac{3}{7} ; \frac{27}{35}$$

Step-by-Step Solution

Answer

No, \(\frac{27}{35}\) is not a solution of the equation.

Step 1: Understand the Problem

The objective is to find out whether \(\frac{27}{35}\) is a solution of the equation \(\frac{2}{9} y+\frac{1}{3} y=\frac{3}{7}\).

Step 2: Substitute the Value For y

Replace \(y\) with \(\frac{27}{35}\) in the equation. Then, the equation becomes \(\frac{2}{9} * \frac{27}{35} + \frac{1}{3} * \frac{27}{35}\)

Step 3: Simplify the Equation

Compute the values of each term, when you multiply and simplify, you get \(\frac{2}{5} + \frac{9}{35} = \frac{7}{10}\)

Step 4: Compare With Expected Value

Upon simplifying the equation, the end result is \(\frac{7}{10}\). But according to the original equation, \(\frac{2}{9} y+\frac{1}{3} y\) has to equal \(\frac{3}{7}\). So, we derive that \(\frac{27}{35}\) is indeed not a solution to the given equation.

Key Concepts

Algebraic Solution Verification

Understanding how to verify an algebraic solution is an essential skill in mastering algebra. When given a certain value for a variable, like in the exercise where we are trying to verify if \(\frac{27}{35}\) is the solution for the equation \(\frac{2}{9} y+\frac{1}{3} y=\frac{3}{7}\), the process involves substituting the value into the original equation and simplifying the result.

To ensure that students fully grasp this concept, a key piece of advice is to remember that every term containing the variable should be replaced with the given value. After substitution, they should simplify the terms carefully, making sure not to mix up the order of operations: firstly resolving multiplication or division, then proceeding with addition or subtraction.

• Substitute the given number into the equation in place of the variable.
• Simplify the equation with the new numerical values.
• Compare the simplified result with the expected value from the original equation.
• If they match, the given number is a solution; if they don't, it is not.

For example, when \(\frac{27}{35}\) is substituted for \(y\) in the exercise, the result after simplification does not match the expected \(\frac{3}{7}\), indicating that \(\frac{27}{35}\) is not the correct solution.

Fractional Equations

Fractional equations, those which include fractions, can sometimes intimidate students, but they follow the same principles as other algebraic equations. The key challenge is often the simplification of fractions and finding a common denominator.

To help prevent common mistakes, it's helpful to encourage students to focus on:

• Identifying the least common denominator (LCD) for all the fractions involved.
• Multiplying each term by the LCD to eliminate fractions.
• Proceeding to solve the simplified equation as they would with integers or whole numbers.

In this particular exercise, since we are verifying a solution, eliminating fractions wasn't necessary. However, understanding how to compute with fractions was still crucial. The multiplying of \(\frac{2}{9}\) and \(\frac{1}{3}\) by \(\frac{27}{35}\) involved controlling numerators and denominators separately and finding the simplest form for resulting fractions, highlighting the importance of a clear grasp on fraction operations.

Substitution Method

The substitution method is a powerful tool for solving equations and variables, not only within a single equation but also in systems of equations. It involves replacing the variable with its value, or another expression, and is especially useful in equations with fractions or those where variables are found in multiple terms.

As we implement the substitution method:

• Always replace the variable with its value in parentheses to clearly distinguish the substituted part from the rest of the equation.
• Simplify cautiously, as fractions require attention to multiplying or dividing numerators and denominators correctly.
• Look out for opportunities to reduce fractions before multiplying to make calculations simpler.

For those who struggle, practicing substitution with simpler numbers before applying it to more complex fractions can help build confidence and understanding. In the provided exercise, proper application of the substitution method helped us to check the proposed solution methodically and determine its validity.