Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation and check your proposed solution. Begin your work by rewriting each equation without fractions. \(\frac{2 y}{3}-\frac{3}{4}=\frac{5}{12}\)

Step-by-Step Solution

Answer

The solution to the given equation is \(y = \frac{7}{4}\)

Step 1: Eliminate fractions

In this case, you need to find the least common multiple (LCM) between the denominators 3, 4, and 12, which is 12. Once found, multiply each term of the equation by this LCM. The equation becomes:\(12 *\frac{2y}{3} - 12* \frac{3}{4} = 12* \frac{5}{12}\)This simplifies to:\(8y - 9 = 5\)

Step 2: Solve for y

Now, isolate the term with \(y\) on one side of the equation. To do this, add 9 to both sides of the equation, resulting in:\[8y = 14\]Finally, divide both sides by 8 to solve for \(y\):\[y = \frac{7}{4}\]

Step 3: Checking the solution

Now, to verify if this value for \(y\) is correct, substitute \(\frac{7}{4}\) back into the original equation and verify whether both sides are indeed equal. \(\frac{2(7/4)}{3}-\frac{3}{4}= \frac{6}{4}-\frac{3}{4}= \frac{3}{4}\)As both left-hand side and right-hand side indeed equal to \(\frac{5}{12}\), the solution is verified as correct.

Key Concepts

Fractions in Algebra

When dealing with algebraic equations involving fractions, it is often necessary to eliminate the fractions to simplify the solution process. Fractions can make equations more complex due to the presence of multiple denominators. Understanding how to manage these fractions is crucial. In algebra, we can simplify equations by finding a way to rewrite them without the fractions.

• Fractions are essentially division problems, with a numerator (top number) and a denominator (bottom number).
• In algebra, when you have terms with fractions, you can eliminate them by finding a common denominator or multiplying all terms by the least common multiple (LCM) of all denominators involved.
• This process transforms the equation into a simpler form, making it easier to solve.

In the given exercise, the equation is \[\frac{2y}{3} - \frac{3}{4} = \frac{5}{12}\]. To solve, you need to eliminate the fractions by finding the LCM of 3, 4, and 12, which is 12. This allows you to multiply each term by 12, simplifying the equation and allowing you to continue with basic algebraic operations.

Least Common Multiple

The Least Common Multiple (LCM) is a key concept in solving equations with fractions. The LCM of two or more numbers is the smallest number that is a multiple of all of them. In the context of simplifying algebraic fractions, finding the LCM allows you to easily eliminate fractions by ensuring all denominators are the same.

• To find the LCM of a set of numbers, list the multiples of each number until you find the smallest common one. Alternatively, use the prime factorization method to efficiently find the LCM.
• Using the LCM helps in rewriting equations without fractions, by multiplying each term of the equation so that the denominators cancel out.
• For the exercise, the LCM of 3, 4, and 12 is 12, which allows you to multiply each part of the equation by 12. This eliminates the denominators, simplifying the equation considerably.

Once the fractions are removed, solving the equation becomes much more straightforward.

Checking Solutions

After solving an equation, especially one involving fractions, it is essential to verify that your solution is correct. This ensures that no mistakes were made during the process.

• To check your solution, substitute the value back into the original equation. If both sides of the equation are equal, your solution is correct.
• In the exercise, after solving the equation for \(y\) and finding \(y = \frac{7}{4}\), substitute \(\frac{7}{4}\) back into the original equation \(\frac{2y}{3} - \frac{3}{4} = \frac{5}{12}\).
• Calculate each side of the equation separately; if they are the same, you have verified your solution.
• Checking solutions is crucial as it acts as a confirmation of your accuracy and understanding.

This process not only confirms the solution but also enhances your grasp of solving equations with fractions, reinforcing the steps involved in achieving the right answer.