Problem 36 - Introductory Algebra for College Students

Question

Solve each equation and check your proposed solution in Exercises. Begin your work by rewriting each equation without fractions. $$\frac{3 y}{4}-\frac{2}{3}=\frac{7}{12}$$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{3y}{4}-\frac{2}{3}=\frac{7}{12}\) is \(y = \frac{5}{3}\)

Step 1: Eliminating Fractions

The first step involves eliminating fractions. To do this, find a common multiple for the denominators 4, 3 and 12, which is 12. Multiply every term of the given equation by 12 to eliminate the fractions and simplify the equation into \(9y - 8 = 7\).

Step 2: Rearrange for \(y\)

The second step is to solve for \(y\). To do this, you should add 8 to both sides of the equation to isolate \(9y\) on one side resulting in an equation as \(9y = 15\).

Step 3: Solve for \(y\)

After obtaining \(9y = 15\), divide both sides by 9 to solve for \(y\). The result will be \(y = \frac{15}{9}\). Simplifying the fraction, you arrive at \(y = \frac{5}{3}\).

Step 4: Checking the Solution

To confirm the validity of the solution, substitute \(y = \frac{5}{3}\) into the initial equation. The left side becomes \(\frac{3 * \frac{5}{3}}{4}-\frac{2}{3}\) which will simplify to \(\frac{7}{12}\), equal to the right side of the initial equation, verifying the solution is correct.

Key Concepts

Eliminating Fractions

To solve equations with fractions, the first task is often to eliminate the fractions entirely. This simplifies the equation and makes it easier to handle. You start by finding a common multiple for all the denominators in the equation. For example, in the equation \( \frac{3y}{4} - \frac{2}{3} = \frac{7}{12} \), the denominators are 4, 3, and 12. The least common multiple (LCM) of these numbers is 12.

Once you've found the common multiple, multiply every term in the equation by this number. This action allows you to clear out the fractions, resulting in an equation that's simpler to solve.

After performing this multiplication, the equation \( \frac{3y}{4} - \frac{2}{3} = \frac{7}{12} \) transforms into \( 9y - 8 = 7 \). This technique effectively removes all the fractions, making it possible to engage in straightforward algebraic manipulations.

Common Denominators

The concept of finding common denominators is fundamental when dealing with fractions, especially in equations. A common denominator allows you to combine or compare fractions by giving them the same base.

For instance, when working with the fractions \( \frac{3y}{4} \), \( \frac{2}{3} \), and \( \frac{7}{12} \), it becomes necessary to identify the least common multiple of the denominators (4, 3, and 12), which is 12.

Upon determining that 12 is the LCM, you multiply each term in the equation by it. This step eliminates the fractions, paving the way for easier computation. Using common denominators is handy because it standardizes the fractions, making it simpler to add, subtract, or equate them.

Finding a common denominator isn't just useful in removing fractions; it ensures accuracy in calculations and is pivotal for success in solving complex equations involving fractions.

Checking Solutions

Once a solution is obtained for an equation, it's important to verify the solution to ensure that it is accurate. This step helps in confirming that no mistakes were made in the solving process.

To check the solution, substitute the value you've found back into the original equation. If the equation holds true, the solution is verified as correct.

For example, after solving the equation \( 9y - 8 = 7 \) and finding \( y = \frac{5}{3} \), you should return to the original fractional equation: \( \frac{3y}{4} - \frac{2}{3} = \frac{7}{12} \). Substitute \( y = \frac{5}{3} \) into the equation and see if both sides equal: \( \frac{3 \times \frac{5}{3}}{4} - \frac{2}{3} = \frac{7}{12} \).

When the left side simplifies to \( \frac{7}{12} \), matching the right side, you confirm the solution is correct. This step is crucial because it helps catch any errors and reassures you of the accuracy of your work.