Problem 35 - 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. $$\frac{2 y}{3}-\frac{3}{4}=\frac{5}{12}$$

Step-by-Step Solution

Answer

The solution is \(y = \frac{7}{4}\).

Step 1: Eliminate fractions

To eliminate fractions, find a number that both 3 and 4 divide into. 12 is a good choice here. So, multiply each term by 12: \(12*\frac{2 y}{3} - 12*\frac{3}{4} = 12*\frac{5}{12} \). This simplifies to \(8y - 9 = 5\).

Step 2: Rearrange the equation

This is a simple equation that just needs to be rearranged into y = ... format. To do this, add 9 to both sides to get \(8y = 14\).

Step 3: Solve for y

Then, divide both sides by 8 to find the value of y: \(y = \frac{14}{8} = \frac{7}{4} \).

Step 4: Check the solution

Finally, check that the solution works by substituting \(\frac{7}{4}\) back into the original equation: \(\frac{2*\frac{7}{4}}{3} - \frac{3}{4} \). This simplifies to \(\frac{5}{12}\), which matches the other side of the equation, so the solution is correct.

Key Concepts

Fractions in Equations

Working with fractions in equations can sometimes seem intimidating, but it's all about maintaining balance. The first step is to eliminate fractions to make the problem more straightforward to solve.
In the given equation \(\frac{2y}{3} - \frac{3}{4} = \frac{5}{12}\), the goal is to get rid of the fractions by finding a common denominator.
This common denominator should be a multiple of all denominators involved. Here, it's 12 because 12 is the least common multiple of 3, 4, and itself.

• Multiply each term by the common denominator (in this case, 12).
  
• This helps to convert each fraction into a whole number, simplifying the equation significantly.

By removing the fractions, the equation becomes much simpler, transforming into one without fractions: \(8y - 9 = 5\). Once this is done, solving the equation becomes a series of basic algebraic steps.
This method significantly reduces complexity and allows you to focus on the primary algebraic manipulation needed to find the solution.

Algebraic Manipulation

Algebraic manipulation involves rearranging terms and performing operations to isolate the variable.
Once the fractions are eliminated, you're left with the equation \(8y - 9 = 5\), which needs to be solved for \(y\).

• Start by moving terms to one side of the equation to isolate the variable term.
  
• Add 9 to both sides of the equation, balancing it to maintain equality: this results in \(8y = 14\).
  
• The final step in isolating \(y\) is to divide both sides by 8, leading to the solution \(y = \frac{14}{8} = \frac{7}{4}\).

These manipulations are crucial as they simplify complex expressions and bring clarity to the solution process.
Understanding how to rearrange equations through addition, subtraction, multiplication, and division is foundational in algebra.

Checking Solutions

Checking solutions is an essential last step in solving equations, ensuring that the solution derived is indeed correct. This is particularly important in ensuring that no mistakes were made during simplification or algebraic manipulation.
To check, substitute the solution \(y = \frac{7}{4}\) back into the original equation \(\frac{2y}{3} - \frac{3}{4} = \frac{5}{12}\):

• Replace \(y\) with \(\frac{7}{4}\). The equation becomes \(\frac{2*\frac{7}{4}}{3} - \frac{3}{4}\).
• Calculate \(\frac{14}{12} - \frac{9}{12} = \frac{5}{12}\).
• The result, \(\frac{5}{12}\), matches the original right hand side.

Since the original equation balances when the solution is substituted back, the solution \(\frac{7}{4}\) is verified as correct.
Checking solutions helps in catching errors early and reinforces the understanding and application of algebraic principles.