Problem 36 - Algebra 1

Question

Solve the equation. $$\frac{1}{3} y=82$$

Step-by-Step Solution

Answer

The solution to the equation is y = 246.

Step 1: Identify the equation

The equation given is \(\frac{1}{3} y = 82\) which needs to be solved for y.

Step 2: Multiply both sides by reciprocal

To get rid of the fraction \(\frac{1}{3}\) on the left hand side, multiply both sides by its reciprocal which is 3. We get: 3 * \(\frac{1}{3} y\) = 3 * 82

Step 3: Solve for y

Multiplying gives us y = 246.

Key Concepts

Inverse Operations

Inverse operations are the operations that reverse the effect of the original operation. They are essential in solving equations because they allow you to isolate the variable you're solving for. For example, if an equation involves addition, you'd typically use subtraction to reverse it.

When solving linear equations, inverse operations help "undo" parts of the equation. In the given problem,

• the operation you needed to undo was the multiplication of \(\frac{1}{3}\) with \(y\).
• To remove this fraction, you used the inverse operation by multiplying by its reciprocal, which is 3. This is because multiplication and division are inverse operations to each other.

By applying inverse operations, you transform the equation into a simpler form that directly solves for the unknown variable. This is a vital skill in algebra and helps maintain the balance of equations while working towards finding a solution.

Fractions in Equations

Fractions often appear in algebraic equations, and they can initially seem complicated. However, they can be simplified or eliminated through strategic operations. Handling fractions correctly is vital to make the equation easier to work with and ultimately solve.

In the equation \(\frac{1}{3}y = 82\), the fraction \(\frac{1}{3}\) is multiplied by the variable \(y\). To eliminate the fraction, you multiply both sides of the equation by the fraction's reciprocal. This is an effective method because:

• The reciprocal essentially cancels the fraction on one side of the equation. In this case, multiplying by 3 cancels \(\frac{1}{3}\), turning it to 1 \(y\).
• It allows you to work with whole numbers instead of fractions, which simplifies calculations.

Dealing with fractions requires careful consideration of reciprocal and consistent application across the equation to maintain balance and solve for the variable.

Equation Solving Steps

Solving equations involves a series of steps that help you systematically find the solution. Each step is critical and builds on the previous one until you isolate the variable. The procedure usually includes:

• Identifying the equation: Understand what needs to be solved. In this example, the goal was to find the value of \(y\) in the equation \(\frac{1}{3}y = 82\).
• Applying inverse operations: Choose the appropriate inverse operations to isolate the variable. Multiplying both sides by 3 was used here to eliminate the fraction.
• Solving for the variable: Perform the operations and solve for the unknown. After multiplying, the result gave \(y = 246\).

These steps provide a structured approach to handling different types of equations. By following these guidelines, solving any linear equation can become a more organized and manageable process, enhancing your problem-solving skills in algebra.