Problem 21 - Prealgebra

Question

Solve each equation. $$a+\frac{1}{3}=-\frac{2}{3}$$

Step-by-Step Solution

Answer

The solution is \( a = -1 \).

Step 1: Identify the equation

The equation given is \( a + \frac{1}{3} = -\frac{2}{3} \). Our goal is to solve for \( a \).

Step 2: Isolate the variable

To isolate \( a \), we need to get rid of \( \frac{1}{3} \) on the left side of the equation. We do this by subtracting \( \frac{1}{3} \) from both sides of the equation.

Step 3: Perform the subtraction

Subtract \( \frac{1}{3} \) from both sides: \[ a + \frac{1}{3} - \frac{1}{3} = -\frac{2}{3} - \frac{1}{3} \] Simplifying both sides, we have: \[ a = -\frac{2}{3} - \frac{1}{3} \] On the right side, combine the fractions by finding a common denominator: \[ a = -\frac{2}{3} - \frac{1}{3} = -\frac{3}{3} = -1 \]

Step 4: Solution

The isolated solution for \( a \) is then \( a = -1 \).

Key Concepts

Isolating the Variable

When solving an equation, one of the primary goals is to isolate the variable. This means rearranging the equation so that the unknown variable stands alone on one side, usually the left, of the equation. This process makes it easier to determine the value of the variable. Consider the equation provided:
Given: \( a + \frac{1}{3} = -\frac{2}{3} \)
In this equation, our target is to isolate "a". By doing so, we will have something like \( a = \text{some number} \). This process generally involves eliminating any numbers or operations that are attached to the variable. In our example, this involves removing \( \frac{1}{3} \) from the left side of the equation.

Steps to isolate the variable include:

• Identify the terms on the side with the variable.
• Decide on the mathematical operation needed (e.g., subtraction) to remove those terms.
• Perform this operation on both sides of the equation to maintain balance.

Combining Like Terms

Combining like terms is a vital part of simplifying algebraic expressions and solving equations. In the context of our problem, we are focusing specifically on the right side of the equation. After subtracting \( \frac{1}{3} \) from both sides, we need to simplify:
Initially, we subtract directly: \( a + \frac{1}{3} - \frac{1}{3} = -\frac{2}{3} - \frac{1}{3} \)
Here we see that the left side becomes simply \( a \). On the right side, both terms \(-\frac{2}{3} \) and \(-\frac{1}{3} \) are similar; they are both fractions. When terms are like this, we can combine them.

Simply add the coefficients:

• Identify common mathematical components (e.g., denominators).
• Add or subtract the numerators while the denominator remains the same.

This gives us: \( a = -\frac{3}{3} \)

Subtracting Fractions

Subtracting fractions can sometimes seem complex initially. However, by remembering a few basic concepts, it becomes quite manageable. Let's consider the subtracting process directly from our example's equation:
After isolating and simplifying, we had: \( a = -\frac{2}{3} - \frac{1}{3} \)

To subtract these fractions:

• Ensure a common denominator, which allows for a straightforward operation.
• Subtract the numerators directly: \( -2 - 1 \), giving \( -3 \).
• The denominator, "3," stays constant: thus \( -\frac{3}{3} \).

This then separates the equation properly and allows for further simplification.

Finding a Common Denominator

To effectively add or subtract fractions, you require a common denominator. This is a critical aspect of fraction operations. For our equation, the fractions in question already share a denominator, which facilitates the process:
Given: \( -\frac{2}{3} - \frac{1}{3} \)

Why do we need a common denominator? It ensures that all fractions are on the same "level," making subtraction (or addition) possible without inconsistencies. Think of it like having the same unit while measuring different objects.

Steps to ensure common denominators include:

• Identify the denominators of all involved fractions.
• Multiply or adjust terms to equate the denominators.
• Perform the arithmetic operation on the numerators alone.

In our case, the common denominator was "3," simplifying our task and leading directly to the solution.