Problem 28 - Algebra 1

Question

Solve the equation. $$\frac{2}{5}=a-\frac{1}{5}$$

Step-by-Step Solution

Answer

The solution of the equation is \(a = \frac{3}{5}\)

Step 1: Setup the equation

The given equation is \(\frac{2}{5} = a - \frac{1}{5}\)

Step 2: Add fraction to both sides

To isolate \(a\), add \(\frac{1}{5}\) to both sides: \(\frac{2}{5} + \frac{1}{5} = a - \frac{1}{5} + \frac{1}{5}\)

Step 3: Simplify both sides

Rearrange the terms to obtain: \(\frac{3}{5} = a\)

Key Concepts

Fractions in Equations

Fractions can often be seen in equations and, at first glance, they might seem a bit intimidating. However, they hold the same properties as regular numbers and can be manipulated in the same way.
When dealing with fractions, always remember that any operation you perform on one side of the equation must be performed on the other side as well. This maintains the balance of an equation.
In our example, we have the fraction \( \frac{2}{5} \). Here, \( \frac{1}{5} \) is subtracted from the variable \( a \).

• Fractions truly shine in algebra because they are representations of division.
• When adding or subtracting fractions, it is crucial to ensure that the denominators (the bottom parts) are the same.
• In our case, since both fractions already have the same denominator, it simplifies our calculation.

Handling fractions correctly can be key to easily solving an equation without errors. Remember, practice makes handling fractions easier!

Isolation of Variables

An essential step in solving any equation is isolating the variable you're solving for. This means rearranging the equation until the variable stands alone on one side of the equation.
By isolating the variable, you effectively solve the equation, giving you the value that makes the equation true.
In the equation \( \frac{2}{5} = a - \frac{1}{5} \), we want to determine the value of \( a \).

• To bring \( a \) on its own, you need to perform operations that eliminate other terms from its side of the equation.
• In our scenario, adding \( \frac{1}{5} \) to both sides allows the \( -\frac{1}{5} \) on the right to cancel out.
• This leaves \( a \) neatly isolated, giving us \( a = \frac{3}{5} \).

Isolating variables clears the path to solve for unknowns efficiently, and it's a skill that will be useful in countless math problems.

Arithmetical Operations

Arithmetical operations—such as addition, subtraction, multiplication, and division—are the basic tools you will use in solving equations.
Each equation may require different operations, but understanding how to properly apply them is fundamental. Let's look at how they play into solving our example equation.

• Addition: In our problem, we added \( \frac{1}{5} \) to both sides to keep the equation balanced while moving terms around.
• Subtraction: The original equation had subtraction with \( a - \frac{1}{5} \). Correctly handling this allowed us to know what to add to isolate \( a \).
• While multiplication and division aren't directly applied here, they can be crucial in more complex equations involving fractions.

Understanding and carefully applying these operations will make solving equations more straightforward. Operations are the language through which we communicate and solve mathematical problems, whether simple or complex.