Problem 40 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ \frac{1}{3}+x=\frac{2}{3} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{1}{3}\)

Step 1: Subtract \(\frac{1}{3}\) from both sides

Subtract \(\frac{1}{3}\) from the left side of the equation and also from the right side to maintain the balance of the equation. This gives you \(x = \frac{2}{3} - \frac{1}{3}\)

Step 2: Simplify

The subtraction of two fractions with the same denominator is simply the subtraction of the numerators over the common denominator. Therefore, \(x = \frac{1}{3}\)

Key Concepts

Fractions Subtraction

When dealing with algebra, one often encounters problems requiring the subtraction of fractions. Fractions represent pieces of a whole, and can be thought of as pie charts or pizza slices which are divided into equal parts. When subtracting fractions, if the denominators (bottom numbers) are the same, you simply subtract the numerators (top numbers) and keep the denominator as it is.

For example, in the exercise \( \frac{2}{3} - \frac{1}{3} \), both fractions share the same denominator, 3. Subtracting 1 from 2 gives us 1, and the denominator remains unchanged. So, the result is \( \frac{1}{3} \). It's essential to note that if the fractions had different denominators, one would first need to find a common denominator before proceeding with the subtraction. Simplifying fractions after subtracting is also an important step to find the simplest form of the result.

Equation Balancing

Equation balancing is a core principle in algebra, which ensures that an equation remains equal on both sides after any operation. This principle stems from the fundamental property of equality; if you perform the same operation on both sides of an equation, the two sides remain equal.

In our exercise, to isolate the variable x on one side, we subtract \( \frac{1}{3} \) from both sides because what you do to one side, you must do to the other to maintain balance. This concept is critical to solving algebraic equations as it allows us to modify equations until we find the value of the unknown variable while keeping the equation valid. Remember, balance is key in algebra; whatever you add, subtract, multiply, or divide on one side must be done on the opposite side as well.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that makes equations easier to work with. It involves reducing equations to their simplest form without changing their value. This could mean combining like terms, reducing fractions, or performing arithmetic operations to simplify the expression.

In our given problem, after subtracting the fractions, we simplify \( \frac{2}{3} - \frac{1}{3} \) to \( \frac{1}{3} \). Simplification in this context meant combining two fractions with a common denominator. It's a process that often makes the next steps of problem-solving clearer and more manageable. In other cases, simplification might involve factoring expressions, cancelling terms, or using exponent rules. Being proficient at simplification can significantly aid in recognizing patterns and understanding deeper algebraic concepts.