Problem 70 - Prealgebra

Question

Apply the distributive property, then find the LCD and simplify. $$\frac{2}{3} x-\frac{5}{6} x$$

Step-by-Step Solution

Answer

Simplified expression is \(-\frac{1}{6}x\).

Step 1: Apply the Distributive Property

First, apply the distributive property to simplify the expression, though in this case, the expressions are already simple fractions with the same variable. The variable portion \(x\) is common, so we can write it as \( (\frac{2}{3} - \frac{5}{6})x \).

Step 2: Find the Least Common Denominator (LCD)

The fractions \(\frac{2}{3}\) and \(\frac{5}{6}\) have denominators 3 and 6, respectively. The least common denominator (LCD) of 3 and 6 is 6.

Step 3: Convert Fractions to Have a Common Denominator

Convert \(\frac{2}{3}\) to an equivalent fraction with the LCD of 6. Multiply both the numerator and the denominator by 2 to get \(\frac{4}{6}\). Now the expression inside the parentheses becomes \(\frac{4}{6} - \frac{5}{6}\).

Step 4: Simplify the Fractions

Now that the fractions have a common denominator, subtract the numerators: \(\frac{4}{6} - \frac{5}{6} = \frac{4 - 5}{6} = \frac{-1}{6}\).

Step 5: Final Simplification

Multiply the simplified fraction by the common variable: \((\frac{-1}{6})x = \frac{-1}{6}x\). This is the simplified form of the expression.

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that allows us to break down expressions to simple parts. It makes handling complicated expressions easier. The mathematical formula for the distributive property is:

• For any numbers or expressions, if you have: \( a(b + c) = ab + ac \)

This property tells us that we can multiply a single term by each term inside a set of parentheses. It helps in situations like our problem where two fractions are involved.
In our exercise, though the expression \( \frac{2}{3}x - \frac{5}{6}x \)is already simple, the distributive property allows us to factor out the common variable, \(x\). This means we can write it as: \( (\frac{2}{3} - \frac{5}{6})x \).
This makes it easier to deal with the coefficients separately from the variable.

Least Common Denominator

When working with fractions, especially when you want to add or subtract them, you need a common denominator. The least common denominator (LCD) is the smallest possible denominator that two or more fractions can share when combined.
Here's how it works:

• Look at the denominators of your fractions.
• Find the smallest number that each denominator can divide into evenly.
• Use this number to convert each fraction to equivalent fractions with a common denominator.

For example, we have the fractions \(\frac{2}{3}\) and \(\frac{5}{6}\), with denominators 3 and 6, respectively.
The smallest number that both 3 and 6 divide into is 6.
Thus, 6 is the least common denominator, allowing you to combine or compare the fractions efficiently.

Fraction Simplification

Simplifying fractions is essential for making calculations easier. It involves reducing fractions to their simplest form. If you're working with similar denominators in algebraic expressions, simplification helps keep things neat. Here is how you can simplify:

• Ensure the fractions have a common denominator.
• Subtract or add numerators while keeping the denominator constant.
• Write the result as a single fraction.

In our example, after converting the fractions to \(\frac{4}{6}\) and \(\frac{5}{6}\),you subtract the numerators: \(4 - 5 = -1\).
The simplified form of the expression then becomes \(\frac{-1}{6}\).
This process significantly simplifies the expressions, making final calculations far more straightforward.

Algebraic Expressions

Algebraic expressions are combinations of letters (variables) and numbers (coefficients or constants) arranged with arithmetic operations. These expressions represent various mathematical relationships and can be manipulated through addition, subtraction, multiplication, division, and other operations to find solutions. Here’s a closer look:

• Variables like \(x\) in our exercise represent unknowns we are solving for.
• Coefficients are the numerical parts that multiply the variables, such as \(\frac{2}{3}\) and \(\frac{5}{6}\).
• Constants are standalone numbers added or subtracted in expressions.

Algebraic expressions are written to usually solve for a value or simplify an equation. For our exercise, we reduce the expression \(\frac{2}{3}x - \frac{5}{6}x \)through distribution, LCM and fraction simplification, resulting in a simplified algebraic form: \(\frac{-1}{6}x \).