Problem 102 - Intermediate Algebra

Question

Simplify each expression. $$\frac{1}{12}(y-12 x)-\frac{1}{3}(y-3 x)$$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{1}{4}y\).

Step 1: Distribute the Fractions

Distribute the fractions into the expressions inside the parentheses. For \( \frac{1}{12}(y-12x) \), distribute \( \frac{1}{12} \) across \( y \) and \( -12x \). For \( \frac{1}{3}(y-3x) \), distribute \( \frac{1}{3} \) across \( y \) and \( -3x \).\[\frac{1}{12} \cdot y - \frac{1}{12} \cdot 12x - \frac{1}{3} \cdot y + \frac{1}{3} \cdot 3x\]

Step 2: Simplify Individual Terms

Simplify each of the terms from the distributions.\[\frac{1}{12}y - x - \frac{1}{3}y + x\]

Step 3: Combine Like Terms

Combine like terms to simplify the expression further. Group the \( y \) terms together and the constant terms together.\[\left(\frac{1}{12} - \frac{1}{3}\right)y + \left(-x + x\right)\]

Step 4: Simplify Fractions

Calculate the difference of the fractions for the \( y \) terms by finding a common denominator:\[\frac{1}{3} = \frac{4}{12}\]Therefore,\[\frac{1}{12} - \frac{4}{12} = -\frac{3}{12} = -\frac{1}{4}\]Combine this with the \( x \) terms:\[-y\]

Step 5: Final Simplified Expression

Combine the simplified terms to give the final expression. The \( x \) terms cancel each other out:\[-\frac{1}{4}y\]

Key Concepts

Distribution in Algebra

Distribution in algebra is a useful technique that helps manage the operations of multiplication over an addition or subtraction within a parenthesis. It follows the distributive property, which states that for any real numbers a, b, and c:

• \[a(b + c) = ab + ac\]
• \[a(b - c) = ab - ac\]

In the problem given, distribution is applied to spread the fractions across the terms inside the parentheses. For example, the term \( \frac{1}{12}(y - 12x) \) becomes \( \frac{1}{12} \cdot y - \frac{1}{12} \cdot 12x \). This step-by-step application ensures each part of the expression is adjusted equally.
It's important to pay attention to the signs: adding and subtracting terms directly impacts the final subtraction results.
Distributing needs careful attention to detail, especially when working with variables and coefficients like fractions.

Combining Like Terms

Combining like terms is a simplifying step where terms that have the same variable component are combined.

• For example, in the expression \( \frac{1}{12}y - x - \frac{1}{3}y + x \), the like terms here are \( \frac{1}{12}y \) and \( -\frac{1}{3}y \) (both contain the variable \( y \)).
• Another set of like terms are \( -x \) and \( x \), which are constants associated with x.

To combine these effectively:- Align the terms with the same variables.- Remember their coefficients (numbers in front of the variables) can be added or subtracted accordingly.- Like here, choosing to combine \( \frac{1}{12}y \) with \( -\frac{1}{3}y \), gives a coefficient of \( \frac{1}{12} - \frac{4}{12} = -\frac{1}{4} \).
Ultimately, once simplified, you should have a single term representing all the similar variable components together, making the expression more concise.

Fraction Operations in Algebra

Working with fractions in algebra involves careful manipulation of the numerical parts of expressions.

• Key to mastering this is understanding how to manage fraction addition, subtraction, and simplification.
• Another crucial aspect is finding a common denominator when combining fractions.

For instance, in the expression involving \( \frac{1}{12}y - \frac{1}{3}y \), reducing fractions to have a common denominator transforms \( \frac{1}{3} \) to \( \frac{4}{12} \).
This enables straightforward subtraction: \( \frac{1}{12} - \frac{4}{12} = -\frac{3}{12} \).
Finally, always simplify fractions to their lowest terms.
In this case, \( -\frac{3}{12} \) simplifies to \( -\frac{1}{4} \). Each step should maintain balance by ensuring all variables and constants in the equation are considered fairly.
Handling fractions correctly is crucial for succinct and precise algebraic expressions.