Problem 48 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{60 y-108}{12} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{60 y-108}{12}\) is \(5y - 9\).

Step 1: Identify Common Factors

Look for common factors between the numerator (60y-108) and the denominator 12. In this case, both 60 and 108 can be divided evenly by 12, which means 12 is a common factor.

Step 2: Simplify using Common Factors

We will now simplify the numerator by dividing each coefficient by 12. When 60y is divided by 12, the result is 5y. Similarly, dividing -108 by 12 results in -9.

Step 3: Write the Simplified Expression

After performing the divisions, we combine the simplified terms to form a single rational expression. So, 60y - 108 divided by 12 simplifies to 5y - 9.

Key Concepts

Understanding Common Factors

To simplify a rational expression, it's essential to understand common factors. A common factor is a number that divides exactly into two or more other numbers. In the context of algebra, we look for numerical values that can divide both the coefficients (numerical parts) of terms and the variables (letters) if applicable.

Identifying common factors simplifies the expression by breaking it down into its most basic form. This not only makes calculations easier but also reveals the most streamlined version of an expression. For instance, in the given exercise, both 60 and 108 share a common factor of 12. By dividing both by this number, we reduce the complexity of our original expression, paving the way towards simplification.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing them to their simplest form. It's much like simplifying regular fractions, but with the addition of variables. The goal is to make the expression as straightforward as possible.

Steps to simplify an algebraic fraction typically include:

• Finding common factors and canceling them out
• Dividing both the numerator and the denominator by these common factors
• Ensuring that the final expression contains no factors other than one that are common to the numerator and denominator

Through simplification, we achieve a result that keeps the original value of the expression intact but in an easier to work with form. In practice, as seen in the given exercise, the process involves reducing the numerator by using common factors and leaving the denominator, when it is a common factor itself, as one.

Dividing Numerical Coefficients

When simplifying expressions, dividing numerical coefficients can dramatically simplify your work. A numerical coefficient is a number that multiplies a variable in an algebraic term. In our given exercise, '60' and '108' are numerical coefficients that are being divided by 12.

This division is a crucial step because it reduces the size of the coefficients, thus simplifying the algebraic fraction as a whole. As long as you divide each term's coefficient by the same non-zero number, the expression's value remains unchanged. It is crucial, however, to perform the same operations to both, otherwise you change the expression's value. The end goal here is to get to the simplest form of the expression where no further division by a common factor is possible.