Problem 51 - Algebra 1

Question

Simplify the expression. $$\frac{45-5 x}{5}$$

Step-by-Step Solution

Answer

After simplification, the expression becomes \( 9 - x \)

Step 1: Distribute the Denominator

Commence by dividing each term of the numerator with the denominator individually. Hence: \( \frac{45}{5} - \frac{5x}{5} \)

Step 2: Simplify Each Term

Next, perform individual division operations. Resulting in: \( 9 - x \)

Key Concepts

Distributing the Denominator

When simplifying algebraic expressions that involve fractions, distributing the denominator is a common technique used to streamline the process. This method entails dividing each term in the numerator by the denominator separately. Imagine you have a delicious pie (the numerator) that you want to share evenly among your friends (the denominator). You would cut slices of the pie corresponding to each term before handing them out.

The operation looks something like this in mathematical terms: Given a general expression like \( \frac{a + b}{c} \), you would distribute the denominator (c) across each term in the numerator (a and b), resulting in \( \frac{a}{c} + \frac{b}{c} \). When applied to the exercise \( \frac{45 - 5x}{5} \), we divide both 45 and -5x by 5, transforming the expression into separate, more manageable parts.

Division of Polynomials

Moving on to the division of polynomials, it's useful to think of it as breaking down larger algebraic expressions into simpler components. Unlike numerical division, dividing polynomials can involve variables along with numbers. The strategy remains the same, however: each term in the polynomial is divided by the divisor.

In the case of the exercise, we're not dealing with a complete polynomial, but the concept is similar. The term -5x in the numerator is a monomial being divided by the number 5. You perform this division as you would with any numbers, treating the variable x as a constant multiplier. For instance, \( \frac{5x}{5} \) simplifies to x, as the fives cancel each other out.

Simplifying Terms

The final step in the process is simplifying terms, which involves performing the division and reducing each separated term to its simplest form. It's similar to tidying up your room – putting everything in its right place to make it look neat and organized.

Considering our exercise, when you perform the division on individual terms - \( \frac{45}{5} \) and \( \frac{5x}{5} \) - you're left with the numbers 9 and -1, since 45 divided by 5 equals 9, and 5 divided by 5 is 1 (multiplying by the variable x). What remains is a crisp and uncluttered expression, 9 - x, without any unnecessary baggage. This is the essence of simplifying terms – get to the point with the least number of elements.