Simplify (5(x-3))/(2x-6)
The problem provided is an algebraic expression simplification task. Specifically, you're asked to simplify the fraction (5(x-3))/(2x-6) by identifying and factoring out common factors in the numerator and the denominator to reduce the expression to its simplest form. The simplification often involves recognizing patterns that indicate a common factor or applying basic algebraic rules such as the distributive property.
$\frac{5 \left(\right. x - 3 \left.\right)}{2 x - 6}$
Answer
Solution:
Step 1: Extract the common factor from the denominator
Extract the common factor of $2$ from the expression $2x - 6$.
Step 1.1: Identify the common factor in $2x$
Write the term $2x$ as $2 \cdot x$.
$$\frac{5(x - 3)}{2(x - 3)}$$
Step 1.2: Identify the common factor in $-6$
Express $-6$ as $2 \cdot (-3)$.
$$\frac{5(x - 3)}{2 \cdot (x - 3)}$$
Step 1.3: Rewrite the denominator with the common factor
Combine the terms in the denominator to show the common factor clearly.
$$\frac{5(x - 3)}{2(x - 3)}$$
Step 2: Eliminate the common terms
Remove the common term $(x - 3)$ from the numerator and denominator.
Step 2.1: Cancel the common term
Strike out the common term $(x - 3)$ in both the numerator and the denominator.
$$\frac{5 \cancel{(x - 3)}}{2 \cancel{(x - 3)}}$$
Step 2.2: Simplify the expression
After canceling out the common term, the simplified form is:
$$\frac{5}{2}$$
Step 3: Represent the result in various forms
The simplified expression can be represented in different ways.
Exact Form:
$$\frac{5}{2}$$
Decimal Form:
$$2.5$$
Mixed Number Form:
$$2 \frac{1}{2}$$
Knowledge Notes:
To simplify a rational expression like $\frac{5(x-3)}{2x-6}$, we need to look for common factors in the numerator and the denominator that can be canceled out. In this case:
Factorization: The denominator $2x-6$ can be factored by taking out the common factor of $2$, resulting in $2(x-3)$. This shows that $x-3$ is a common term in both the numerator and the denominator.
Cancellation: If a term is present in both the numerator and the denominator, it can be canceled out according to the property $\frac{a}{a} = 1$ for any non-zero $a$. This simplifies the expression.
Simplified Form: After canceling the common terms, we are left with a simplified form of the expression. In this case, it is $\frac{5}{2}$.
Equivalent Representations: The simplified result can be expressed in various forms, such as an exact fraction, a decimal, or a mixed number, depending on the context or preference.
Algebraic Manipulation: The process involves algebraic manipulation, which requires understanding of factorization, properties of fractions, and simplification techniques.
By following these steps, we can simplify algebraic expressions and represent them in various forms.
