Simplify (2t)/(3t-12)-8/(3t-12)
The problem provided requires you to simplify a mathematical expression that consists of two fractions being subtracted. Both fractions have the same denominator of (3t-12), but different numerators: one has 2t and the other has 8. The question is asking for a simplification of the expression, combining the two fractions into one by performing the subtraction of their numerators, resulting in a single, simpler rational expression.
$\frac{2 t}{3 t - 12} - \frac{8}{3 t - 12}$
Answer
Solution:
Step 1:
Write the expressions with a common denominator: $\frac{2t - 8}{3t - 12}$.
Step 2:
Extract the common factor from the numerator.
Step 2.1:
Take out the factor of $2$ from $2t$: $\frac{2(t) - 8}{3t - 12}$.
Step 2.2:
Extract the factor of $2$ from $-8$: $\frac{2t + 2 \cdot (-4)}{3t - 12}$.
Step 2.3:
Combine the factored terms in the numerator: $\frac{2(t - 4)}{3t - 12}$.
Step 3:
Factor out the common term from the denominator.
Step 3.1:
Factor $3$ from $3t$: $\frac{2(t - 4)}{3(t) - 12}$.
Step 3.2:
Factor $3$ from $-12$: $\frac{2(t - 4)}{3t + 3 \cdot (-4)}$.
Step 3.3:
Combine the factored terms in the denominator: $\frac{2(t - 4)}{3(t - 4)}$.
Step 4:
Eliminate the common term $(t - 4)$.
Step 4.1:
Cancel out the $(t - 4)$ term: $\frac{2(\cancel{t - 4})}{3(\cancel{t - 4})}$.
Step 4.2:
Simplify the expression: $\frac{2}{3}$.
Step 5:
Present the final result in various formats.
Exact Form: $\frac{2}{3}$ Decimal Form: $0.666\ldots$
Knowledge Notes:
When simplifying algebraic fractions, the goal is to combine like terms and reduce the expression to its simplest form. Here are the relevant knowledge points and detailed explanations:
Common Denominator: When subtracting fractions with the same denominator, you can combine the numerators over the common denominator. This is because fractions with the same denominator can be directly combined by performing the operation indicated (addition, subtraction, etc.) on the numerators.
Factoring: Factoring is the process of breaking down an expression into products of other expressions that, when multiplied together, give the original expression. In this case, we factored out common terms from both the numerator and the denominator to simplify the fraction.
Cancelling Common Factors: If a factor appears in both the numerator and the denominator, it can be cancelled out. This is because any number divided by itself equals one, and multiplying or dividing by one does not change the value of an expression.
Simplified Form: The simplest form of a fraction is when the numerator and the denominator have no common factors other than 1. This means the fraction cannot be reduced any further.
Latex Formatting: In the solution, Latex is used to format mathematical expressions for clarity. It allows for the proper display of fractions, factoring, and cancellation.
By applying these concepts, we were able to simplify the given algebraic fraction to its simplest form, $\frac{2}{3}$.
