Problem 19 - Algebra 1: Concepts and Skills

Question

SUBTRACTING RATIONAL EXPRESSIONS. Simplify the expression. $$ \frac{8+6 t}{3 t}-\frac{5 t-6}{3 t} $$

Step-by-Step Solution

Answer

\[\frac{14 + t}{3t}\]

Step 1: Identify the common denominator

The common denominator for both fractions is \(3t\). Having the same denominator is important since it allows addition or subtraction of the numerators directly.

Step 2: Subtract the numerators

Since there is a common denominator, subtract the numerators from each other. The equation becomes: \[\frac{(8+6t) - (5t-6)}{3t}\]

Step 3: Simplify the numerator

Perform the subtraction inside the brackets in the numerator. Be aware of the negative sign in front of the second bracket. Distributing the negative to each term in the second bracket, we get: \[\frac{8 + 6t -5t +6}{3t}\]Combine like terms in the numerator to simplify the expression: \[\frac{(8 + 6) + (6t - 5t)}{3t}\]This simplifies to: \[\frac{14 + t}{3t}\]

Step 4: Final result

The final result for the simplified expression is: \[\frac{14 + t}{3t}\]This is a simplified rational expression.

Key Concepts

Algebraic Fractions

Algebraic fractions are a type of fraction where both the numerator and the denominator contain algebraic expressions, which means they include variables like \(x\) or \(t\), along with numbers. These fractions work much like numerical fractions, but with the added complexity of dealing with variables. For example, in the expression \(\frac{8+6t}{3t}\), the numerator is \(8+6t\) and the denominator is \(3t\). Both are algebraic because they include the variable \(t\).
Algebraic fractions often require operations such as addition, subtraction, multiplication, and division. To manage these operations, familiarity with basic rules of algebra, such as distributing and combining like terms, is essential. Understanding these rules is crucial when performing operations like subtraction, which can result in new simplified forms.

Common Denominators

A common denominator is a shared divisor between two fractions that allows us to perform addition or subtraction. For general fractions like \(\frac{a}{b}\) and \(\frac{c}{d}\), finding a common denominator means finding a number that both \(b\) and \(d\) divide evenly.
In the given problem, both algebraic fractions \(\frac{8+6t}{3t}\) and \(\frac{5t-6}{3t}\) already have a common denominator, which is \(3t\).
Having this common denominator is key, as it enables us to directly subtract the numerators. Without a common denominator, we would need to find equivalents for the fractions that do share a denominator, which can involve multiplying parts of the fractions by certain factors to obtain the desired common base.

Numerator Subtraction

Numerator subtraction is the process of subtracting one algebraic expression from another, assuming the fractions involved have common denominators. When subtracting numerators, it is important to overlap terms correctly and remember that subtraction is equivalent to adding the opposite.
In the exercise, subtracting \(\frac{5t-6}{3t}\) from \(\frac{8+6t}{3t}\) involves taking the expression \((8+6t)\) and subtracting the expression \((5t-6)\). This results in the combined numerator \((8+6t)-(5t-6)\).
One must be careful with negative signs, as these can change the signs of terms being subtracted. Expanding the expression will then allow for combining like terms, such as \(6t\) and \(-5t\), to achieve simplification.

Rational Expression Simplification

Simplifying a rational expression means reducing it to its simplest form. This often involves performing operations on the numerator and the denominator to eliminate redundancies or collapse it to fewer terms.
In this case, following the subtraction of numerators, the expression \(\frac{8 + 6t -5t +6}{3t}\) is reconstructed. By combining like terms, the expression simplifies to \(\frac{14 + t}{3t}\).
Simplification aims to make the expression more compact or easier to interpret without changing the inherent value. This may involve factoring, canceling numbers, or reducing coefficients and similar variables. Having a simplified rational expression is particularly useful for further algebraic manipulations or solving equations where this expression might appear.