Simplify 6/(6k-9)-6/2

The problem involves simplifying a mathematical expression that contains fractions with variables. You are asked to find a simpler form of the given expression which consists of two fractions being subtracted from one another. The first fraction is 6 divided by $(6 k - 9)$ and the second fraction is 6 divided by 2. In order to simplify such an expression, common algebraic techniques such as finding a common denominator and combining like terms would typically be employed.

$\frac{6}{6 k - 9} - \frac{6}{2}$

Answer

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Solution:

Step 1: Break down each term for simplification.

Step 1.1: Identify and remove common factors between $6$ and $6 k - 9$ .

Step 1.1.1: Extract the factor of $3$ from $6$ . $\frac{3 \times 2}{6 k - 9} - \frac{6}{2}$ Step 1.1.2: Eliminate identical factors.

Step 1.1.2.1: Take out a factor of $3$ from $6 k$ . $\frac{3 \times 2}{3 (2 k) - 9} - \frac{6}{2}$ Step 1.1.2.2: Extract a factor of $3$ from $- 9$ . $\frac{3 \times 2}{3 (2 k) + 3 \times (- 3)} - \frac{6}{2}$ Step 1.1.2.3: Factor out $3$ from $3 (2 k) + 3 (- 3)$ . $\frac{3 \times 2}{3 (2 k - 3)} - \frac{6}{2}$ Step 1.1.2.4: Cancel out the common factor of $3$ . $\frac{3 \times 2}{3 (2 k - 3)} - \frac{6}{2}$ Step 1.1.2.5: Rewrite the simplified expression. $\frac{2}{2 k - 3} - \frac{6}{2}$ Step 1.2: Divide $6$ by $2$ . $\frac{2}{2 k - 3} - 3$ Step 1.3: Multiply $- 1$ by $3$ . $\frac{2}{2 k - 3} - 3$

Step 2: Convert $- 3$ into a fraction with the same denominator by multiplying by $\frac{2 k - 3}{2 k - 3}$ .

Step 3: Simplify the terms.

Step 3.1: Combine $- 3$ with $\frac{2 k - 3}{2 k - 3}$ . $\frac{2}{2 k - 3} - \frac{3 (2 k - 3)}{2 k - 3}$ Step 3.2: Merge the numerators over a common denominator. $\frac{2 - 3 (2 k - 3)}{2 k - 3}$

Step 4: Streamline the numerator.

Step 4.1: Apply the distributive property. $\frac{2 - 3 \times 2 k + 3 \times 3}{2 k - 3}$ Step 4.2: Multiply $2$ by $- 3$ . $\frac{2 - 6 k + 3 \times 3}{2 k - 3}$ Step 4.3: Multiply $- 3$ by $3$ . $\frac{2 - 6 k + 9}{2 k - 3}$ Step 4.4: Sum up $2$ and $9$ . $\frac{11 - 6 k}{2 k - 3}$

Step 5: Refine by factoring.

Step 5.1: Factor out $- 1$ from $- 6 k$ . $\frac{- 1 (6 k) + 11}{2 k - 3}$ Step 5.2: Express $11$ as $- 1 \times (- 11)$ . $\frac{- 1 (6 k) - 1 \times (- 11)}{2 k - 3}$ Step 5.3: Factor out $- 1$ from the entire numerator. $\frac{- 1 (6 k - 11)}{2 k - 3}$ Step 5.4: Condense the expression.

Step 5.4.1: Represent $- 1 (6 k - 11)$ as $- 1 \times (6 k - 11)$ . $\frac{- 1 (6 k - 11)}{2 k - 3}$ Step 5.4.2: Place the negative sign in front of the fraction. $- \frac{6 k - 11}{2 k - 3}$

Knowledge Notes:

Common Factor: A number that divides exactly into two or more other numbers. In this problem, $3$ is a common factor of $6$ and $6 k - 9$ .
Simplifying Fractions: The process of reducing the numerator and denominator to their smallest whole numbers. This can involve factoring out common factors.
Distributive Property: A property that allows you to multiply a sum by multiplying each addend separately and then add the products. It is expressed as $a (b + c) = a b + a c$ .
Negative Numbers: When factoring out a negative number, it is equivalent to multiplying by $- 1$ . This can change the signs of the terms within the parentheses.
Common Denominator: When combining fractions, they must have the same denominator. If they do not, you can find a common denominator by multiplying the denominators together or by finding an equivalent fraction for one or both fractions.
LaTeX: A typesetting system used for formatting mathematical expressions, as seen in the solution steps above.

Simplify 6/(6k-9)-6/2

Answer

Solution:

Knowledge Notes:

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