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 $(6k-9)$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
Solution:
Step 1: Break down each term for simplification.
Step 1.1: Identify and remove common factors between $6$ and $6k - 9$.
Step 1.1.1: Extract the factor of $3$ from $6$. $\frac{3 \times 2}{6k - 9} - \frac{6}{2}$ Step 1.1.2: Eliminate identical factors.
Step 1.1.2.1: Take out a factor of $3$ from $6k$. $\frac{3 \times 2}{3(2k) - 9} - \frac{6}{2}$ Step 1.1.2.2: Extract a factor of $3$ from $-9$. $\frac{3 \times 2}{3(2k) + 3 \times (-3)} - \frac{6}{2}$ Step 1.1.2.3: Factor out $3$ from $3(2k) + 3(-3)$. $\frac{3 \times 2}{3(2k - 3)} - \frac{6}{2}$ Step 1.1.2.4: Cancel out the common factor of $3$. $\frac{\cancel{3} \times 2}{\cancel{3}(2k - 3)} - \frac{6}{2}$ Step 1.1.2.5: Rewrite the simplified expression. $\frac{2}{2k - 3} - \frac{6}{2}$ Step 1.2: Divide $6$ by $2$. $\frac{2}{2k - 3} - 3$ Step 1.3: Multiply $-1$ by $3$. $\frac{2}{2k - 3} - 3$
Step 2: Convert $-3$ into a fraction with the same denominator by multiplying by $\frac{2k - 3}{2k - 3}$.
Step 3: Simplify the terms.
Step 3.1: Combine $-3$ with $\frac{2k - 3}{2k - 3}$. $\frac{2}{2k - 3} - \frac{3(2k - 3)}{2k - 3}$ Step 3.2: Merge the numerators over a common denominator. $\frac{2 - 3(2k - 3)}{2k - 3}$
Step 4: Streamline the numerator.
Step 4.1: Apply the distributive property. $\frac{2 - 3 \times 2k + 3 \times 3}{2k - 3}$ Step 4.2: Multiply $2$ by $-3$. $\frac{2 - 6k + 3 \times 3}{2k - 3}$ Step 4.3: Multiply $-3$ by $3$. $\frac{2 - 6k + 9}{2k - 3}$ Step 4.4: Sum up $2$ and $9$. $\frac{11 - 6k}{2k - 3}$
Step 5: Refine by factoring.
Step 5.1: Factor out $-1$ from $-6k$. $\frac{-1(6k) + 11}{2k - 3}$ Step 5.2: Express $11$ as $-1 \times (-11)$. $\frac{-1(6k) - 1 \times (-11)}{2k - 3}$ Step 5.3: Factor out $-1$ from the entire numerator. $\frac{-1(6k - 11)}{2k - 3}$ Step 5.4: Condense the expression.
Step 5.4.1: Represent $-1(6k - 11)$ as $-1 \times (6k - 11)$. $\frac{-1(6k - 11)}{2k - 3}$ Step 5.4.2: Place the negative sign in front of the fraction. $-\frac{6k - 11}{2k - 3}$
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 $6k - 9$.
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 $a(b + c) = ab + ac$.
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.
