Simplify a/(2a-b)+(3a-b)/(b-2a)
This problem is asking for the simplification of a given algebraic expression that consists of two rational expressions or fractions. The expression involves variables 'a' and 'b'. The task is to perform algebraic manipulations such as combining the fractions, which involves finding a common denominator and then simplifying the result by factoring, canceling out terms, or rearranging terms to reach the simplest form of the expression. The difficulty arises from the fact that the denominators of the two fractions appear to be opposites of each other (2a-b and b-2a), which will need to be considered when finding a common denominator.
$\frac{a}{2 a - b} + \frac{\left(\right. 3 a - b \left.\right)}{b - 2 a}$
Answer
Solution:
Step:1
Refine the expressions.
Step:1.1
Extract $-1$ from the term $b$ to get $\frac{a}{2a - b} + \frac{3a - b}{-1(-b) - 2a}$.
Step:1.2
Pull out $-1$ from the term $-2a$ to rewrite as $\frac{a}{2a - b} + \frac{3a - b}{-1(-b) - (-2a)}$.
Step:1.3
Remove $-1$ from the entire expression $-1(-b) - (-2a)$ to simplify to $\frac{a}{2a - b} + \frac{3a - b}{-1(-b + 2a)}$.
Step:1.4
Streamline the given expression.
Step:1.4.1
Transfer the negative sign from the denominator to the numerator in $\frac{3a - b}{-1(-b + 2a)}$ to obtain $\frac{a}{2a - b} + \frac{-(3a - b)}{-b + 2a}$.
Step:1.4.2
Rearrange the terms to align them as $\frac{a}{2a - b} + \frac{-(3a - b)}{2a - b}$.
Step:1.5
Merge the numerators over a shared denominator to form $\frac{a - (3a - b)}{2a - b}$.
Step:2
Refine the numerator.
Step:2.1
Utilize the distributive property to expand $\frac{a - (3a) - (-b)}{2a - b}$.
Step:2.2
Multiply $3$ by $-1$ to get $\frac{a - 3a - (-b)}{2a - b}$.
Step:2.3
Resolve the double negative.
Step:2.3.1
Multiply $-1$ by $-1$ to simplify to $\frac{a - 3a + 1b}{2a - b}$.
Step:2.3.2
Multiply $b$ by $1$ to maintain $\frac{a - 3a + b}{2a - b}$.
Step:2.4
Subtract $3a$ from $a$ to get $\frac{-2a + b}{2a - b}$.
Step:3
Eliminate the common factors in the numerator and denominator.
Step:3.1
Extract $-1$ from $-2a$ to rewrite as $\frac{-2a + b}{2a - b}$.
Step:3.2
Extract $-1$ from $b$ to get $\frac{-2a - 1(-b)}{2a - b}$.
Step:3.3
Remove $-1$ from the entire expression $-2a - 1(-b)$ to simplify to $\frac{-(2a - b)}{2a - b}$.
Step:3.4
Represent $-(2a - b)$ as $-1(2a - b)$ to get $\frac{-1(2a - b)}{2a - b}$.
Step:3.5
Cancel out the common term to simplify to $\frac{-1(\cancel{2a - b})}{\cancel{2a - b}}$.
Step:3.6
Divide $-1$ by $1$ to obtain the final result $-1$.
Knowledge Notes:
To solve the given problem, we apply several algebraic techniques:
Factoring: This involves taking out a common factor from terms to simplify expressions. In this problem, we factored out $-1$ from various terms to help simplify the expression.
Combining Like Terms: When terms have the same variable raised to the same power, they can be combined by adding or subtracting their coefficients.
Distributive Property: This property is used to expand expressions of the form $a(b + c)$, which becomes $ab + ac$. It is also used to simplify expressions where a term is subtracted from a parenthesis.
Cancellation: If the same term appears in both the numerator and the denominator of a fraction, they can be cancelled out, as they are essentially a factor of 1.
Negative Signs: Careful attention must be paid to negative signs, especially when they are factored out or distributed across terms. A double negative turns into a positive.
These algebraic techniques are fundamental to simplifying complex expressions and solving equations in algebra.
