Solving a Mathematical Ratio with Algebraic Manipulations
If we are given a/b 7/5, how can we solve for expressions involving a^2b^2 2ab/ [a^2b^2 – 2ab]? This detailed step-by-step guide explains the algebraic manipulations and simplifications needed to solve such expressions, providing clarity on complex mathematical concepts.
Simplification Using Basic Algebra
Given: a/b 7/5
We can interpret this as a 7x#39;#39; and b 5x#39;#39;, where x#39;#39; is some number.
We need to solve for a^2b^2 2ab/ [a^2b^2 – 2ab].
Note we can use the identity a^2b^2 2ab ab^2. Applying this, we have:
a^2b^2 2ab a^2b^2 – 2ab ab^2So, the expression becomes:
(frac{var{2ab}}{var{ab^2}}) ( frac{2ab}{ab^2})Substituting ab 35x#39;#39;' (combination of previous steps) into the expression, we get:
(frac{2(35x#39;#39;)}{(35x#39;#39;)^2}) ( frac{7#39;#39;}{1225x#39;#39;^2}) ( frac{70}{1225x#39;#39;^1}) ( frac{14#39;#39;^1}{245#39;#39;^2}) ( frac{140}{2450}) ( frac{140 ÷ 70}{2450 ÷ 70}) ( frac{2}{35}) ( 1/35 Note that x#39;#39; gets cancelled out.Therefore, the simplified answer is 36.
Using Componendo and Dividendo
a/b 7/5 can be used in the identity:
( frac{var{ab}}{var{a-b}} frac{7n}{5n} ) (implies frac{ab}{a-b} frac{7}{5})Squaring both sides:
( left( frac{ab}{a-b} right)^2 left( frac{7}{5} right)^2 ) (implies left( frac{ab}{a-b} right)^2 frac{49}{25}) (implies ab/a-b^2 frac{49}{25} implies 36)Alternative Solution
Given: a/b 7/5
Expressing in terms of a common term:
( frac{var{a}var{b}^2var{a}var{b}}{[var{a}var{b}^2-2var{a}var{b}]} ) ( frac{var{a}var{b}var{a}var{b}}{var{a}var{b}(var{a}var{b}-2)} ) ( frac{var{ab}}{var{ab}-2} ) ( frac{7/5 cdot 5/7(7/5 cdot 5/7-1)}{1} )Simplifying further:
( frac{7/5 (7/5 - 1)}{1} ) ( frac{7/5 cdot 2/5}{1} ) ( frac{14/25}{1} ) ( frac{14/25}{1} ) ( 36 )Thus, the solution is 36.