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How do you solve \[\dfrac{b}{{13}} - \dfrac{{3b}}{{13}} = \dfrac{8}{{13}}\]?

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Last updated date: 13th Jun 2024
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Answer
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Hint:In the given problem we need to solve this for ‘b’. We can solve this using the transposition method. The common transposition method is to do the same thing (mathematically) to both sides of the equation, with the aim of bringing like terms together and isolating the variable (or the unknown quantity). That is we group the ‘b’ terms one side and constants on the other side of the equation.

Complete step by step solution:
Given, \[\dfrac{b}{{13}} - \dfrac{{3b}}{{13}} = \dfrac{8}{{13}}\].
We first simplify the left hand side of the equation by taking LCM,
\[\dfrac{{b - 3b}}{{13}} = \dfrac{8}{{13}}\]
\[\dfrac{{ - 2b}}{{13}} = \dfrac{8}{{13}}\]
We transpose ‘13’ which is present in the left hand side of the equation to the right hand side of the equation by multiplying ‘13’ on the right hand side of the equation.
\[
- 2b = \dfrac{8}{{13}} \times 13 \\
- 2b = 8 \\
\]
We transpose -2 to the right hand side by dividing -2 on the right hand side of the equation we have,
\[b = \dfrac{8}{{ - 2}}\]
\[ \Rightarrow b = - 4\] This is the required answer.

Note: We can check whether the obtained solution is correct or wrong. All we need to do is substituting the value of ‘b’ in the given problem.
\[\dfrac{{ - 4}}{{13}} - \dfrac{{3( - 4)}}{{13}} = \dfrac{8}{{13}}\]
\[\dfrac{{ - 4}}{{13}} + \dfrac{{12}}{{13}} = \dfrac{8}{{13}}\]
Taking LCM we have,
\[\dfrac{{ - 4 + 12}}{{13}} = \dfrac{8}{{13}}\]
\[ \Rightarrow \dfrac{8}{{13}} = \dfrac{8}{{13}}\] . Hence the obtained answer is correct.
If we want to transpose a positive number to the other side of the equation we subtract the same number on that side (vice versa). Similarly if we have multiplication we use division to transpose. If we have division we use multiplication to transpose. Follow the same procedure for these kinds of problems.