Question

How do you solve \[3y\left( {4y - 1} \right) - 2y\left( {6y - 5} \right) = 9y - 8\left( {3 + y} \right)\]?

Answer 1

Hint: Here in the question, we have to solve the given equation with respect to the variable ‘y’ by using the transposition method. To solve this first we have to remove the parentheses by multiplying the outside term into the parenthesis and next we have to isolate the variable ‘y’ by using the simple arithmetic operation and on further simplification we get the required solution.

Complete step by step answer:
Consider the given variable equation:
\[ \Rightarrow 3y\left( {4y - 1} \right) - 2y\left( {6y - 5} \right) = 9y - 8\left( {3 + y} \right)\]-------(1)
Here, y is the variable.
we need to solve this for ‘y’. 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 ‘y’ terms one side and constants on the other side of the equation.
Now, remove the parentheses in equation (1) by multiplying the outside terms
\[ \Rightarrow 12{y^2} - 3y - 12{y^2} + 10y = 9y - 24 - 8y\]
On simplification we get
\[ \Rightarrow 7y = y - 24\]
Subtract both side by y, then
\[ \Rightarrow 7y - y = y - 24 - y\]
\[ \Rightarrow 6y = - 24\]
Divide both side by 6
\[ \Rightarrow y = \dfrac{{ - 24}}{6}\]
On simplification, we get
\[ \Rightarrow y = - 4\]
Hence, the required solution is \[y = - 4\].
We can check whether the obtained solution is correct or wrong. All we need to do is substituting the value of ‘x’ in the given problem.
\[ \Rightarrow 3\left( { - 4} \right)\left( {4\left( { - 4} \right) - 1} \right) - 2\left( { - 4} \right)\left( {6\left( { - 4} \right) - 5} \right) = 9\left( { - 4} \right) - 8\left( {3 + \left( { - 4} \right)} \right)\]
\[ \Rightarrow - 12\left( { - 16 - 1} \right) + 8\left( { - 24 - 5} \right) = - 36 - 8\left( {3 - 4} \right)\]
\[ \Rightarrow - 12\left( { - 17} \right) + 8\left( { - 29} \right) = - 36 - 8\left( { - 1} \right)\]
\[ \Rightarrow 204 - 232 = - 36 + 8\]
\[ \Rightarrow - 28 = - 28\]
Hence the obtained answer is correct.

Note: We know that the product of two negative numbers is a positive number. Product of a negative number and a positive number gives negative number (vice versa). 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.