Question

Solve the linear equation \[\dfrac{{3y - 2}}{{2y + 4}} = \dfrac{7}{6}\]

Answer 1

Hint: In this, we need to solve for ‘y’. First, we cross multiply it and we simplify it. After simplification we will have a simple linear equation with one variable, we solve this using the transposition method. That is by shifting or grouping the variable terms on one side of the equation and constants on the other side of the equation.

Complete step-by-step solution:
Given, \[\dfrac{{3y - 2}}{{2y + 4}} = \dfrac{7}{6}\].
Cross multiplying it we have,
\[6\left( {3y - 2} \right) = 7\left( {2y + 4} \right)\]
Now expanding the brackets we have,
\[18y - 12 = 14y + 28\]
We transpose $‘-12’$ which is present in the left-hand side of the equation to the right-hand side of the equation by adding ‘12’ on the right-hand side of the equation.
\[18y = 14y + 28 + 12\]
Similarly, we transpose 14y to the left-hand side of the equation by subtracting 14y on the left-hand side of the equation.
\[18y - 14y = 28 + 12\]
Thus we have separated the variables and the constants. Now we can apply the mathematical operation.
\[4y = 40\]
Divide the whole equation by 4
\[y = \dfrac{{40}}{4}\].
\[ \Rightarrow y = 10\]. This is the required result.

Note: We can check whether the obtained solution is correct or wrong. All we need to do is substituting the value of ‘y’ in the given problem.
\[\dfrac{{3y - 2}}{{2y + 4}} = \dfrac{7}{6}\]
\[\dfrac{{3\left( {10} \right) - 2}}{{2\left( {10} \right) + 4}} = \dfrac{7}{6}\]
\[\dfrac{{30 - 2}}{{20 + 4}} = \dfrac{7}{6}\]
\[\dfrac{{28}}{{24}} = \dfrac{7}{6}\]
Simplifying we have,
\[ \Rightarrow \dfrac{7}{6} = \dfrac{7}{6}\].
That is LHS=RHS. Hence the obtained is correct.
In the above, we did the transpose of addition and subtraction. 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.