Question

Solve the following equation: \[\dfrac{{\left( {3y + 4} \right)}}{{\left( {2 - 6y} \right)}} = - \dfrac{2}{5}\]

Answer 1

Hint: To solve the given equation we will first cross multiply. Then we will simplify the equation obtained on cross multiplication. Since the given equation has only one variable we will find the value of the variable by taking constant terms to one side of the equation and by further simplification.

Complete step by step solution:
Given equation is \[\dfrac{{\left( {3y + 4} \right)}}{{\left( {2 - 6y} \right)}} = - \dfrac{2}{5}\]
On cross multiplying we get
\[ \Rightarrow 5 \times \left( {3y + 4} \right) = - 2 \times \left( {2 - 6y} \right)\]
On multiplying and simplifying both the sides we get
\[ \Rightarrow \left( {15y + 20} \right) = \left( { - 4 + 12y} \right)\]
We know that an equation remains valid if we add or subtract the same number both in the Left Hand Side (LHS) and Right Hand Side (RHS).
Therefore, we should subtract a number from both sides such that only the term containing \[y\] remains on the Left Hand Side. Since, \[20\] is present on the LHS we will subtract both the sides by \[20\], so get
\[ \Rightarrow 15y + 20 - 20 = - 4 + 12y - 20\]
On simplifying we get,
\[ \Rightarrow 15y = 12y - 24\]
Now, we will take \[12y\] from Right Hand Side of the equation to the Left Hand Side, therefore we get
\[ \Rightarrow 15y - 12y = - 24\]
On solving we get
\[ \Rightarrow 3y = - 24\]
As we know that equation will remain valid if we divide both the Left Hand Side (LHS) and Right Hand Side (RHS) by the same number.
Therefore, on dividing both the sides by \[3\] we get
\[ \Rightarrow y = \dfrac{{ - 24}}{3}\]
On simplifying, we get
\[ \Rightarrow y = - 8\]
Putting this obtained value of \[y\] in LHS of the given equation we get \[\dfrac{{\left( {3 \times \left( { - 8} \right) + 4} \right)}}{{\left( {2 - 6 \times \left( { - 8} \right)} \right)}}\] i.e., \[ - \dfrac{2}{5}\], which is the RHS of this given equation. So, the obtained value of \[y\] is correct.
So, the correct answer is “ \[y = - 8\]”.

Note: Here, we have only one variable i.e., \[y\]. In these types of questions if there are \[n\] variables in an equation then there should be a minimum of \[n\] different equations, to get the value of all the variables. By substituting the values of variables in different equations we can get the values of different variables.