Question

How do you solve \[\dfrac{y}{2}=\dfrac{y}{3}-1\]?

Answer 1

Hint: Any equation can be solved by taking all the constants to one side and all the unknowns to the other side of the equation. The constant side must be solved step-by-step to get through the solution. We can use the distributive property and do the addition, subtraction, multiplication and division operations wherever necessary in such a way to simplify the equation.

Complete step by step answer:
As per the given question, we are provided with an equation which is to be simplified to get the solution of the equation. A solution is that which when substituted back into the equation, both the sides of the equation will be equal. Here, the given equation is \[\dfrac{y}{2}=\dfrac{y}{3}-1\].
In the given equation, we have fractions. To get rid of these fractions for better simplification, we have to multiply both sides of the equation by the LCM (least common multiple) of the denominators of the fractions.
Here, we have to find the LCM of 2 and 3. As we know that LCM of 2 and 3 is 6, we multiply both the sides of the equation by 6. Then, we get
\[\Rightarrow 6\times \dfrac{y}{2}=6\times \left( \dfrac{y}{3}-1 \right)\to 3y=2y-6\]
Now, we have isolated y by subtracting \[2y\] from both sides of the equation. Then, we get
\[\Rightarrow 3y-2y=2y-6-2y\]
As we know that the subtraction of \[2y\] from \[3y\] is \[y\] and that of \[2y\] from \[2y\] is zero, we can rewrite the equation as
\[\Rightarrow y=0-6\to y=-6\]

\[\therefore y=-6\] is the required solution of \[\dfrac{y}{2}=\dfrac{y}{3}-1\].

Note: We can rather solve the given equation by shifting \[\dfrac{y}{3}\] to the left-hand side of the equation and then simplifying on the left-hand side to get the coefficient of the variable y. And, dividing the right-hand side constant with the coefficient of variable y, we can get the solution as \[\dfrac{y}{2}=\dfrac{y}{3}-1\to y\left( \dfrac{1}{6} \right)=-1\to y=-6\]. This is a three-step solution which involves only three steps to go towards the solution. We should avoid calculation mistakes to get the correct solution.