Question

If $2y + \dfrac{5}{3} = \dfrac{{26}}{3} - y$ , then $y = $ ?
(A) $1$
(B) $\dfrac{2}{3}$
(C) $\dfrac{6}{5}$
(D) $\dfrac{7}{3}$

Answer 1

Hint:
Consider the given equation and try to solve it using transformation. Start with losing the denominator by multiplying the equation with the same number. Now transpose the variable terms and constant on different sides of the equation. Perform algebraic operations to get the required value.

Complete step by step solution:
Here in this problem, we are given an equation in a single variable ‘y’ which is $2y + \dfrac{5}{3} = \dfrac{{26}}{3} - y$ . And using this equation, we need to find the value of the unknown variable ‘y’. There are four options given, and we have to find one correct option from it.
The given equation can be solved by using some easy transformations and manipulations in the equation and thus the unknown will have a constant value.
Let’s start with multiplying the whole equation by $3$ to remove the denominator in some expressions.
$ \Rightarrow 2y + \dfrac{5}{3} = \dfrac{{26}}{3} - y \Rightarrow 6y + 5 = 26 - 3y$
Now we can transpose $ - 3y$ from the right-hand side to the left side, which reverses its sign
$ \Rightarrow 6y + 5 = 26 - 3y \Rightarrow 6y + 3y + 5 = 26$
Similarly, we can transpose $5$ from the left-hand side to the right-hand side, and it will change its sign to negative again. This way we get the terms with a variable on the left side and constants on the right side
\[ \Rightarrow 6y + 3y + 5 = 26 \Rightarrow 6y + 3y = 26 - 5\]
This can be easily solved by simplifying both sides separately. On the left side, we can take the variable ‘y’ common in both the terms and then subtract the coefficients
\[ \Rightarrow 6y + 3y = 26 - 5 \Rightarrow \left( {6 + 3} \right)y = 21 \Rightarrow 9y = 21\]
Let us now divide both sides of the equation by the number $9$ . This will give us:
\[ \Rightarrow \dfrac{{9y}}{9} = \dfrac{{21}}{9} \Rightarrow y = \dfrac{{7 \times 3}}{{3 \times 3}} \Rightarrow y = \dfrac{7}{3}\]
Therefore, we solved the given equation and found the only unknown as $y = \dfrac{7}{3}$

Hence, the option (D) is the correct answer.

Note:
In questions like this, remember that the one linear equation with one variable is always enough to solve it alone and find the value of the involved variable. You just need to perform some transformations in the equation to simplify the variable. The technique of transposing one term to another side is just using opposite operations on both sides with the same term. For example, in an equation $x + a = 3$ transposing ‘a’ from the left side to the right means subtracting ‘a’ from both side, i.e. $x + a = 3 \Rightarrow x + a - a = 3 - a \Rightarrow x = 3 - a$