Question

How do you solve $16x + 9 = 9y - 2x$ for y ?

Answer 1

Hint: In this question, we need to solve for the variable y in the given equation. i.e. we need to isolate the term y and find an expression in terms of x. We make use of some mathematical operations to do this. Firstly, we try to keep the terms containing y on L.H.S. of the equation and take all the other terms which do not contain y to the R.H.S. Then in R.H.S. simplify the terms to obtain the solution and so we obtain an expression for the variable y.

Complete step by step solution:
Given an equation of the form $16x + 9 = 9y - 2x$ ……(1)
Here it is mentioned that we need to solve for the variable y. i.e. we need to find an expression for y.
We try to do this by taking the terms which do not contain the variable y to the other side and simplify it.
Firstly, we keep the terms only containing y in L.H.S. of the equation.
So subtract $9y$ on both sides of the equation (1), we get,
$ \Rightarrow 16x + 9 - 9y = 9y - 2x - 9y$
Combining like terms and simplifying we get,
$ \Rightarrow 16x + 9 - 9y = 9y - 9y - 2x$
$ \Rightarrow 16x + 9 - 9y = 0 - 2x$
$ \Rightarrow 16x + 9 - 9y = - 2x$
Move all terms which do not contain the variable y to R.H.S.
So we subtract $16x$ and 9 on both sides of the equation we get,
$ \Rightarrow 16x + 9 - 9y - 16x - 9 = - 2x - 16x - 9$
Rearranging the terms, we get,
$ \Rightarrow 16x - 16x + 9 - 9 - 9y = - 2x - 16x - 9$
 Combining the like terms $16x - 16x = 0$
Combining the like terms $9 - 9 = 0$
Combining the like terms $ - 2x - 16x = - 18x$
Hence we get,
$ \Rightarrow 0 + 0 - 9y = - 18x - 9$
$ \Rightarrow - 9y = - 18x - 9$
Dividing by 9 on both sides, we get,
$ \Rightarrow \dfrac{{ - 9y}}{9} = \dfrac{{ - 18x - 9}}{9}$
Simplifying this we get,
$ \Rightarrow - y = - \dfrac{{18}}{9}x - \dfrac{9}{9}$
$ \Rightarrow - y = - 2x - 1$
Multiplying on both sides by -1, we get,
$ \Rightarrow ( - 1)( - y) = ( - 1)( - 2x - 1)$
So finally we obtain the expression as,
$ \Rightarrow y = 2x + 1$

Hence the solution for y in the equation $16x + 9 = 9y - 2x$ is given by $y = 2x + 1$.

Note: Since it is mentioned to solve for the variable y, we found the expression for the variable y. In a similar manner we can also find an expression for the variable x.
We can verify whether the solution we obtained is correct or not by substituting back the value of y in the given equation. If the equation satisfies, then the obtained value of is the required solution for a given problem.
It is important to know the following basic facts.
An equation remains unchanged or undisturbed if its satisfies the following conditions.
(1) If L.H.S. and R.H.S. are interchanged.
(2) If the same number is added on both sides of the equation.
(3) If the same number is subtracted on both sides of the equation.
(4) When both L.H.S. and R.H.S. are multiplied by the same number.
(5) When both L.H.S. and R.H.S. are divided by the same number.