Question

How do you solve the following system of equations $3x + 2y = 11$ and $x - 2 = - 4y$?

Answer 1

Hint: We will find the value of x from the second equation in terms of x and then put it in the first equation. After that we will get the value of y and then put that in the second equation to get the value of x.

Complete step-by-step solution:
We are given that we are required to solve the system of equations $3x + 2y = 11$ and $x - 2 = - 4y$.
We will use substitution to solve the same.
Let us term the given equation $3x + 2y = 11$ as the equation number 1 and the given equation $x - 2 = - 4y$ as equation number 2.
Consider $x - 2 = - 4y$:
Re – arranging the terms, we will get:-
$ \Rightarrow x = 2 - 4y$ ………………(3)
Putting this value in equation number 1, we will then obtain the following expression with us:-
$ \Rightarrow 3\left( {2 - 4y} \right) + 2y = 11$
Simplifying the left hand side of the above expression, we will then obtain the following equation:-
$ \Rightarrow 6 - 12y + 2y = 11$
Simplifying the left hand side of the above expression further, we will then obtain the following equation:-
$ \Rightarrow 6 - 10y = 11$
Taking 6 from addition in the left hand side of the above expression to subtraction in the right hand side, we will then obtain the following expression:-
$ \Rightarrow - 10y = 11 - 6$
Simplifying this further, we will then obtain the following equation:-
$ \Rightarrow - 10y = 5$
Thus, we get: $y = - \dfrac{1}{2}$
Putting this in equation number 3, we will then obtain the following equation:-
$ \Rightarrow x = 2 - 4\left( { - \dfrac{1}{2}} \right)$
Simplifying the calculations, we will then obtain the following equation:-
$ \Rightarrow x = 4$
Hence, the answer is $x = 4$ and $y = - \dfrac{1}{2}$.

Note: The students must note that you may use alternate methods for solving the equations other than using the substitution method.
Alternate Way:
We are given that we are required to solve $3x + 2y = 11$ …………(1) and $x - 2 = - 4y$ ………(2)
Multiplying the equation 2 by 3, we will then obtain the following equations respectively:
$ \Rightarrow 3x - 6 = - 12y$ ……………(3)
Subtracting the equation number 3 from equation number 1, we will then obtain the following equation:-
$ \Rightarrow \left\{ {3x + 2y} \right\} - \left\{ {3x - 6} \right\} = 11 - ( - 12y)$
Simplifying the equation, we will then obtain the following equation:-
$ \Rightarrow 2y + 6 = 11 + 12y$
Thus, we have $y = - \dfrac{1}{2}$
Therefore, by putting this in equation number 2, we get $x = 4$.
Hence, the answer is $x = 4$ and $y = - \dfrac{1}{2}$.