Question

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

Answer 1

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

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

Hence, the answer is $x = 3$ and $y = 1$.

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 $x + y = 4$ …………(1) and $x - y = 2$ ………(2)
Adding the equation number 1 and equation number 2, we will then obtain the following equation:-
$ \Rightarrow \left\{ {x + y} \right\} + \left\{ {x - y} \right\} = 4 + 2$
Simplifying the equation, we will then obtain the following equation:-
$ \Rightarrow 2x = 6$
Thus, we have $x = 3$
Therefore, by putting this in equation number 2, we get $y = 1$.
Hence, the answer is $x = 3$ and $y = 1$