Question

How do you solve the following system: $3x - y = 12,5x + 2y = 20$ ?

Answer 1

Hint:
We will first find the value of $y$ from $3x - y = 12$ and then put that value of y in $5x + 2y = 20$ and thus, we get an equation in $x$ which can be solved for $x$ and thus we have the value of $y$ as well.

Complete step by step solution:
We are given that we are required to solve the following system: $3x - y = 12,5x + 2y = 20$.
Let us assume $3x - y = 12$ to be equation number 1 and $5x + 2y = 20$ to be equation number 2.
Now, consider equation number 1 which is: $3x - y = 12$.
Now, taking $3x$ from addition in the left hand side to subtraction in right hand side, we will then obtain the following equation with us:-
$ \Rightarrow - y = 12 - 3x$
Multiplying the above equation by – 1, we will then obtain the following equation with us:-
$ \Rightarrow y = - 12 + 3x$ ……………(3)
Putting this value of $y$ in equation number 2, we will then obtain the following equation with us:-
$ \Rightarrow 5x + 2\left( { - 12 + 3x} \right) = 20$
Now, simplifying the brackets in the above equation, we will then obtain the following equation:-
$ \Rightarrow 5x - 24 + 6x = 20$
Now, simplifying the left hand side in the above equation, we will then obtain the following equation:-
$ \Rightarrow 11x - 24 = 20$
Taking 24 from subtraction in the left hand side to addition in the right hand side, we will then obtain the following equation:-
$ \Rightarrow 11x = 44$
Thus, we have x = 4.
Putting this in equation number 3, we will then obtain the following equation:-
$ \Rightarrow y = - 12 + 3 \times 4$
Simplifying the right hand side in the above equation, we will then obtain the following equation:-
$ \Rightarrow y = 0$

Hence, the answer is $x = 4$ and $y = 0$.

Note:
The students must note that there is an alternate way to solve the system of equations given as follows:-
Let us assume $3x - y = 12$ to be equation number 1 and $5x + 2y = 20$ to be equation number 2.
Multiplying the equation number 1 by 2, we will then obtain the following equation:-
$ \Rightarrow 6x - 2y = 24$ …………….(3)
Adding the equation number 2 and 3, we will then obtain the following equation:-
$ \Rightarrow \left( {5x + 2y} \right) + \left( {6x - 2y} \right) = 20 + 24$
Simplifying the above equation, we will then obtain the following equation:-
$ \Rightarrow 11x = 44$
Thus, we have x = 4.