Question

How do you solve the system $x + 2y = 13$ and $ - 2x - 3y = - 18$ using substitution?

Answer 1

Hint: We will find the value of x from the first equation and then put it in the second equation. After that we will get the value of y and then put that in the first 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 $x + 2y = 13$ and $ - 2x - 3y = - 18$ using substitution.
Let us terms the given equation $x + 2y = 13$ as equation number 1 and the given equation $ - 2x - 3y = - 18$ as equation number 2.
Consider $x + 2y = 13$:
Re – arranging the terms, we will get:-
$ \Rightarrow x = 13 - 2y$ ………………(3)
Putting this value in equation number 2, we will then obtain the following expression with us:-
$ \Rightarrow - 2\left( {13 - 2y} \right) - 3y = - 18$
Simplifying the left hand side of the above expression, we will then obtain the following equation:-
$ \Rightarrow - 26 + 4y - 3y = - 18$
Simplifying the left hand side of the above expression further, we will then obtain the following equation:-
$ \Rightarrow - 26 + y = - 18$
Taking 26 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 y = 26 - 18$
Simplifying this further, we will then obtain the following equation:-
$ \Rightarrow y = 8$
Thus, we get: $y = 8$
Putting this in equation number 3, we will then obtain the following equation:-
$ \Rightarrow x = 13 - 2(8)$
Simplifying the calculations, we will then obtain the following equation:-
$ \Rightarrow x = - 3$
Hence, the answer is $x = - 3$ and $y = 8$.

Note: The students must note that you may use alternate methods for solving the equations other than using the substitution method if not mentioned in the question.
Alternate Way:
We are given that we are required to solve $x + 2y = 13$ …………(1) and $ - 2x - 3y = - 18$ ………(2)
Multiplying the equation 1 by 3 and 2 by 2, we will then obtain the following equations respectively:
$ \Rightarrow 3x + 6y = 39$ ……………(3)
$ \Rightarrow - 4x - 6y = - 36$ ……………(4)
Adding the equation number 4 and equation number 3, we will then obtain the following equation:-
$ \Rightarrow \left\{ {3x + 6y} \right\} + \left\{ { - 4x - 6y} \right\} = 39 - 36$
Simplifying the equation, we will then obtain the following equation:-
$ \Rightarrow - x = 3$
Thus, we have $x = - 3$
Therefore, by putting this in equation number 1, we get $y = 8$.
Hence, the answer is $x = - 3$ and $y = 8$.