Question

How do you solve this system of equations $2x + y = - 2$ and $5x + 3y = - 8$ ?

Answer 1

First of all, find the value of y from the first equation and then put it in the second equation. After that, the value of x will be obtained and thus putting that in equation 1, we have the value of y.

Complete step-by-step solution:
We are given that we are required to solve the system of equations $2x + y = - 2$ and $5x + 3y = - 8$.
Let us terms the given equation $2x + y = - 2$ as equation number 1 and the given equation $5x + 3y = - 8$ as equation number 2.
Consider $2x + y = - 2$:
Re-arranging the terms, we will get:-
$\Rightarrow y = - 2 - 2x$ ………………(3)
Putting this value in equation number 2, we will then obtain the following expression with us:-
$\Rightarrow 5x + 3( - 2 - 2x) = - 8$
Simplifying the left hand side of the above expression, we will then obtain the following equation:-
$\Rightarrow 5x - 6 - 6x = - 8$
Simplifying the left hand side of the above expression further, we will then obtain the following equation:-
$\Rightarrow - x - 6 = - 8$
Taking 6 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:-
$\Rightarrow - x = - 8 + 6$
Simplifying this further, we will then obtain the following equation:-
$\Rightarrow x = 2$
Thus, we get: $x = 2$
Putting this in equation number 3, we will then obtain the following equation:-
$\Rightarrow y = - 2 - 2(2)$
Simplifying the calculations, we will then obtain the following equation:-
$\Rightarrow y = - 6$
Hence, the answer is $x = 2$ and $y = - 6$.

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 $2x + y = - 2$ …………(1) and $5x + 3y = - 8$ ………(2)
Multiplying the equation 1 by 3, we will then obtain the following equations respectively:
$\Rightarrow 6x + 3y = - 6$ ……………(3)
Subtracting the equation number 3 from equation number 2, we will then obtain the following equation:-
\Rightarrow \left\\{ {5x + 3y} \right\\} - \left\\{ {6x + 3y} \right\\} = - 8 + 6
Simplifying the equation, we will then obtain the following equation:-
$\Rightarrow - x = - 2$
Thus, we have $x = 2$
Therefore, by putting this in equation number 1, we get $y = - 6$.
Hence, the answer is $x = 2$ and $y = - 6$.