Question

How do solve the following system of equations: $ - 2x + y = 1$ and $ - 4x + y = - 1$?

Answer 1

Hint: Choose either of the two equations, say (i), and find the value of one variable, say $y$, in terms of the other, i.e., $x$. Then, substitute the value of $y$, obtained in the other equation, i.e., (ii) to get an equation in $x$ and solve the equation to get the value of $x$. Then, substitute the value of $x$ obtained in the expression for $y$ in terms of $x$ obtained to get the value of $y$. The values of $x$ and $y$ obtained constitute the solution of the given system of two linear equations.
Formula used:
Method of Elimination by Substitution:
In this method, we express one of the variables in terms of the other variable from either of the two equations and then this expression is put in the other equation to obtain an equation in one variable.

Complete step-by-step solution:
The given system of equations is
$ - 2x + y = 1$…(i)
$ - 4x + y = - 1$…(ii)
Choose either of the two equations, say (i), and find the value of one variable, say $y$, in terms of the other, i.e., $x$.
$y = 1 + 2x$
Substitute the value of $y$, obtained in the other equation, i.e., (ii) to get an equation in $x$.
$ - 4x + 1 + 2x = - 1$
Solve the equation to get the value of $x$.
$ - 2x = - 1 - 1$
$ \Rightarrow - 2x = - 2$
$\therefore x = 1$
Substitute the value of $x$ obtained in the expression for $y$ in terms of $x$ obtained to get the value of $y$.
$y = 1 + 2\left( 1 \right)$
$ \Rightarrow y = 3$
The values of $x$ and $y$ obtained constitute the solution of the given system of two linear equations.
Hence, the solution of the given system of equations is $x = 1$, $y = 3$.

Note: We can also find the solution of a given system by Method of Elimination by equating the coefficients.
Method of Elimination by equating the coefficients:
In this method, we eliminate one of the two variables to obtain an equation in one variable which can be easily solved. Putting the value of this variable in any one of the given equations, the value of another variable can be obtained.
Step by step solution:
The given system of equations is
$ - 2x + y = 1$…(i)
$ - 4x + y = - 1$…(ii)
We can eliminate the $y$ variable by subtracting the two equations.
So, subtract equation (ii) from (i), we get
$ \Rightarrow 2x = 2$
Now, divide both sides of the equation by $1$.
$\therefore x = 1$
Now, substitute the value of $x$ in equation (i) and find the value of $y$.
$y = 1 + 2\left( 1 \right)$
$ \Rightarrow y = 3$
Hence, the solution of the given system of equations is $x = 1$, $y = 3$.