Question

How do you solve $3x - y = 9$ and $2x + y = 1$ ?

Answer 1

Hint: In order to solve this question, we solve the two equations simultaneously. We add both the equations to cancel out the like variable and make the equation homogeneous. Once we have added both the equations, we cancel out $y$ and add the values of the $x$ variable. Then once we have found the value of the variable $x$ , we solve further to get the value of the variable $y$.

Complete Step by Step Solution:
In this question, we are given two equations which we need to solve in order to find the value of the variables $x$ and $y$. In order to do so, we need to solve the two equations simultaneously.
The two equations are given as-
$3x - y = 9$…………………… equation (1)
 $2x + y = 1$………………….. equation (2)
First, we need to adjust any one equation in such a way so that the coefficient of one of its variable matches with the coefficient of the same variable of the other equation, thus they get canceled out when we add or subtract, depending on the sign.
Here we find that variable $y$ has the same coefficient in both the equations, i.e $1$ and since the signs are different, we simply add the two equations to cancel out $y$
On adding equation (1) and (2), we get:
\[\begin{array}{*{20}{c}}
  {3x - y = 9} \\
  {\underline {2x + y = 1} } \\
  {\,\,\,5x{\text{ }}\,\,\,\,\,{\text{ = 10}}}
\end{array}\]
Thus,
$ \Rightarrow 5x = 10$
$ \Rightarrow x = 2$
Substituting the value of $x$ , in equation (1), we get:
$ \Rightarrow 3\left( 2 \right) - y = 9$
$ \Rightarrow 6 - y = 9$
On adding $ - 6$ to both sides, we get:
$ \Rightarrow - y = 3$
$ \Rightarrow y = - 3$

Thus, we have found our required values of $x=2$ and $y=-3$.

Note: An alternate way of doing this question is – by the method of substitution
The two equations are given as:
$3x - y = 9$…………………… equation (1)
 $2x + y = 1$………………….. equation (2)
We can also write equation (2) as:
$ \Rightarrow y = 1 - 2x$
On substituting this value of $y$ in equation (1), we get:
$ \Rightarrow 3x - \left( {1 - 2x} \right) = 9$
Bow our equation has become homogeneous, i.e. it contains only one variable, thus we can solve it easily.
$ \Rightarrow 3x - 1 + 2x = 9$
$ \Rightarrow 5x - 1 = 9$
On adding $ + 1$ to both the sides, we get:
$ \Rightarrow 5x = 10$
$ \Rightarrow x = 2$
Placing this value of $x$ in $y = 1 - 2x$, we find the value of $y$ :
$ \Rightarrow y = 1 - 2\left( 2 \right)$
$ \Rightarrow y = - 3$
Hence we get the required answer.