Question

Solve
$3x-2y\ =\ 1$
$2x+y\ =3$

Answer 1

Hint: Equalise the coefficient of $x$ or $y$ of both the given equations.
Multiply equation $\left( 1 \right)$ by $2$ and equation $\left( 2 \right)$ by $3$, then by subtracting the equation determine the value of $''y''$ and the value of $''x''$

Complete step-by-step solution:
As per question,
We have,
$3x-2y\ =1...\left( 1 \right)$
And, $2x+y\ =3...\left( 2 \right)$
Here, as in above both equation maximum power of variables is $1,$ So, the both of the questions are linear equations, and due to presence of two $x$ and $y$, it is $1$ linear equation with two variables.
Here,
For determining the solution of given equation,
We will make any one of two variables same in both thee given equation,
Like here in equation $\left( 1 \right)$ the coefficient of $x$ is $3$ and in second equation the coefficient of $x$ is $2$,
So, in order to equalise the value of coefficient we will multiply first equation by $2$ and second equation by $3$,
The obtained equation will be,
$2(3x-2y)\ =2\times 1...\left( 1 \right)$
$3\left( 2x+y \right)\ =\ 3\times 3...\left( 4 \right)$
$\Rightarrow \ 6x-4y\ =2...\left( a \right)$
$\And \ 6x+3y\ =\ 9...\left( b \right)$
Now subtracting $\left( b \right)$ from $\left( a \right)$ ,
$\left( 6x-4x\ =2 \right)-\left( 6x+3y\,=9 \right)$
$-7y\ =\ -7$
$\Rightarrow \ y\ =\ \dfrac{7}{7}\ =1$
Hence, we got value of $y=1,$
Now putting the value of $y=1,$ in any of equation $(1)\ \And \left( 2 \right),$
So, putting the value of $y=1$ in $\left( 1 \right)$ ,
We will get,
$3x-2y\,=1$
$\Rightarrow 3x-\left( 2\times 1 \right)\,=1$
$\Rightarrow 3x-2\ =1$
$\Rightarrow 3x\ =3$
$\Rightarrow x\ =\dfrac{3}{3}$
$\Rightarrow x\ =1$

Hence, value of $\left( x,y \right)$ will be $\left( 1,1 \right)$

Additional Information: (1) We can also solve the above question by substituting method,
Like, $3x-2y\ =1$
$\Rightarrow 3x\,=1+2y\ $
$\Rightarrow x\ =\ \dfrac{1+2y}{3}-\left( a \right)$
And then putting the value of $''x''$ obtained in second equation,
As,
$2x+y\ =3$
$\Rightarrow 2\times \left( \dfrac{1+2y}{3} \right)+y=3$
$\Rightarrow \dfrac{2+4y}{3}+y=3$
$\Rightarrow \dfrac{2+4y+3y}{3}=3$
$\Rightarrow 2+7y=9$
$\Rightarrow 7y=9-2$
$\Rightarrow 7y=7$
$\Rightarrow y=\dfrac{7}{7}=1$
Hence, $x=\ \dfrac{1+2y}{3}=\dfrac{1+\left( 2\times 1 \right)}{3}=\dfrac{1+2}{3}=1.$

Note: Here, during making the coefficient of variable equal we can consider any of two variable, we can either equalize the coefficient of $x$ in both equation or, coefficient of $'y'$ in both equation
(2) If we multiply/divide the each value of the equation by a same number then the value of the equation doesn’t affect it.