Question

Use elimination method for solving the following equations: $2x + y = 7,2x - 3y = 3$.

Answer 1

Hint: In elimination method, we eliminate one of the variables using the given pair of equations. For that, multiply the equations with suitable numbers so as to make the coefficients same. If the coefficients are of the same sign, then subtract one from another and if they are of different sign, add both of them. Thus we get an equation with a single variable. Solving, we get the value of that variable. Substituting this value in either of the equations, we get the value of the next variable.

Complete step-by-step answer:
Given two equations: $2x + y = 7,2x - 3y = 3$
We can see that the coefficient of variable $x$ is $2$ in both equations. So we can eliminate the variable $x$.
Consider$
  2x + y = 7 - - - - (i) \\
  2x - 3y = 3 - - - (ii) \\
\ $
Subtracting equation $(ii)$ from equation $(i)$, we have
$\Rightarrow$$2x + y - (2x - 3y) = 7 - 3$
Simplifying we get,
$\Rightarrow$$2x - 2x + y - - 3y = 4$
$ \Rightarrow y + 3y = 4$
So we have, $4y = 4$
Dividing both sides by $4$ we get,
$\Rightarrow$$y = \dfrac{4}{4} = 1$
Now we got the value of one of the variables.
Using this we can find the value of $x$.
For that substitute $y = 1$ in equation $(i)$.
We have,
$\Rightarrow$$2x + 1 = 7$
Subtracting $1$ from both sides we get,
$\Rightarrow$$2x + 1 - 1 = 7 - 1$
Simplifying we get,
$\Rightarrow$$2x = 6$
Dividing both sides by $2$ we get,
$\Rightarrow$$x = \dfrac{6}{2} = 3$
So we have the solution for the pair of equations.
$\Rightarrow$$x = 3,y = 1$

Therefore the value of x is 3 and the value of y is 1.

Note: Here we have eliminated $x$ using the two equations. Similarly we can eliminate $y$ also. For that, we have to make the coefficients of $y$ same in two equations. Now the first equation has a coefficient of $y$ as $1$ and the second equation has a coefficient of $y$ as $3$.
Multiply equation $(i)$ by $3$. We get,
$6x + 3y = 21 - - - (iii)$
Now add equations $(ii)$ and $(iii)$. We get,
$2x - 3y + 6x + 3y = 3 + 21$
$ \Rightarrow 8x = 24$
Dividing by $8$ we get, $x = 3$.
Substituting in equation $(i)$ we get,
$6 + y = 7$
Simplifying we get, $y = 1$