Question

How do we solve the system $2x + 2y = 7$ and $x - 2y = - 1$ using substitution?

Answer 1

To solve this question, first we will pick one of the equations and work on it to keep aside any one variable at a side. And then substitute the value of that variable in another equation then we will get the value of the other equation. And finally substitute the value of the second variable in the first equation and that’s how we will get the values of both of the equations.

Step by step solution:-
Given equations:
$2x + 2y = 7$ ….eq(1)
$x - 2y = - 1$ ….eq(2)
To solve this system, by using substitution method, we will go through step by step as follows:-
Step-1 : According to the Substitution Method, first we will pick one equation and keep aside the value of $x$ or $y$ .
So, we are picking the equation(2):
$\therefore x - 2y = - 1$
To do $x$ alone in L.H.S, add $+ 2y$ in both the sides:
$\Rightarrow x - 2y + 2y = - 1 + 2y$
Now, in L.H.S, cancel out $- 2y$ and $+ 2y$ , we get:
$\because x = 2y - 1$
Now, we have the value: $x = 2y - 1$ .
Step-2 : Substitute $2y - 1$ for $x$ in the eq(1) and solve for $y$ while keeping the equation balanced.
$\therefore 2x + 2y = 7 \\\ \Rightarrow 2(2y - 1) + 2y = 7 \\\ \Rightarrow 4y - 2 + 2y = 7 \\\ \Rightarrow 6y = 7 + 2 \\\ \Rightarrow 6y = 9 \\\ \because y = \dfrac{9}{6} = \dfrac{3}{2} \\\$
Now, we have the value for $y$ is $\dfrac{3}{2}$ .
Step-3 : Substitute $\dfrac{3}{2}$ for $y$ in the Equation(2) and solve for $x$ :-
$\therefore x = 2y - 1 \\\ \Rightarrow x = 2(\dfrac{3}{2}) - 1 \\\ \Rightarrow x = 3 - 1 = 2 \\\$
Hence, the value of $x$ and $y$ are 2 and $\dfrac{3}{2}$ respectively.
Note:- We can also solve this question by the Elimination Method. In this method, we can work directly with both the equations and eliminate one of the variables to find the value of the other variable. Again, substitute the value of the found variable in any of the equations. And that’s how we will get the values of both the variables.