Question

How do you solve the system of equation $2x + y = 2$ and $x - 3y = - 27$?

Answer 1

Hint: In this question, a linear equation of two variables is given. And we have to solve these two linear equations with the help of the substitution method. To solve the equations using the substitution method, we should follow the below steps:
Select one equation and solve it for one of its variables.
On the other equation, substitute for the variable that we get from the first step.
Solve the new equation.
Substitute the value that we found into any equation involving both variables and solve for the other variable.
Check the solution in both original equations.

Complete step by step solution:
Here, we want to solve the equations using the substitution method.
$ \Rightarrow 2x + y = 2$ ...(1)
$ \Rightarrow x - 3y = - 27$ ...(2)
The first step is to select one equation and solve it for one of its variables.
Let us take equation (1), it is already in the form of x variable.
 $ \Rightarrow 2x + y = 2$
Let us subtract 2x on both sides.
$ \Rightarrow 2x - 2x + y = 2 - 2x$
That is equal to,
$ \Rightarrow y = 2 - 2x$
In the second step, on the other equation, substitute for the variable that we get from the first step.
 Substitute $y = 2 - 2x$ in equation (2).
$ \Rightarrow x - 3y = - 27$
$ \Rightarrow x - 3\left( {2 - 2x} \right) = - 27$
That is equal to,
$ \Rightarrow x - 6 + 6x = - 27$
Let us add the left-hand side.
$ \Rightarrow 7x - 6 = - 27$
Add 6 on both sides.
$ \Rightarrow 7x - 6 + 6 = - 27 + 6$
That is equal to,
$ \Rightarrow 7x = - 21$
Now, divide by 7 into both sides.
$ \Rightarrow \dfrac{{7x}}{7} = \dfrac{{ - 21}}{7}$
So,
$ \Rightarrow x = - 3$
Now, put the value of x in equation (1).
$ \Rightarrow 2x + y = 2$
Put the value of x is equal to -3.
$ \Rightarrow 2\left( { - 3} \right) + y = 2$
That is equal to
$ \Rightarrow - 6 + y = 2$
Add 6 on both sides.
 $ \Rightarrow - 6 + 6 + y = 2 + 6$
That is equal to,
$ \Rightarrow y = 8$

Hence, we find the value of x is -3 and the value of y is 8.

Note:
To check whether our answer is correct or not, feed the x and y values in each equation.
Let us take equation (1) and put the values.
$ \Rightarrow 2x + y = 2$
put $x = - 3$ and $y = 8$.
$ \Rightarrow 2\left( { - 3} \right) + 8 = 2$
That is equal to,
$ \Rightarrow - 6 + 8 = 2$
Therefore,
 $ \Rightarrow 2 = 2$
Now, let us take equation (2) and put the values.
$ \Rightarrow x - 3y = - 27$
put $x = - 3$ and $y = 8$.
$ \Rightarrow - 3 - 3\left( 8 \right) = - 27$
That is equal to,
$ \Rightarrow - 3 - 24 = - 27$
Therefore,
$ \Rightarrow - 27 = - 27$