Question

How do you solve $2x + 2y = 3$ and $x - 4y = - 1$?

Answer 1

Hint: In this question, a linear equation of two variables is given. Here we will use the substitution method to solve these two linear equations. 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 + 2y = 3$......................(1)
 $ \Rightarrow x - 4y = - 1$ ………………....(2)
Let us subtract 2x on both sides.
$ \Rightarrow 2x + 2y - 2x = 3 - 2x$
That is equal to,
$ \Rightarrow 2y = 3 - 2x$
Let us divide both sides by 2.
$ \Rightarrow y = \dfrac{{3 - 2x}}{2}$
In the first step, on the other equation, substitute for the variable that we get from the first step.
 Substitute $y = \dfrac{{3 - 2x}}{2}$ in equation (2).
 $ \Rightarrow x - 4y = - 1$
$ \Rightarrow x - 4\left( {\dfrac{{3 - 2x}}{2}} \right) = - 1$
That is equal to,
$ \Rightarrow x - 2\left( {3 - 2x} \right) = - 1$
Now, let us remove the bracket.
$ \Rightarrow x - 6 + 4x = - 1$
Therefore,
$ \Rightarrow 5x - 6 = - 1$
Let us add 6 on both sides.
$ \Rightarrow 5x - 6 + 6 = - 1 + 6$
That is equal to,
 $ \Rightarrow 5x = 5$
Let us divide both sides by 5.
$ \Rightarrow \dfrac{{5x}}{5} = \dfrac{5}{5}$
So,
$ \Rightarrow x = 1$
Now, put the value of x in equation (1).
$ \Rightarrow 2x + 2y = 3$
Put the value of x is equal to 1.
$ \Rightarrow 2\left( 1 \right) + 2y = 3$
That is equal to
$ \Rightarrow 2 + 2y = 3$
Let us subtract 2 on both sides.
$ \Rightarrow 2 + 2y - 2 = 3 - 2$
So,
$ \Rightarrow 2y = 1$
Divide both sides by 2.
$ \Rightarrow \dfrac{{2y}}{2} = \dfrac{1}{2}$
So,
$ \Rightarrow y = \dfrac{1}{2}$

Hence, we find the value of $x$ is $1$ and the value of $y$ is $\dfrac{1}{2}$ using substitution methods.

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 + 2y = 3$
Put $x = 1$ and $y = \dfrac{1}{2}$ .
 $ \Rightarrow 2\left( 1 \right) + 2\left( {\dfrac{1}{2}} \right) = 3$
That is equal to,
$ \Rightarrow 2 + 1 = 3$
So,
$ \Rightarrow 3 = 3$
Now, let us take equation (2) and put the values.
 $ \Rightarrow x - 4y = - 1$
Put $x = 1$ and $y = \dfrac{1}{2}$ .
 $ \Rightarrow 1 - 4\left( {\dfrac{1}{2}} \right) = - 1$
That is equal to,
$ \Rightarrow 1 - 2 = - 1$
So,
$ \Rightarrow - 1 = - 1$