Question

How do you solve $2x - 8y = 8$ and $ - x + 4y = 4$

Answer 1

Hint: We are given a pair of linear equations and we have to find the value of x and y by using the given equation. We can solve the equation by the method of elimination or by using the method of substitution for the method of substitution. First we will find the value of one variable in the form of another for example we will find the value of x in terms of y then substitute that value in another equation. Then we will solve the equation and find the value of that variable. After that, substitute the value of that variable in the other equation and find the value of the remaining one variable.

Complete step by step solution:
We are given a pair of linear equations $2x - 8y = 8$ and $ - x + 4y = 4$ by applying the method of substitution we will find the value of both variables. We will solve the first equation for $x$:

$ \Rightarrow 2x - 8y = 8$

Adding $8y $ to both sides:

$ \Rightarrow 2x - 8y + 8y = 8 + 8y$

On proper rearrangement we will get:

$ \Rightarrow 2x = 8 + 8y$

Now, we will divide both sides by 2.

$ \Rightarrow x = \dfrac{8}{2} + \dfrac{8}{2}y$

$ \Rightarrow x = 4 + 4y$

Now we will substitute the value of $x$ in the second equation and solve for$y$:

$ \Rightarrow - \left( {4 + 4y} \right) + 4y = 4$

On applying the distributive property, we get:

$ \Rightarrow - 4 - 4y + 4y = 4$

On combining like terms, we get:

$ \Rightarrow - 4 = 4$

Therefore, the statement $ - 4 = 4$ is false for all real numbers which means the system of equations has no solution.

Hence the system of equations has no solution.

Note: The students please note that such types of questions can be solved by another method described here.

Alternate method:

We are given two equations i.e.

  $2x - 8y = 8$ …(1)
  $ - x + 4y = 4$ ….(2)

Here, ${a_1} = 2$, ${a_2} = - 1$, ${b_1} = - 8$, ${b_2} = 4$, ${c_1} = 8$ and ${c_2} = 4$

Now, substitute the values into the expression, $\dfrac{{{a_1}}}{{{a_2}}}$, $\dfrac{{{b_1}}}{{{b_2}}}$ and $\dfrac{{{c_1}}}{{{c_2}}}$
$ \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{2}{{ - 1}}$

$ \Rightarrow \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{ - 8}}{4} = \dfrac{{ - 2}}{1}$

$ \Rightarrow \dfrac{{{c_1}}}{{{c_2}}} = \dfrac{8}{4} = \dfrac{2}{1}$

Thus, the relation between $\dfrac{{{a_1}}}{{{a_2}}}$, $\dfrac{{{b_1}}}{{{b_2}}}$ and $\dfrac{{{c_1}}}{{{c_2}}}$ is

$ \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$

Thus, the system of equations has no solution.