Question

Solve:
$x + y = 2xy$
$x - y = 6xy$
A. $x = - \dfrac{1}{2},y = \dfrac{1}{7}$
B. $x = - \dfrac{1}{2},y = \dfrac{1}{4}$
C. $x = - \dfrac{1}{2},y = \dfrac{1}{2}$
D. $x = - \dfrac{1}{2},y = \dfrac{1}{5}$

Answer 1

Hint: Linear equations can be defined as the equations with unknown variables present in it, with the degree of 1. If the above example is considered, that is a linear equation with 2 variables, we have to make use of simultaneous equations and solve it. That is first solve to find the value of x, and then substitute the value of x in one of the original equations to get the value of y. Basic methods of addition, subtraction and multiplication are only used. Simultaneous equations means when 2 or more equations are solved simultaneously with the help of mathematical equations, i.e. Addition, subtraction, multiplication and division.

Complete step by step answer:
Equations given in the question:
$x + y = 2xy\, and\, x - y = 6xy$
To find: x and y
Let us start with finding the value of y first by using simultaneous equations.
We are going to add the 2 equations given in the question, to get the value of y.
Adding $x + y = 2xy\, and\, x - y = 6xy$, we get,
\[x + y + \,x - y = 6xy + 2xy\]
\[2x = 8xy\]
As we want the value of y, we take all other values to one side.
Therefore, dividing by 8x on both sides, we get,
$y = \dfrac{{2x}}{{8x}} = \dfrac{1}{4}$
Therefore, $y = \dfrac{1}{4}$
Now we have the value of one unknown variable that is y, so finding out another unknown variable will be easy. You just have to substitute the value of y in one of the original equations.
Substituting $y = \dfrac{1}{4}$ in $x + y = 2xy$, we get,
$x + \dfrac{1}{4} = 2x \times \dfrac{1}{4} = \dfrac{x}{2}$
As we want the value of x, we take all other values to one side.
\[x - \dfrac{x}{2} = - \dfrac{1}{4}\]
\[\dfrac{x}{2} = - \dfrac{1}{4}\]
\[x = - \dfrac{2}{4}\]
Therefore, $x = - \dfrac{1}{2}$

So, the correct answer is “Option B”.

Note:
Addition, subtraction, all these methods are to be performed correctly, most of the mistakes happen because of the wrong sign. Just to be sure about your answer, you can put the value of x in the original equation and compare.