Question

How do you solve the system of equations $2x - y = 6$ and $4x - 2y = - 3$ ?

Answer 1

Hint: In the given question, we need to solve two simultaneous equations in two variables. There are various methods to solve two given equations in two variables like substitution method, cross multiplication method, elimination method, matrix method and many more. The equations given in the question can be solved using any one of the above mentioned methods easily. But we will solve the equations using the substitution method as mentioned in the question.

Complete step by step answer:
In the question, we are given a couple of simultaneous linear equations in two variables.
$2x - y = 6$
$\Rightarrow 4x - 2y = - 3$
In the substitution method, we substitute the value of one variable from an equation into another equation so as to get an equation in only one variable. So, we have, $2x - y = 6$. We will find the value of y in terms of x. So, shifting the terms from one side of equation to another, we get,
$ \Rightarrow y = 2x - 6$

Now putting the value of y obtained from one equation into another.
$4x - 2y = - 3$
$ \Rightarrow 4x - 2\left( {2x - 6} \right) = - 3$
Opening the brackets, we get,
$ \Rightarrow 4x - 4x + 12 = - 3$
Now, cancelling the like terms with same magnitude and opposite signs, we get,
$ \Rightarrow 12 = - 3$
Now, we know that the expression that we have obtained is not true mathematically. This means that there is no solution for the given system of equations.

Hence, the system of equations $2x - y = 6$ and $4x - 2y = - 3$ is inconsistent.

Note: A equation consisting of 2 variables having degree one is known as Linear Equation in two variables. Standard form of Linear Equation in two variables is $ax + by + c = 0$ where a, b and c are the real numbers and a, b which are coefficients of x and y respectively are not equal to 0. The system of equations having no solutions is called an inconsistent system of equations. The straight lines represented by the given equations are parallel to each other and don’t meet at any point on the Cartesian plane.