Question

How do you know when the system of equations is inconsistent?

Answer 1

Hint: In this question, we have to find how the system of equations is inconsistent. As we know, a system of equations is those many equations that share the same variables. For example, $ax+by=0$ and $cx+dy=0$ , which implies both the equation have common variables that are x and y, so when we solve both the equations together, we get the value of x and y. inconsistent means when the system of the equation have no solutions, which means both the x and y cancel out, and thus we did not get any value of the variables. Thus, in this problem, we take 2 equations and solve them using the substitution method. After that, we get no solutions for the equations, which is the required solution of the problem.

Complete step by step answer:
According to the question, we have to find when the system of equations is inconsistent.
A system of equations is inconsistent when we do not get any solution for the equations, which means there are no values for the variables that exist to satisfy those equations.
Therefore, we will explain this through an example.
Let us say we have a system of equation, that is
$6x+2y=10$ ---------- (1) , and
$-6x-2y=12$ ------- (2)
Now, we will use the substitution method to do the same.
The equations given in the problem are: So, we will first rewrite the equation (1) in terms of x, which is
$6x+2y=10$
Now, subtract 6x on both sides of the above equation, we get
$\Rightarrow 6x+2y-6x=10-6x$
As we know, the same terms with opposite signs cancel out each other, therefore we get
$\Rightarrow 2y=10-6x$ -------- (3)
Now, we will put the value of equation (3) into equation (2), we get
$\Rightarrow -6x-(10-6x)=12$
Now, we will open the brackets of the above equation, we get
$\Rightarrow -6x-10+6x=12$
As we know, the same terms with opposite signs cancel out each other, therefore we get
$\Rightarrow -10=12$
Thus, we see that in the above equation, the variable x cancels out, therefore we are left with the equation $-10=12$ which is not true.
Therefore, for the equations $6x+2y=10$ and $-6x-2y=12$, we do not get any value of x and y, which implies the system of equations is called inconsistent, that is there are no solutions for both the equations.
Hence, we now know how to find whether the system of equations is inconsistent or not.

Note:
While solving this problem, do not confuse the definition of inconsistent and consistent. One of the alternative methods to find about the inconsistency of a system of equation is, solve the example using the cross multiplication method; we will multiply the variable of the numerator of each side by the denominator of the other side, to get the required result of the problem, which is no solution for the problem.