# when you solve a system of equations by the substitution method, how do you determine whether the...

## Question:

when you solve a system of equations by the substitution method, how do you determine whether the system of equations is inconsistent?

## Inconsistent equations

Inconsistent set of equations are those that are parallel and do not have a common point of meeting. Such a set of equations cannot be solved as there is no solution possible.

Suppose you have two sets of parallel equations with unequal constants {eq}c_1 {/eq} and {eq}c_2 {/eq} as follows:

(eq. 1): {eq}ax + by = c_1 {/eq}

(eq. 2): {eq}ax + by = c_2 {/eq}

We can write (eq. 1) as:

{eq}x = \ \frac{c_1 - by}{a} {/eq}

We can replace this value of x in (eq. 2) as:

{eq}a\ \frac{c_1 - by}{a} + by = c_2 {/eq}

=> {eq}c_1 - by + by = c_2 {/eq}

=> {eq}c_1 = c_2 {/eq}

This is not possible since the constants are not equal.

Hence, when solving two equations we reach a point that no variables are left and two unequal numbers are equating to zero, we can claim that the set of equations are inconsistent.