Today we will learn about types of simultaneous equations that cannot be solved.
Let us take a look at the following equations:
$4x + 7y = 18$
$12x + 21y = 72$
If we try to solve this using normal methods, we can multiply the first equation by $3$ and subtract from the second equation.
We get:
$(12x + 21y) - 3(4x + 7y) = 72 - 3 \times 18$
$12x + 21y - 12x - 21y = 72 - 54$
$0 = 18$
Now, we can see that if we try to eliminate $1$ variable, both the variables get eliminated and we are left with an impossible equation, because $0$ can never be equal to $18$.
If we plot these two equations on a graph paper, we will get two parallel lines, which never meet.
Therefore, there is no coordinate point or values of $x$ and $y$ that will satisfy both the equations simultaneously. Hence, these equations have $0$ solutions.
These equations are called $\underline{Inconsistent\ Equation\ System}$
Let us take a look at another type of equations.
$5x + 3y = 21$ $...eqn\ (i)$
$15x + 9y = 63$ $...eqn\ (ii)$
In this case also, if we try to eliminate any one of the variables, then both will get eliminated, and we will be left with the equation $0 = 0$. This is not an impossible equation, but this is a trivial equation, and no matter what the value of $x$ and $y$, $0 = 0$ is always true.
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If we plot these two equations on a graph paper, we will obtain the same line. It is easy to see that these two are the same equation, since we can obtain equation $(ii)$ by multiplying equation $(i)$ by $3$.
Cases like this, where one equation can be obtained from the other one by multiplying or dividing both sides by a constant, are called $\underline{Dependent\ Equation\ System}$.
They can have infinitely many solutions. In our example the values:
$(3, 2)$, $(6, -3)$, $(9, -8)$ and many more integer as well as fractional values will satisfy both the equations.
Now, let us summarise what we learnt:
- When two equations are such that one can be derived from the other, they represent the same equation. These are called $\underline{Dependent\ Equation\ System}$ and can have infinitely many solutions.
- When two equations that cannot be directly derived from one another, but do not have a solution because they represent parallel lines, they have $0$ solutions. These are called $\underline{Inconsistent\ Equation\ System}$
- To be able to solve a given set of equations, they must be $\underline{independent}$ and $\underline{consistent}$ set of equations, containing the same number of equations as there are unknowns.