Conditions For Infinite And No Solutions

We are all well aware of every equation as well as expression. And we almost calculate it daily in mathematics. So, let’s just have a quick refresh of the meanings the words have, all over again before we dive into the topic. An expression that has an equal sign (=), in the middle, is termed an equation.  For instance, 5+2 = 7. An expression that consists of some variables for example u or v and terms which are constant and are conjoined with each other by using operators which are algebraic. For instance, 2u + 4v – 9 where u and v represent variables and 9 is representing a constant. There is usually only one answer to a given equation which can be solved by using various methods. But it is also possible that the equation can have more solutions than one or also no solution or its opposite infinite number of solutions. When an equation is having no answer that is called a no solution equation on the other end when an equation has a value for the given variable that will result in making a true equation is called an infinite solution

Details on Infinite Solution and No solution

The total count of variables that an equation has, determines the number of answers it will result in. And based upon that, answers can be clubbed into three categories, that are: 

1. Unique Solution (having only 1 type of solution).
2. No Solutions (which have no solutions)
3. Infinite Solutions (which have many solutions)

But how will we get to know if the given answer to our calculated equation is having an infinite solution? Here’s the answer, there’s a very easy process to see if our solution is not non-infinite. An infinite solution has equal value on both sides. An infinite solution example, 6u + 2v – 8 = 12u +4v – 16. If we solve the given equation with an infinite solutions formula or methodology, we’ll get equal figures on both sides, therefore, it is termed an infinite solution. N. It is generally displayed by the use of the sign” ∞ “.

Condition for Infinite Solution 

An equation with an infinite solution would produce an infinite number of solutions until and unless it satisfies some given conditions for the infinite solutions. An infinite solution could be made if the given lines do coincide and then they have to be at the same y-intercept. The two lines which are having a similar y-intercept and also the slope is the exactly same line. Stating in a simple sentence, we could conclude that if the same line is being shared between the two lines, then the solution would give an infinite answer. Hence, a procedure would be consistent only if the solution of the equations has an infinite solution.

For example, showing equations with infinite solutions:  

  v = u + 3                           — 1

  5v = 5u + 15                    — 2

If we multiply 5 to the equation to the given equation, we will achieve equation 2 and by dividing equation 2 by 5, we will be getting the exact equation which will be at first.

Condition for No Solution:

Example

Find the value of x and y

-4u + 10v = 6

2u – 5v = 3

Solution:

A.T.Q  -4u + 10v = 6 …(i)

2u – 5v = 3 …(ii)

Dividing (i) by 2 and reducing it.

We obtain -2u + 5v = 3

Solving for u

u = (5/2 )v – (3/2)

Substituting u in (ii)

2[(5/2) v – (3/2)] – 5v = 3

5v – 3 – 5v = 3

0 = 3 + 3

By getting, 0 = 6. Hence this solution is not true.

Therefore, the system explained for this equation has no solution.

Conclusion

As mentioned above there is also a unique solution which means having only one solution. In the set of one equation which is linear simultaneous, a unique solution comes compulsorily only if, (a) the digit of not known and the digits of that equation are matching, (b) all the equations are having a consistent nature, and lastly (c) there is negative linear dependence among any of the two equations, which is, all the calculations are independent.