Infinite Solutions

We all are well acquainted with equations and expressions. We solve it almost daily in mathematics. Let's just quickly refresh the meanings of the terms once again before we dig in.  An equation is an expression which has an equal to sign (=) in between. For example, 4+3 = 7. And an expression consists of variables like x or y and constant terms which are conjoined together using algebraic operators. For example, 2x + 4y - 9 where x and y are variables and 9 is a constant. As far as we look there is usually one solution to an equation. But it is not impossible that an equation cannot have more than one solution or infinite number of solutions or no solutions at all. Having no solution means that an equation has no answer whereas infinite solutions of an equation means that any value for the variable would make the equation true. 

What Are Infinite Solutions?

The total number of variables in an equation determines the number of solutions it will produce. And on the basis of this, solutions can be grouped into three types, they are: 

  1. Unique Solution (which has only 1 solution).

  2. No Solutions (having no solutions)

  3. Infinite Solutions ( having many solutions )

But how would you know if the solution to your solved equation is an infinite solution? Well, there is a simple way to know if your solution is an infinite solution. An infinite solution has both sides equal. For example, 6x + 2y - 8 = 12x +4y - 16. If you simplify the equation using an infinite solutions formula or method, you’ll get both sides equal, hence, it is an infinite solution. Infinite represents limitless or unboundedness. It is usually represented by the symbol ” ∞ “.

Conditions For Infinite Solution

An equation will produce an infinite solution if it satisfies some conditions for infinite solutions. An infinite solution can be produced if the lines are coincident and they must have the same y-intercept. The two lines having the same y-intercept and the slope,  are actually the exact same line. In simpler words, we can say that if the two lines are sharing the same line, then the system would result in an infinite solution. Hence, a system will be consistent if the system of equations has an infinite number of solutions.

For example, consider the following equations. 

 y = x + 3

 5y = 5x + 15

If we multiply 5 to equation 1, we will achieve equation 2 and on dividing equation 2 with 5, we will get the exact first equation.  

Infinite Solutions Example

What is an example of an infinite solution? This is the question we were waiting for so long. But in order to solve systems of an equation in two or three variables, it is important to understand whether an equation is a dependent one or an independent, whether it is a consistent equation or an inconsistent equation. A consistent pair of linear equations will always have unique or infinite solutions.


Example 1) Here are two equations in two variables. 

a1x + b1y = c1 ——- (1)

a2x + b2y = c2 ——- (2)

If (a1/a2) = (b1/b2) = (c1/c2) 

Then the equation is a consistent and dependent equation which has infinitely many solutions.


Example 2) Here are few equations with infinite solutions -6x + 4y = 2

3x - 2y = -1

Now if we multiply the second equation by -2, we will get the first equation. 

-2(3x-2y) = -2(-1)

-6x + 4y = 2

Therefore, the equations are equivalent and will share the same graph. So, the solution that will work for one equation would also work for other equations as well. Hence, they are infinite solutions to the system.

 

Example 3) x-10+x = 8+2x-18

Now, here is how we proceed 

                    x-10+x = 8+2x-18

                     2x-10 = 2x-10

                      -2x=-2x

                __________________

                     -10 =-10

Since -10 = -10 we are left with a true statement  and we can say that it is an infinite solution.


Example 4) Let us take another example: x+2x+3+3=3(x+2)


                                                                   x+2x+3+3=3(x+2)

                                                                   3x+6=3x+6

                                                                   -3x    = -3x

                                                                     6=6


The coefficients and the constants match after combining the like terms. This gives us a true statement. Therefore, there can be called infinite solutions.  


Example 5) Consider 4(x+1)=4x+4 as an equation.


                                   4(x+1)=4x+4 

                                   4x+4=4x+4

We can see that in the final equation, both sides are equal. Therefore, it is an infinite solution.


FAQ (Frequently Asked Questions)

1. What are the conditions of an infinite solution in matrices?

In order to solve matrices, just think about it as systems of linear equations.

If there are 3 unknowns, then you would need 3 equations. However, if one of the equations would turn out to be a linear combination of the others, then basically it might be just “useless” that is because it is redundant and will offer you with no information about how to resolve the system.

Consider an example:

x1 + x2 = 1

2x1 + x2 + x3= 10

4x1 + 3x2 + x3 = 21

So, your matrix is basically

1 1 0

2 1 1

4 3 1

We can see how the third row turns out to be a linear combination of the first and second rows. (2*R1 + R2)


It would not be wrong if we say that there are infinitely many solutions. Since there is not enough information as one of the rows is redundant. Thus, we can also call this a “singular” matrix.


Now to determine singularity, we can take the determinant of the matrix and see that the determinant of a singular matrix is 0. If you doubt, then just google about it for more information. In case you have a row of zeros, then it is a linear combination of any rows (0*R1 + 0*R2 + 0*R3 +…). Therefore, any square matrix having a row of zeros will be singular and it will consist of infinitely many solutions. By taking the determinant, you can arrive at the same conclusion.