How can you tell if an equation has infinite solutions

There are some equations with no solutions, or infinitely many. Equations with no solutions won't be equal when simplified. Equations with infinite solutions will simplify to the same constant on both sides.

Table of Contents Show

What are Infinite Solutions?
Conditions for Infinite Solution
Infinite Solutions Example
How do you know if there is infinite solutions to an equation?
How do you tell if a system has infinite solutions or no solutions?

Here are a few possibilities:

The equation simplifies to the point that it no longer contains a variable, but expresses a true equation, e.g. #0 = 0#. For example: #2x+2 = 2(x+1)# simplifies in this way.
The equation has an identifiable solution and is periodic in nature. For example: #tan^2 x + tan x - 5 = 0# has infinitely many solutions since #tan x# has period #pi#.
The equation has a piecewise behaviour and simplifies within at least one of the intervals to a true equation without variables. For example: #abs(x+1)+abs(x-1) = 2#, which simplifies suitably for #x in [-1, 1]#.
The equation has more than one variable and does not force uniqueness. For example: #x^2+y^2=1# has infinitely many solutions, but #x^2+y^2=0# has one solution (assuming #x, y in RR#).

Note that it may be extremely difficult to determine the number of solutions in the case of Diophantine equations - equations where the values of the variables are limited to integers or to positive integers.

For example, Euler conjectured that the equation:

#x^4+y^4+z^4 = w^4#

had no non-trivial solutions, but Noam Elkies found one in 1988, hence there are an infinite number of non-trivial solutions, since any solution can be multipled by a fourth power.

In Mathematics, we come across equations and expressions. An equation is an expression with an equal sign used in between. An expression is made up of variables and constant terms conjoined together using algebraic operators. An algebraic equation can have one or more solutions. The solution of the equation or the values of variables in the equation must satisfy the equation. In this article, we are going to discuss the equations with infinite solutions, and the condition for the infinite solution with examples.

What are Infinite Solutions?

The number of solutions of an equation depends on the total number of variables contained in it. Thus, the system of the equation has two or more equations containing two or more variables. It can be any combination such as

2 equations in 3 variables
6 equations in 4 variables, and so on

Depending on the number of equations and variables, there are three types of solutions to an equation. They are

Unique Solution (One solution)
No solution
Infinite Solutions (Many solutions)

The term “infinite” represents limitless or unboundedness. It is denoted by the letter” ∞ “.

To solve systems of an equation in two or three variables, first, we need to determine whether the equation is dependent, independent, consistent, or inconsistent. If a pair of the linear equations have unique or infinite solutions, then the system of equation is said to be a consistent pair of linear equations. Thus, suppose we have two equations in two variables as follows:

a1x + b1y = c1 ——- (1)

a2x + b2y = c2 ——- (2)

The given equations are consistent and dependent and have infinitely many solutions, if and only if,

(a1/a2) = (b1/b2) = (c1/c2)

Conditions for Infinite Solution

An equation can have infinitely many solutions when it should satisfy some conditions. The system of an equation has infinitely many solutions when the lines are coincident, and they have the same y-intercept. If the two lines have the same y-intercept and the slope, they are actually in the same exact line. In other words, when the two lines are the same line, then the system should have infinite solutions. It means that if the system of equations has an infinite number of solution, then the system is said to be consistent.

As an example, consider the following two lines.

Line 1: y = x + 3
Line 2: 5y = 5x + 15

These two lines are exactly the same line. If you multiply line 1 by 5, you get the line 2. Otherwise, if you divide the line 2 by 5, you get line 1.

Infinite Solutions Example

Example:

Show that the following system of equation has infinite solution: 2x + 5y = 10 and 10x + 25y = 50

Solution:

Given system of the equations is 2x + 5y = 10 and 10x + 25y = 50

2x + 5y = 10 ………….(1)

10x + 25y = 50 ………..(2)

By comparing with linear system, we get

a1x + b1y = c1

a2x + b2y = c2

=> a1 = 2, b1 = 5, c1 = 10, a2 = 10, b2 = 25 and c2 = 50

Now, the ratios are:

(a1/a2) = 2/10 = 1 / 5

(b1/b2) = 5 /25 = 1/5

(c1/c2) = 10/50 = 1/5

(a1/a2) = (b1/b2) = (c1/c2)

Therefore, the given system of equation has infinitely many solutions.

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How do you know if there is infinite solutions to an equation?

Well, there is a simple way to know if your solution is infinite. 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.

How do you tell if a system has infinite solutions or no solutions?

A linear system has many (infinite) solutions when the two lines are the same (such as y=x+3 and 2y=2x+6 ). And a linear system has no solution when the lines never intersect (in other words, they're parallel; their slopes are equal).