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” **∞ “**.

## Equations with Infinite solutions

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:

a_{1}x + b_{1}y = c_{1} ——- (1)

a_{2}x + b_{2}y = c_{2} ——- (2)

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

(a_{1}/a_{2}) = (b_{1}/b_{2}) = (c_{1}/c_{2})

## 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

a_{1}x + b_{1}y = c_{1}

a_{2}x + b_{2}y = c_{2}

=> a_{1} = 2, b_{1} = 5, c_{1} = 10, a_{2} = 10, b_{2} = 25 and c_{2 }= 50

Now, the ratios are:

(a_{1}/a_{2}) = 2/10 = 1 / 5

(b_{1}/b_{2}) = 5 /25 = 1/5

(c_{1}/c_{2}) = 10/50 = 1/5

(a_{1}/a_{2}) = (b_{1}/b_{2}) = (c_{1}/c_{2})

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

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