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An equation always has two sides to it which are the right-hand side and the left and side. An infinite solution would always have both sides equal to each other.

We know that an equation will arrive at one solution but is it possible to have more than one solution in other types of equations that are not linear? Is it possible to have no solutions or infinite solutions? No matter what value is assigned to the variable is it possible for the equation to be true. Infinite solutions mean that any value for the variable would make the equation true.

So, let’s see solving equations with infinite solutions is in this article.

### Types of Solutions

There are many types of solutions such as;

**1. No solution:** It is non-consistent and independent.

For example: Rank of A is not equal to Rank of A:B.

**2. Unique solution:** It is consistent and dependent.

For example: Rank of A is equal to Rank of A:B is also equal to the number of variables or unknown.

**3. Infinite solution:** It is consistent and independent.

For example, A’s Rank is equal to the Rank of A: B but less than several variables.

### Solving Equations with Infinite Solutions:

Take a look at the equations below,

2x + 3 = 2x +3

We can observe that there are variables on the sides of the equation. Hence, subtracting 2x from both sides, the result of the equation is,

3 = 3

This clarifies that no matter what value we substitute for x, the equation will also be true. We can put any value and experiment in equations like this to verify this is true.

Also, note that two times the number plus three is equal to itself in the original equation. So there are infinite solutions. Sometimes the symbol ∞ , which is called infinity, is used to represent infinite solutions.

-2(x+3) = -2x – 6

-2x – 6 = -2x -6

-6 = -6

Here again, we can see a statement that is always true. Hence there are infinite solutions.

Below is an example of creating Multi-step infinite solutions equations. In case we are required to create a false math statement for no solutions. We need a statement that will always be true. Consider the example below:

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

3x + 6 = 3x + 6

6 = 6

The coefficients have matched after we combine like terms and used the distributive property, but the constant has matched in this case. Hence the statement that six equals six is a true equation. Therefore, infinite solutions exist. Let’s look at one more example:

4(x +1) = 4x + 4

4x + 4 = 4x + 4

As you notice, we can stop the equation here since both sides are the same. Four times a number plus four is equal to four times that number plus four. Therefore there are infinite solutions.

### Conditions of Infinite solutions:

- When an equation system is represented on a graph, the intersection of two straight lines is the answer.
- If we have multiple solutions, we’ll have multiple points of overlapping of lines. Thus, any point on the given line can be a possible solution. Hence, if two lines have the y-intercept same and even the slope. They are ideally in the same line.
- We can say that a system has infinite solutions when the number of variables exceeds the number of non-zero rows regarding a matrix. Hence it is said to be consistent.
- Since 0 = 0, which is equal to x, such a system also has infinite solutions.
- On solving the equation, the same simplified expression is shown on both sides.
- They form linear equations when represented on a graph.
- The term infinite means it has no limit or boundary. It is denoted as ‘∞’.
- The line is concurrent and has an infinite solution if the condition is a/m = b/n = c/p

#### Conclusion

Hope the examples helped you to understand the concept well. This way you would be able to identify when an equation will have infinite solutions.

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