How do you find infinitely many solutions?

You find infinitely many solutions when an equation or system of equations simplifies to a statement that is always true, meaning the variables can take on an infinite number of values that satisfy the condition.

Understanding Infinite Solutions

An equation has infinitely many solutions when, after simplification, both sides are identical. This indicates that the equation represents the same line or relationship regardless of the variable's value. This contrasts with equations that have a single solution or no solution.

Identifying Infinite Solutions in a Single Equation

Simplification: Simplify the equation as much as possible by combining like terms and performing necessary operations.
Equality: If, after simplification, the left side of the equation is identical to the right side, you have infinitely many solutions. For example:
```
6x + 2y - 8 = 12x + 4y - 16
```
Divide the entire equation by 2:
```
3x + y - 4 = 6x + 2y - 8
```
Rearranging the terms (although not strictly necessary for this simple example) to isolate 'y':
```
y = -3x + 4
2y = -6x + 8
y = -3x + 4
```
Since both equations are equivalent, this indicates infinite solutions. Every x value will provide a y value that satisfies both equations, because they represent the same line.

Identifying Infinite Solutions in a System of Equations

A system of equations has infinitely many solutions when the equations represent the same line. There are multiple ways to determine this:

Graphical Method: Graph both equations. If they overlap completely, they represent the same line and have infinite solutions.
Algebraic Method (Substitution or Elimination):
1. Solve one equation for one variable.
2. Substitute that expression into the other equation.
3. If the resulting equation is a true statement (e.g., 0 = 0), the system has infinitely many solutions.
Scalar Multiple: If one equation is a scalar multiple of the other (i.e., you can multiply one equation by a constant to get the other equation), then the equations represent the same line, resulting in infinite solutions.

For example, consider this system of equations:
```
x + y = 5
2x + 2y = 10
```
Notice that the second equation is simply the first equation multiplied by 2. Therefore, they represent the same line and have infinitely many solutions.

Example

Solve for x:

2(x + 3) = 2x + 6

Distribute the 2: 2x + 6 = 2x + 6
Subtract 2x from both sides: 6 = 6

Since we arrive at a true statement, the equation has infinitely many solutions.

Conclusion

In essence, identifying infinite solutions involves simplifying equations or systems of equations to reveal if they are inherently true statements or if they represent the same line. When the equation simplifies to an identity (e.g., 0=0, 6=6) or when the system's equations are just multiples of each other, you have infinitely many solutions.

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