A system of equations with infinite solutions is when two or more equations represent the same line or plane. This means they have all the same solutions.
Here's an example:
- Equation 1: x + y = 1
- Equation 2: 2x + 2y = 2
Explanation
Notice that Equation 2 is simply Equation 1 multiplied by 2. Therefore, they are equivalent equations. Graphically, they would be the same line. Any (x, y) pair that satisfies x + y = 1 will also satisfy 2x + 2y = 2, and vice versa. Since there are infinitely many points on a line, there are infinitely many solutions to this system of equations.
How to Identify Infinite Solutions
You can identify a system with infinite solutions by:
- Algebraically: If you manipulate one equation and can obtain the other equation, they are dependent and have infinite solutions.
- Graphically: If the graphs of the equations are the same line (or plane in 3D), then they have infinite solutions.
Another Example
Here's another example of a system of equations with infinite solutions:
- Equation 1: 3x - y = 4
- Equation 2: 6x - 2y = 8
In this case, Equation 2 is simply Equation 1 multiplied by 2. Therefore, they represent the same line.
