A system of equations is said to have infinite solutions when there are countless values that satisfy all the equations in the system simultaneously. This typically occurs in the context of linear equations.
Here are some key points to understand about infinite solutions:
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Dependent Equations: The equations in the system are not independent; instead, they are related in such a way that one equation can be derived from the others. This means that at least one of the equations can be expressed as a linear combination of the others.
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Geometric Interpretation: In two or three dimensions, infinite solutions can often be visualized:
- In two-dimensional space (2D), if you have two lines that are coincident (they lie on top of each other), then every point on that line is a solution to the system.
- In three-dimensional space (3D), if you have two planes that coincide, every point on that plane is a solution.
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Free Variables: In the context of solving systems, infinite solutions usually indicate that there are free variables in the system. A free variable is one that can take on any value, leading to multiple parameterized solutions.
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Matrix Representation: When using matrix methods, a system that has infinite solutions will typically yield a row-echelon form with at least one row of zeros, indicating that not all variables are determined. This occurs in systems that are underdetermined.
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Example: Consider the system of equations: \[ x + y = 5 \] \[ 2x + 2y = 10 \] The second equation is simply a multiple of the first. Thus, any combination of \(x\) and \(y\) that satisfies \(x + y = 5\) is a solution, leading to infinitely many pairs \((x, y)\) such as \((1, 4)\), \((2, 3)\), \((0, 5)\), etc.
In summary, a system with infinite solutions signifies a lack of unique solutions due to redundancy or dependence among the equations.