In the context of systems of equations, particularly linear equations, the term "infinite solutions" refers to a scenario where there are an unlimited number of solutions that satisfy all of the equations in the system. This typically indicates that the equations are dependent, meaning that one or more of the equations can be derived from the others.
Algebraically:
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Linear Equations: For example, consider the following system of two linear equations: \[ y = 2x + 3 \] \[ 2y = 4x + 6 \] The second equation can be simplified to be the same as the first equation, indicating that they describe the same line. Thus, every point (x, y) on that line is a solution, leading to an infinite number of solutions.
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Consistent and Dependent: The system is described as consistent and dependent, suggesting that each equation does not provide new information compared to the others.
Graphically:
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Intersection of Lines: Graphically, if you were to plot the equations on a Cartesian plane, both equations would represent the same line. This means that instead of intersecting at a single point or having no points of intersection (which would indicate no solutions), they overlap completely.
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Visual Representation: For example:
- If you plot the equation \(y = 2x + 3\), you'll see a line with a slope of 2 that crosses the y-axis at (0, 3).
- When you plot the equation \(2y = 4x + 6\), after simplifying it, you’ll find that it also plots the exact same line, leading to total overlap.
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Conclusion: Therefore, all points on this line are solutions to the system of equations. This graphical representation shows how infinitely many points (solutions) exist where the lines coincide.
Summary:
- Infinite Solutions: Indicates that there are an unlimited number of solutions due to dependence between equations.
- Graphically: This is represented by the overlap of lines on a graph, meaning every point along that line is a solution to the system.
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