An infinite number of solutions often arises in linear systems when equations are dependent, essentially representing the same line. You can create such a system by manipulating a single equation.
Here's how:
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Generate Equivalent Equations: The core principle is to create multiple equations that, despite appearing different, represent the exact same relationship between the variables.
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Multiply or Divide by a Constant: This is the simplest method. If you have an equation like
x + y = 5, you can multiply both sides by any non-zero constant. For example:2(x + y) = 2(5)becomes2x + 2y = 10-1(x + y) = -1(5)becomes-x - y = -5
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Rearrange the Equation: Alter the equation's form without changing its fundamental meaning.
x + y = 5can be rearranged toy = -x + 5. Both equations represent the same line.
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Creating a System of Equations: A system with infinite solutions requires at least two equations that are dependent (essentially representing the same line). Example:
x + y = 5 2x + 2y = 10 // This equation is simply the first one multiplied by 2.This system has infinite solutions because any solution that satisfies the first equation will automatically satisfy the second equation. Graphically, both equations represent the same line.
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Slope-Intercept Form Example: You can express the same line in different slope-intercept forms (y = mx + b) which will highlight the dependency.
y = -x + 5 2y = -2x + 10 // Second equation is first multiplied by 2
Key Characteristics of Infinite Solutions:
- Dependent Equations: The equations in the system are multiples of each other or can be transformed into one another through algebraic manipulation.
- Graphical Representation: When graphed, the equations in the system represent the same line.
- Solution Form: Solutions can be expressed in parametric form (e.g., x = t, y = 5 - t, where 't' can be any real number), reflecting the infinite possibilities.
In essence, creating infinite solutions involves presenting the same linear relationship in multiple disguises.
