Yes, a system of two linear equations can indeed have an infinite number of solutions. This occurs when the two equations represent the same line. In such cases, every point on the line is a solution to both equations.
How to Tell by Looking at the Equations:
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Identical Ratios: Two equations of the form \( ax + by = c \) are equivalent (and thus represent the same line) if the coefficients are proportional. Specifically, for equations \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \), the system has infinitely many solutions if:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
If you can simplify both equations to the same form or one equation can be derived algebraically from the other, they are identical.
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Example: Consider the two equations:
- \( 2x + 3y = 6 \)
- \( 4x + 6y = 12 \)
The second equation can be simplified to the first by dividing through by 2. Thus, both equations describe the same line.
How to Tell from the Graph:
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Graphical Representation: When you graph both lines represented by the equations, if they coincide completely (i.e., lie on top of each other), then the system has an infinite number of solutions.
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Visual Characteristics: If the lines are parallel but not coincident, this would instead indicate that there are no solutions (i.e., the system is inconsistent).
In conclusion, to determine if a system of two linear equations has an infinite number of solutions, you can check the ratios of the coefficients in the equations or graph the equations to see if the lines coincide. If they do, that indicates an infinite number of solutions.