To determine which graph shows that the linear system given by the equations \(-2x + y = 6\) and \(4x + 2y = 12\) has an infinite number of solutions, we first need to analyze the system of equations.
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Rearranging the equations:
- The first equation can be rewritten in slope-intercept form (y = mx + b): \[ y = 2x + 6 \]
- The second equation can also be simplified. Dividing the entire equation by 2 gives us: \[ 2x + y = 6 \] Rearranging this as well gives: \[ y = -2x + 6 \]
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Checking for equivalence: Notice that the second equation is equivalent to the first one when rearranged. Since both equations represent the same line, the system has an infinite number of solutions.
Next, we will identify the graph that represents the same line twice, indicating infinite solutions.
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Graph 1: Shows two points (0,6) and (-3,0). This does not depict two lines or infinite solutions.
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Graph 2: Shows two points (3,0) and (0,-6). This does not show infinite solutions.
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Graph 3: Line 1 contains points (0,6) (3,0) and Line 2 contains points (0,6) and (-6,0). If both lines coincide or are represented by the same equation, this suggests infinite solutions.
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Graph 4: Shows one line through (0,0) and another through points (0,6) and (-3,0). This does not represent infinite solutions.
The correct answer is Graph 3, which includes two representations of the same line, indicating that the system has an infinite number of solutions.
1 answer