To determine which statement is true about the system of linear equations, let's analyze the second equation represented by the two points given: \((-12, 20)\) and \((4, 12)\).
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Find the slope of the line using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1) = (-12, 20)\) and \((x_2, y_2) = (4, 12)\): \[ m = \frac{12 - 20}{4 + 12} = \frac{-8}{16} = -\frac{1}{2} \]
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Use the point-slope form to find the equation of the line that passes through one of the points. Using the point \((4, 12)\): \[ y - 12 = -\frac{1}{2}(x - 4) \] Simplifying this gives: \[ y - 12 = -\frac{1}{2}x + 2 \] \[ y = -\frac{1}{2}x + 14 \]
Now we have the equation of the second line in the system: \[ y = -\frac{1}{2}x + 14 \]
Now, let's analyze the conditions given in the statements:
- Statement A: The only solution to the system is \((0, 14)\).
- Statement B: The system has no solution.
- Statement C: The only solution to the system is \((10, 5)\).
- Statement D: The system has an infinite number of solutions.
To determine if there is a unique solution, no solution, or infinite solutions, we need to compare this line with the line represented by the first equation (not specified in your problem).
If the first equation is either parallel to the second equation (which has a slope of \(-\frac{1}{2}\), meaning they are parallel), then there is no solution, leading us to option B.
If both equations are the same (meaning they have the same slope and y-intercept), then there would be an infinite number of solutions, leading us to option D.
If neither option applies, we would need to graph both equations to find any intersection point that represents a unique solution.
Since we don't know the specific equation of the first line, we cannot definitively choose the correct statement without that information.
If you have the equation of the first line, we can check for:
- intersection points for a unique solution,
- parallel lines for no solutions,
- or identical lines for infinite solutions.
Please provide the equation of the first line for further analysis.
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