To determine the correct statement about the system of linear equations, we first need to analyze the two lines represented by the equations.
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Identify the first equation: While the first line is not provided visually here, we can assume it is given in the problem context.
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Find the equation of the second line using the points (−12, 20) and (4, 12).
- First, calculate the slope (m) of the line using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the points: \[ m = \frac{12 - 20}{4 - (-12)} = \frac{-8}{16} = -\frac{1}{2} \]
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Use point-slope form to write the equation of the line. We can use the point (4, 12) for this: \[ y - y_1 = m(x - x_1) \] Substituting the values: \[ y - 12 = -\frac{1}{2}(x - 4) \] Simplifying this gives: \[ y - 12 = -\frac{1}{2}x + 2 \] \[ y = -\frac{1}{2}x + 14 \]
Now we have the second equation: \(y = -\frac{1}{2}x + 14\).
- Comparing the two equations:
- If the first equation's graph represents the same line as the second equation's graph (i.e., is identical), then the system would have an infinite number of solutions.
- If the first and second lines are parallel but not the same line, the system would have no solutions (inconsistent).
- If the first and second lines intersect at exactly one point, then that point would be the unique solution.
Since we don't have the first equation, we cannot definitively conclude from this information alone if the graphs intersect, are parallel, or are identical.
However, we can analyze the answer options based on typical outcomes:
- If the lines cross at (0, 14), (10, 5), or don't intersect at all, we can make some assumptions:
- A. The only solution is (0, 14) indicates they intersect at this specific point,
- C. The only solution is (10, 5) indicates they intersect there,
- B. The system has no solution suggests parallel lines,
- D. The system has an infinite number of solutions suggests they are the same line.
Without the first equation or a visual representation, we can't definitively choose an answer. You must evaluate the situation based on whether the first line and the derived second line are identical, parallel, or intersecting.
If you know more about the first equation's graph characteristics or its points, that could help finalize the conclusion. Here's a summary:
- Check if the first line intersects, is identical, or is parallel to the second.
- Choose between A, B, C, D based on that analysis.
1 answer