To determine the correct statement based on the information provided about the system of equations, let's analyze the situation further.
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Identifying the lines:
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The first line passes through the points (-2, 0) and (0, -3). We can calculate its slope and equation:
- The slope (m) is calculated as: \[ m = \frac{-3 - 0}{0 - (-2)} = \frac{-3}{2} \]
- Using the point-slope form (y - y₁ = m(x - x₁)) with the point (0, -3): \[ y + 3 = -\frac{3}{2}(x - 0) \] Simplifying gives: \[ y = -\frac{3}{2}x - 3 \]
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The second line passes through the points (0, 1) and (2, 0):
- The slope (m) is: \[ m = \frac{0 - 1}{2 - 0} = \frac{-1}{2} \]
- Using the point-slope form with the point (0, 1): \[ y - 1 = -\frac{1}{2}(x - 0) \] Simplifying gives: \[ y = -\frac{1}{2}x + 1 \]
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Finding the intersection:
- According to your description, these two lines intersect at (-4, 3). To confirm this is the solution of the system of equations, we check if (-4, 3) satisfies both equations.
- For the first line \(y = -\frac{3}{2}x - 3\): \[ 3 = -\frac{3}{2}(-4) - 3 \] \[ 3 = 6 - 3 \quad \text{(True)} \]
- For the second line \(y = -\frac{1}{2}x + 1\): \[ 3 = -\frac{1}{2}(-4) + 1 \] \[ 3 = 2 + 1 \quad \text{(True)} \]
- According to your description, these two lines intersect at (-4, 3). To confirm this is the solution of the system of equations, we check if (-4, 3) satisfies both equations.
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Interpreting the results:
- Both lines intersect at the point (-4, 3). Therefore, this point is the only solution to the system of equations.
Given this analysis, we can evaluate the statements:
A. The system has infinite solutions. ➔ False (They intersect at one point). B. The solutions to the system are and . ➔ Incomplete. C. The solution to the system is . ➔ True (but not 100% complete without specifying the coordinates). D. The system has no solution. ➔ False (there is a point of intersection).
The most accurate statement among the provided options is C, indicating that the solution to the system is the point where the two lines intersect, which is (-4, 3).