To match each system of equations with its number of solutions, we need to evaluate the relationships between the lines represented by the equations. Here's how to determine the number of solutions for a system of equations based on the equations provided:
- y = 5x + 2
- y = -x + 1
- y = -3x - 8
- y = -3x + 3
- y = -4x + 1
- -2y = 8x - 2, which can be rewritten as y = -4x + 1.
Now, we can analyze these equations:
Analysis of the Equations:
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y = 5x + 2: This is a line with a slope of 5.
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y = -x + 1: This line has a slope of -1.
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y = -3x - 8: This line has a slope of -3.
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y = -3x + 3: This line has a slope of -3. Since it shares the same slope with the previous line, these two are parallel and do not intersect.
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y = -4x + 1: This line has a slope of -4.
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-2y = 8x - 2: Simplifying, we get y = -4x + 1, which is the same as the previous equation.
Number of Solutions:
- Different slopes - If lines have different slopes, they intersect at one point, resulting in one solution.
- Same slope, different y-intercepts - If two lines have the same slope but different y-intercepts, they are parallel and have no solution.
- Same slope and same y-intercept - If two lines have the same slope and the same y-intercept, they are the same line (infinite solutions).
Grouping:
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One solution:
- y = 5x + 2
- y = -x + 1
- y = -3x - 8
- y = -4x + 1 (when paired with y = -4x +1).
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No solution:
- y = -3x - 8 and y = -3x + 3 (parallel lines).
- y = -4x + 1 (the same as another, no additional solutions it contributed).
Therefore, the mappings of the equations to their respective number of solutions can be summarized as follows:
- Lines with one solution: Lines with different slopes.
- Lines with no solution: y = -3x - 8 and y = -3x + 3.
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