Let's analyze each system of equations:
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System: \( y = 5x + 7 \) and \( 3y - 15x = 18 \)
- The first equation is in slope-intercept form.
- The second equation can be rewritten as \( y = 5x + 6 \) by substituting \(3y = 15x + 18\) and then dividing by 3.
- Since both equations have the same slope (5), but different y-intercepts (7 and 6), they are parallel lines.
- Number of solutions: No solutions.
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System: \( y = \frac{2}{3}x - 1 \) and \( y = \frac{2}{3}x - 2 \)
- Both equations are in slope-intercept form.
- They have the same slope (\(\frac{2}{3}\)) but different y-intercepts (-1 and -2).
- Hence, these lines are also parallel and will never intersect.
- Number of solutions: No solutions.
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System: \( x - 2y = 6 \) and \( 3x - 6y = 18 \)
- The first equation can be rewritten as \( y = \frac{1}{2}x - 3 \).
- The second equation can be simplified to \( y = \frac{1}{2}x - 3 \).
- Both equations represent the same line since they are equivalent.
- Number of solutions: Infinite solutions.
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System: \( 3x + 6 = y \) and \( y = -\frac{3}{1}x - 4 \)
- The first equation can be rewritten as \( y = 3x + 6 \).
- The second equation simplifies to \( y = -3x - 4 \).
- These lines have different slopes (3 and -3) and will intersect at exactly one point.
- Number of solutions: One solution.
Summary:
- y = 5x + 7 and 3y - 15x = 18: No solutions
- y = \frac{2}{3}x - 1 and y = \frac{2}{3}x - 2: No solutions
- x - 2y = 6 and 3x - 6y = 18: Infinite solutions
- 3x + 6 = y and y = -\frac{3}{1}x - 4: One solution
1 answer