To determine the number of solutions for each system of equations, we can analyze the equations and see if they are parallel, identical, or intersecting lines.
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System: \(y = 5x + 7\)
\(3y - 15x = 18\)- Rewrite the second equation: \[ 3y = 15x + 18 \implies y = 5x + 6 \]
- Since the first equation is \(y = 5x + 7\) and now we have \(y = 5x + 6\), they are parallel lines (same slope, different intercepts).
Solution: no solutions.
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System: \(y = 3x + 6\)
\(y = -13x - 4\)- Both are in slope-intercept form with different slopes (3 and -13), so they will intersect at one point.
Solution: one solution.
- Both are in slope-intercept form with different slopes (3 and -13), so they will intersect at one point.
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System: \(x - 2y = 6\)
\(3x - 6y = 18\)- Rewrite the second equation: \[ 3x - 6y = 18 \implies x - 2y = 6 \]
- The second equation is a multiple of the first, so they represent the same line.
Solution: infinite solutions.
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System: \(y = 23x - 1\)
\(y = 23x - 2\)- These lines have the same slope (23) but different y-intercepts (-1 and -2), which means they are parallel.
Solution: no solutions.
- These lines have the same slope (23) but different y-intercepts (-1 and -2), which means they are parallel.
Summary of Solutions:
- \(y = 5x + 7\) and \(3y - 15x = 18\) - no solutions.
- \(y = 3x + 6\) and \(y = -13x - 4\) - one solution.
- \(x - 2y = 6\) and \(3x - 6y = 18\) - infinite solutions.
- \(y = 23x - 1\) and \(y = 23x - 2\) - no solutions.