Drag each pair of equations to show if the system has no solution, one solution, or infinitely many solutions for the following equations: y=6x + 3 3y=18x + 9 y=x + 1 2y=2x + 3 y=3x + 2 y=5x y=4x – 1 4y=16x – 4 y=2x + 1 y=5x

Let’s analyze each pair of equations to determine the number of solutions:

1. Equations:

– ( y = 6x + 3 )

– ( 3y = 18x + 9 ) (This simplifies to ( y = 6x + 3 ))

Solution: Infinitely many solutions (the two equations are identical).

2. Equations:

– ( y = x + 1 )

– ( 2y = 2x + 3 ) (This simplifies to ( y = x + frac{3}{2} ))

Solution: No solution (the lines are parallel and never intersect).

3. Equations:

– ( y = 3x + 2 )

– ( y = 5x )

Solution: One solution (the lines intersect at a single point).

4. Equations:

– ( y = 4x – 1 )

– ( 4y = 16x – 4 ) (This simplifies to ( y = 4x – 1 ))

Solution: Infinitely many solutions (the two equations are identical).

5. Equations:

– ( y = 2x + 1 )

– ( y = 5x )

Solution: One solution (the lines intersect at a single point).

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