The equations y = 3x + 1 and y = 3x + 4 represent parallel lines. How many solutions does the system have?

The equations y = 3x + 1 and y = 3x + 4 represent parallel lines. How many solutions does the system have?

• A. No solution
• B. One solution
• C. Infinitely many solutions
• D. The lines are the same

Answer: (A)

When two linear equations are written in slope-intercept form, if they share the same slope but have different y-intercepts, they are parallel and never cross. A solution to the system would be a point that lies on both lines, so it would have to satisfy both equations at once. Since these lines never meet, there is no such point. That means there are no solutions to the system. If they were the same line, there would be infinitely many solutions, because every point on the line would satisfy both equations. If they intersected at exactly one point, there would be one solution.

When two linear equations are written in slope-intercept form, if they share the same slope but have different y-intercepts, they are parallel and never cross. A solution to the system would be a point that lies on both lines, so it would have to satisfy both equations at once. Since these lines never meet, there is no such point. That means there are no solutions to the system.

If they were the same line, there would be infinitely many solutions, because every point on the line would satisfy both equations. If they intersected at exactly one point, there would be one solution.