The system given by 3x + y = 6 and 6x + 2y = 12 has

The system given by 3x + y = 6 and 6x + 2y = 12 has

• A. No solution
• B. One solution
• C. Infinitely many solutions
• D. The lines intersect at a single point

Answer: (C)

A system of linear equations can have infinitely many solutions when both equations describe the same line. Here, the second equation is exactly twice the first: multiplying 3x + y = 6 by 2 gives 6x + 2y = 12. That means both equations are the same line, so every point that satisfies the first equation also satisfies the second. Therefore there are infinitely many solutions. If they were parallel but not the same line, there would be no solution; if they were two different lines that cross, there would be one solution—but here they coincide, giving infinitely many.

A system of linear equations can have infinitely many solutions when both equations describe the same line. Here, the second equation is exactly twice the first: multiplying 3x + y = 6 by 2 gives 6x + 2y = 12. That means both equations are the same line, so every point that satisfies the first equation also satisfies the second. Therefore there are infinitely many solutions. If they were parallel but not the same line, there would be no solution; if they were two different lines that cross, there would be one solution—but here they coincide, giving infinitely many.