For infinitely many solutions of (6x+(h+1)y=12) and (15x+20y=30), what is (h)?

Correct answer: (h=7)

The direct answer is option C: h=7. For infinitely many solutions, the ratios of corresponding coefficients and constants must be equal: 6/15=(h+1)/20=12/30. Compute 6/15=2/5 and 12/30=2/5, so we need (h+1)/20=2/5. Multiply by 20: h+1=8, hence h=7. With this value, the second equation is 5/2 times the first: 6×5/2=15, 8×5/2=20, and 12×5/2=30, so both equations represent the same line. Option A, h=5, gives h+1=6 and breaks the required ratio. Option B gives 7, and option D gives 9; neither equals 8, so their lines are distinct and parallel rather than coincident. Memory cue: infinite solutions require all three coefficient ratios to match.