Graphical method and consistency MCQs

Q1. A stationery seller models two combo deals as the lines 9x + 1y = 21 and 7x + 1y = 19; how many ordered pairs (x, y) lie on both at the same time?

• A.infinitely many solutions (consistent and dependent; the lines coincide)
• B.exactly two solutions
• C.exactly one solution (consistent; the lines intersect at a single point)
• D.no solution (inconsistent; the lines are parallel and never meet)

Answer: C. exactly one solution (consistent; the lines intersect at a single point)

Compare the coefficient ratios. a1/a2 = 9/7 = 9/7 and b1/b2 = 1/1 = 1. Since 9/7 is not equal to 1, a1/a2 differs from b1/b2, so the graphs are two lines that cut each other at exactly one point. The pair is consistent with a unique solution.

Q2. During a science fair two fundraising teams record their rules as 5x + 2y = 3 and 10x + 4y = 6; what is the nature of the solution shown where the drawn lines meet?

• A.exactly one solution (consistent; the lines intersect at a single point)
• B.no solution (inconsistent; the lines are parallel and never meet)
• C.infinitely many solutions (consistent and dependent; the lines coincide)
• D.exactly two solutions

Answer: C. infinitely many solutions (consistent and dependent; the lines coincide)

Compare all three ratios. a1/a2 = 5/10 = 1/2, b1/b2 = 2/4 = 1/2, c1/c2 = 3/6 = 1/2. They are all equal to 1/2, so one equation is just a multiple of the other and both graphs are the same straight line (coincident). The pair is consistent and dependent with infinitely many solutions.

Q3. On a city map two straight avenues follow the equations 2x + 3y = 11 and 12x + 18y = 67; geometrically how do the two graphs relate and how many common points appear?

• A.no solution (inconsistent; the lines are parallel and never meet)
• B.infinitely many solutions (consistent and dependent; the lines coincide)
• C.exactly one solution (consistent; the lines intersect at a single point)
• D.exactly two solutions

Answer: A. no solution (inconsistent; the lines are parallel and never meet)

Compare all three ratios. a1/a2 = 2/12 = 1/6, b1/b2 = 3/18 = 1/6, c1/c2 = 11/67 = 11/67. Here a1/a2 = b1/b2 = 1/6 but c1/c2 = 11/67 is different, so the graphs are parallel lines that never meet. The pair is inconsistent and has no solution.