Conditions for solvability

The relevant principle is that equations whose all corresponding terms have the same multiplier describe the same line. Multiply 2x+3y=11 by 8. The result is 16x+24y=88, exactly the first equation. In ratio form, 16/2=24/3=88/11=8, so the coefficients of x and y and the constants are all proportional. The two equations therefore do not impose two different conditions; they are duplicate descriptions of one line. Every point satisfying one equation satisfies the other, giving infinitely many solutions. Such a pair is called consistent and dependent, so option A is correct. It is not inconsistent, because the lines are not distinct parallel lines. It is not independent, because there is not one unique intersection point; and “unsolvable” is not appropriate when infinitely many solutions exist.

Use the consistency criterion for two linear equations. The second equation is 2x+3y=11. Multiplying every term by 8 gives 16x+24y=88, which is precisely the first equation. Thus the coefficient ratios and constant ratio are equal: 16/2=24/3=88/11=8. Geometrically, both equations represent the same straight line, so they have infinitely many common points. Algebraically, there is only one independent equation for the two variables, leaving one variable free; this is why the pair is dependent. Because at least one common solution exists—in fact infinitely many—the pair is consistent as well. Therefore option A is the only correct answer. Inconsistent would mean no common solution, and independent would mean two non-proportional lines meeting at one point. Neither description fits these proportional equations.

The governing concept is identification of coincident equations using proportional coefficients. Multiply the second equation, 3x+5y=21, by 6. We obtain 18x+30y=126, exactly the first equation. Therefore 18/3=30/5=126/21=6. Equal ratios for both variable coefficients and the constants show that the two equations have identical standard form after scaling. Their graphs are the same line, not parallel separate lines, so every point on that line is a common solution. Hence the pair has infinitely many solutions and is consistent and dependent. Option A is correct. An inconsistent pair would have equal ratios for the variable coefficients but a different ratio for the constants. A consistent independent pair would have non-proportional coefficients and one unique intersection. Neither alternative describes the given equations.

Multiply the first equation by 2: 2(21x+8y=97) gives 42x+16y=194, which is exactly the second equation. Hence the two equations are dependent and represent the same straight line. Every point on this common line satisfies both equations, so the pair has infinitely many solutions. Option A and option D describe distinct parallel lines, while option C would require non-proportional coefficients.

The governing idea is the graphical interpretation of a pair of linear equations. A line in slope-intercept form is y = mx + c, where m is its slope and c is its y-intercept. If two lines have both the same m and the same c, their equations describe exactly the same geometric line. Every point on that common line satisfies both equations, and a line contains infinitely many points. Therefore, the pair has infinitely many common solutions, making option C correct. Equal slopes with different intercepts would produce distinct parallel lines and no solution. Different slopes would produce one intersection and hence a unique solution. The origin is not necessarily on the common line.