Conditions for solvability

The governing concept is the graphical classification of a pair of linear equations. If both graphs lie exactly on the same line, they are coincident lines. Every point on that common line satisfies the first equation and the second equation, so the pair has infinitely many common ordered pairs and therefore infinitely many solutions. Option B is correct. Intersecting lines meet at one point and produce one solution, whereas parallel distinct lines never meet and produce no solution. Two solutions are not possible for a pair of linear equations in two variables when the lines coincide; the common points are not limited to two. Algebraically, the equations are dependent and one equation is a scalar multiple of the other.

When coefficient ratios are different, the lines meet at one point. In exams, check the ratios of (a) and (b) first.

In this case, the lines are parallel and do not meet. In exams, if the (c) ratio differs, write no solution.

For the pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the condition a₁/a₂ = b₁/b₂ = c₁/c₂ means that all corresponding coefficients have the same ratio. Consequently, one equation is a constant multiple of the other, so both equations represent the same coincident line. Every point on this common line satisfies both equations, and therefore the system has infinitely many solutions. This is the dependent and consistent case. A unique solution occurs when a₁/a₂ is not equal to b₁/b₂, corresponding to intersecting lines. No solution occurs when a₁/a₂ = b₁/b₂ but this common ratio differs from c₁/c₂, corresponding to distinct parallel lines. Hence option C is unambiguous.

Here (2/4=3/6=(-5)/(-10)), so the lines are coincident. In exams, if all ratios match, write infinitely many solutions.

Here (1/2=1/2) but ((-4)/(-9)) is different, so there is no solution. Parallel lines form an inconsistent pair.

Here (3/6 \ne 2/5), so the lines intersect at one point. Different (a) and (b) ratios give a unique solution.

The intersection point satisfies both equations. In a graph, one intersection means a unique solution.

Parallel lines never meet, so there is no common point. In exams, connect parallel lines with no solution.

If two linear equations represent the same line, then every point on that line satisfies both equations. Therefore the system has infinitely many solutions.

Here (5/10=(-1)/(-2)) but (2/3) is different, so no solution exists. Equal first two ratios and a different third ratio give parallel lines.

All three ratios are equal, so both equations represent the same line. This is called a consistent dependent pair.

Here (2/3 \ne (-5)/1), so there is a unique solution. If the first two ratios differ, the third ratio need not be checked.

When all three ratios are equal, both lines lie on each other. These are called coincident lines.

The direct answer is option C: intersecting lines. For a pair \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\), if \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\), the coefficients are not proportional in the same way. In slope form, this means the two lines have different slopes. Lines with different slopes meet at exactly one point, so the pair has one unique solution. Option A, parallel, is wrong because parallel lines have equal slopes and usually \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\). Option B, coincident, is wrong because coincident lines have all three corresponding ratios equal. Option C is correct: unequal coefficient ratios indicate a unique intersection. Option D, inconsistent, describes the equation system's lack of a solution, which occurs for distinct parallel lines, not for intersecting lines. Memory cue: different slopes mean one meeting point and one solution.

Here (1/3=(-2)/(-6)=6/18), so the lines are coincident. Coincident lines have infinitely many common points.

Here, \(a_1/a_2=4/8=1/2\), \(b_1/b_2=1/2\), but \(c_1/c_2=(-3)/(-5)=3/5\). Thus, \(a_1/a_2=b_1/b_2\ne c_1/c_2\), so the two lines are distinct and parallel. Therefore, the pair has no solution and is inconsistent. Option C is incorrect because infinitely many solutions require all three ratios to be equal. Exam tip: if \(a_1/a_2=b_1/b_2\ne c_1/c_2\), the pair is inconsistent.

Here (6/12 \ne 7/15), so there is one unique solution. Different coefficient ratios indicate intersecting lines.

An inconsistent pair has no common solution for the two linear equations. Graphically, the two lines are distinct and parallel. One solution occurs when the lines intersect at one point, while infinitely many solutions occur when the lines coincide. A pair of linear equations cannot have exactly two solutions. Exam tip: remember that inconsistent always means “no solution.”

A pair of linear equations is called consistent when the two equations have at least one common solution. Geometrically, their graphs either meet at one point or lie on the same line. Thus, consistency does not require exactly one solution; a pair may also have infinitely many solutions.

For two lines, intersecting lines give one common point, while coincident lines give infinitely many common points. Both cases are consistent. Parallel distinct lines have no common point and are inconsistent. Therefore, option A is the complete and correct condition. The ratio statement in option D is not a general definition and cannot identify every consistent pair.

Write the equations as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. Here, a₁/a₂ = 9/3 = 3, b₁/b₂ = (−3)/(−1) = 3, and c₁/c₂ = 6/2 = 3. Since all three ratios are equal, both equations represent the same, or coincident, line. Every point on that line satisfies both equations, so the pair has infinitely many solutions. Options A and D describe different conditions.

Here (2/4=1/2) but (5/12) is different, so the lines are parallel. Distinct parallel lines have no solution.

The direct answer is option C: infinitely many solutions. The first equation is 3x-4y=8. Multiplying every term by 2 gives 6x-8y=16, which is exactly the second equation. Thus the two equations do not describe two different lines; they describe the same line. Every point on that line satisfies both equations, so there are infinitely many solutions. Option A, one solution, would require two different lines that intersect once. Option B, no solution, would require distinct parallel lines, usually with equal coefficient ratios but a different constant ratio. Option C is correct because the second equation is simply twice the first. Option D, inconsistent, is wrong because this system is consistent: it has solutions, in fact infinitely many. For example, from 3x-4y=8, choosing suitable values such as x=4,y=1 works, and many other points work too. Memory cue: multiplying an entire equation by a nonzero number does not change its line.

For the first equation, a₁=1 and b₁=3, while for the second equation, a₂=2 and b₂=5. Here, \(a₁/a₂=1/2\) and \(b₁/b₂=3/5\), so the ratios are unequal. Therefore, the two lines intersect at one point and the pair has a unique solution. In fact, the solution is \(x=-8, y=5\). Exam tip: If \(a₁/a₂ \ne b₁/b₂\), a pair of linear equations has a unique solution.

Coincident lines are two equations that represent exactly the same straight line. Every point on that line satisfies both equations, so the pair has infinitely many common solutions. Because at least one common solution exists, the pair is consistent. Because the equations do not determine one unique point, it is also called dependent.

This gives the classification “consistent and dependent,” which is option B. An inconsistent pair would represent distinct parallel lines and have no common solution. A consistent independent pair would represent intersecting lines and have exactly one solution. Therefore, neither A nor C describes coincident lines, and option B follows directly.