Conditions for solvability

Correct answer: C. Infinitely many solutions

The first equation is (2) times the second, so both lines are coincident. Coincident lines have infinitely many solutions.

Correct answer: C. Infinitely many solutions

The second equation is (2) times the first so both lines are coincident. Coincident lines have infinitely many solutions.

Correct answer: C. Inconsistent

Here (3/6=4/8) but (11/25) is different so there is no solution. In exams equal first two ratios and a different third ratio mean inconsistent.

Correct answer: D. There is one unique solution

Here (1/2 \ne 7/9) so the lines intersect at one point. Different coefficient ratios give one unique solution.

Correct answer: A. Consistent and dependent

When all three ratios are equal both equations give the same line. This is called a consistent and dependent pair.

Correct answer: B. Parallel and distinct

Equal first two ratios give the same slope and a different third ratio keeps the lines separate. Such lines have no solution.

Correct answer: C. One unique solution

When the ratios of (a) and (b) are different the lines intersect. Intersecting lines give exactly one common point.

Correct answer: B. Infinitely many solutions

The second equation is (2) times the first so both lines are the same. All points on the same line are solutions.

Correct answer: A. No solution

Here (4/8=1/2) but (6/15) is different so the lines are parallel. Distinct parallel lines give no solution.

Correct answer: C. There are infinitely many solutions

The first equation is (3) times the second so both are the same line. In such questions spotting the common multiplier is the fastest method.

Correct answer: C. Distinct parallel lines

Comparing the coefficients, \(a_1/a_2=2/4=1/2\) and \(b_1/b_2=(-3)/(-6)=1/2\), but \(c_1/c_2=5/7\). Thus, \(a_1/a_2=b_1/b_2\ne c_1/c_2\), which is the condition for distinct parallel lines. Therefore, option C is correct. Option B is incorrect because coincident lines require all three ratios to be equal. Exam tip: If \(a_1/a_2=b_1/b_2\ne c_1/c_2\), the lines are distinct and parallel.

Correct answer: D. One unique solution

The direct answer is option D: one unique solution. For two linear equations \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\), different ratios \(a_1/a_2\) and \(b_1/b_2\) mean the lines have different slopes and meet once. Here \(a_1/a_2=7/5\), while \(b_1/b_2=2/4=1/2\). These are unequal, so the lines are neither parallel nor the same line. They intersect at exactly one point. Option A, infinitely many solutions, would require all three corresponding ratios to be equal. Option B, no solution, would describe distinct parallel lines, usually when the coefficient ratios are equal but the constant ratio differs. Option C, exactly two solutions, is impossible for two distinct linear equations in two variables: two lines can meet once, never, or coincide, but not at exactly two points. Option D is therefore correct. Exam cue: unequal coefficient ratios usually signal a unique solution.

Correct answer: B. No solution

Lines that never meet are parallel so there is no common point. No common point means no solution.

Correct answer: C. Infinitely many solutions

The governing concept is the graphical classification of a pair of linear equations. When two lines completely overlap, they are called coincident lines. Every point on one line also lies on the other, so every such point satisfies both equations simultaneously. A line contains infinitely many points; consequently, the pair has infinitely many ordered-pair solutions. Thus option C is correct. Distinct parallel lines never meet and give no solution, while intersecting lines share exactly one point and give one solution. Option D is not a general description of a coincident pair because the common points need not lie on y = 0; they can lie anywhere on the common line.

Correct answer: A. Consistent and independent

When two lines intersect at exactly one point, the pair has one unique solution. Therefore, it is a consistent and independent pair. An inconsistent pair is represented by distinct parallel lines, whereas a consistent and dependent pair has coincident lines and infinitely many solutions. Exam tip: One intersection point = one unique solution = consistent and independent.

Correct answer: D. Infinitely many solutions

The second equation is (3) times the first so both lines are coincident. Coincident lines give infinitely many solutions.

Correct answer: B. No solution

Here (9/3=6/2) but (21/8) is different so the lines are parallel. Distinct parallel lines have no solution.

Correct answer: A. One unique solution

The governing concept is the ratio test for a pair of linear equations, a x + b y + c = 0 and a₁x + b₁y + c₁ = 0. Here a/a₁ = 3/6 = 1/2, but b/b₁ = 5/11, and these ratios are unequal. Therefore the two lines have different slopes and intersect at exactly one point. A pair with one intersection has one unique solution, so option A is correct. Infinitely many solutions would require all three corresponding ratios to be equal, while no solution would require a/a₁ = b/b₁ but not c/c₁. Since the coefficient ratios already differ, neither of those cases applies.

Correct answer: C. Inconsistent

The direct answer is option C: inconsistent. A pair is called consistent if it has at least one common solution. If it has no common solution, it is inconsistent. On a graph, two linear equations with no common solution are represented by distinct parallel lines: they never meet, so no ordered pair \((x,y)\) satisfies both equations. Option A, consistent, is wrong because consistency means one or more common solutions. Option B, dependent, usually means the two equations represent the same line and therefore have infinitely many common solutions, not zero. Option C is correct because the phrase “no common solution” is exactly the definition of an inconsistent pair. Option D, independent, generally refers to equations that give a unique common solution, so it does not describe this case. Memory cue: no solution means inconsistent; one solution means independent and consistent; infinitely many solutions means dependent and consistent.

Correct answer: B. Same line

Infinitely many common solutions occur only when both equations represent the same line. In exams this can be written as coincident lines.

Correct answer: C. Lines intersecting at exactly one point

Exactly one common solution means that the graphs of the two linear equations share only one point. Therefore, the lines intersect at exactly one point. Coincident lines have infinitely many common points, whereas distinct parallel lines have no common point. Exam tip: one solution means intersecting lines, no solution means distinct parallel lines, and infinitely many solutions mean coincident lines.

Correct answer: C. 4

A pair of linear equations has no solution when the ratios of the coefficients of x and y are equal, but the ratio of the constant terms is different. Thus, k/8 = 3/6 gives k = 4. Also, 6/15 is not equal to 3/6, so the two lines are distinct and parallel. Therefore, 4 is correct. Exam tip: For no solution, check a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Correct answer: C. 4

For a pair of linear equations to have infinitely many solutions, a₁/a₂ = b₁/b₂ = c₁/c₂ must hold. Here, 5/10 = 9/18 = 1/2, so 2/b = 1/2, which gives b = 4. Therefore, option C is correct. Exam tip: For infinitely many solutions, all three corresponding ratios must be equal, not just two of them.

Correct answer: B. 4

For two linear equations to have infinitely many solutions, the condition is \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\). Here, \(\frac{3}{6}=\frac{12}{24}=\frac{1}{2}\), so \(\frac{k}{8}=\frac{1}{2}\), giving \(k=4\). Therefore, option B is correct. Exam tip: infinitely many solutions require all three corresponding ratios to be equal, not just two of them.

Correct answer: C. 3

For two linear equations to have no solution, the condition is \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\). Here, \(\frac{a}{6}=\frac{2}{4}=\frac{1}{2}\), while \(\frac{5}{13}\ne\frac{1}{2}\). Hence, \(a=3\). With the other options, the coefficient ratios are unequal, so the equations have a unique solution. Exam tip: first equate the ratios of the coefficients of x and y, then compare that ratio with the constants.