Conditions for solvability

Correct answer: C. Coincident

When all three ratios are equal, both equations represent the same line. Such a pair has infinitely many solutions.

Correct answer: C. It is inconsistent

If the first two ratios are equal and the third differs, the lines are distinct parallel. Therefore, there is no common solution.

Correct answer: C. 3

For two linear equations to have infinitely many solutions, the ratios of their corresponding coefficients and constant terms must be equal. Here, p/9 = 2/6 = 6/18 = 1/3. Therefore, p/9 = 1/3, giving p = 3, so option C is correct. For example, if p = 2, the ratio of the x-coefficients would be 2/9, not 1/3; hence the equations would not have infinitely many solutions. Exam tip: For infinitely many solutions, always check the equality of all three corresponding ratios.

Correct answer: B. \(q\ne4\)

For two linear equations to have a unique solution, the ratios of the corresponding coefficients must not be equal. Here, the determinant is \(2\times10-5\times q=20-5q\). A unique solution requires \(20-5q\ne0\), which gives \(q\ne4\). When \(q=4\), the two lines have the same slope but different constants, so they are parallel and have no solution. Exam tip: for a unique solution, check that the coefficient determinant is non-zero.

Correct answer: C. (12)

The direct answer is option C, k=12. Infinitely many solutions mean both equations represent the same line, so every coefficient and constant in one equation must have the same multiplier relative to the other. The second equation is x+3y=4. Multiplying it by 4 gives 4x+12y=16, which matches the first equation when k=12. Option A, 8, would give 4x+8y=16 and is not a scalar multiple of the second equation. Option B, 10, also fails because the y coefficient would not be 12. Option C gives exact equality of all three terms and therefore infinitely many solutions. Option D, 14, likewise fails the required multiplier condition. Memory cue: compare the fixed coefficients first; since 4x matches 4 times x and 16 matches 4 times 4, the middle coefficient must be 4 times 3, namely 12.

Correct answer: D. 15

Two linear equations have no solution when \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\). Here, \(\frac{6}{2}=3\) must equal \(\frac{a}{5}\), so \(\frac{a}{5}=3\) and hence \(a=15\). Also, \(\frac{30}{11}\ne3\), confirming that the two lines are distinct and parallel. Therefore, 15 is correct. Exam tip: equate the ratios of the coefficients first, then verify that the ratio of constants is different.

Correct answer: C. Lines are distinct parallel

For the two equations, \(a_1/a_2=8/2=4\) and \(b_1/b_2=12/3=4\), whereas \(c_1/c_2=20/6=10/3\). Thus, \(a_1/a_2=b_1/b_2\ne c_1/c_2\), so the lines are distinct and parallel; they neither intersect nor coincide. Exam tip: compare the three coefficient ratios—if the first two are equal but the third is different, the lines are distinct parallel lines.

Correct answer: A. Same line

Dividing every term of \(9x+6y=15\) by 3 gives \(3x+2y=5\). Hence, both equations represent the same line and have infinitely many common solutions. Option B is incorrect because distinct parallel lines require the coefficients of \(x\) and \(y\) to be proportional but the constants not to be proportional. Exam tip: if \(a_1/a_2=b_1/b_2=c_1/c_2\), the two lines are coincident, or the same line.

Correct answer: C. Consistent and dependent

The slope of a non-vertical line describes its steepness, and the y-intercept tells where it crosses the y-axis. If two lines have the same slope, they are parallel or identical. If their y-intercepts are also the same, they pass through the same point on the y-axis and have the same direction. Consequently, they are not merely parallel; they coincide completely.

Coincident lines have every point in common. Therefore the pair has infinitely many solutions and is called consistent and dependent. Option C is correct. An inconsistent pair has no common point, as with distinct parallel lines. A consistent independent pair has one common point, which does not apply here.

Correct answer: B. No solution

For a line written as y = mx + c, m is its slope and c is its y-intercept. If two lines have the same slope, they have the same direction. Different y-intercepts mean that they cross the y-axis at different points, so they are not the same line. Consequently, they are distinct parallel lines and cannot meet at any common point. The associated pair of linear equations therefore has no solution, making option B correct. A unique solution requires different slopes and an intersection. Infinitely many solutions occur only when both slope and intercept are equal, because then the two equations describe the same line. Thus option C and option D are not separate valid conclusions here; coincident lines require equal intercepts as well.

Correct answer: C. One unique solution

When two lines have different slopes, they cannot be parallel or coincident. Hence, they intersect at exactly one point, giving the pair of linear equations one unique solution. Therefore, option C is correct. Option A represents parallel lines, while option B represents coincident lines. Exam tip: different slopes always indicate a unique solution.

Correct answer: C. \(\frac{2}{6}=\frac{3}{9}=\frac{-11}{-33}\)

For the first equation, a₁=2, b₁=3 and c₁=-11; for the second, a₂=6, b₂=9 and c₂=-33. Thus, \(\frac{a_1}{a_2}=\frac{2}{6}=\frac{1}{3}\), \(\frac{b_1}{b_2}=\frac{3}{9}=\frac{1}{3}\), and \(\frac{c_1}{c_2}=\frac{-11}{-33}=\frac{1}{3}\). Therefore, option C gives the correct relation. The second equation is three times the first, so the two lines are coincident and the pair has infinitely many solutions. Exam tip: Equal values of all three ratios indicate infinitely many solutions.

Correct answer: A. 4/8 = (−5)/(−10) ≠ 7/9

For two equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the relation a₁/a₂ = b₁/b₂ ≠ c₁/c₂ indicates distinct parallel lines and no solution. Here, 4/8 = 1/2. Also, (−5)/(−10) = 1/2, so the first two ratios are equal. However, 7/9 is not equal to 1/2, so the constant-term ratio is different. Therefore the correct relation is 4/8 = (−5)/(−10) ≠ 7/9, which is option A. The equal first two ratios show equal direction or equal slope, while the unequal constant ratio shows that the lines are distinct rather than coincident. Option C would represent coincident lines and infinitely many solutions.

Correct answer: C. One unique solution

Here (5/10 \ne 6/9), so the lines intersect at one point. In this case, the third ratio need not be checked.

Correct answer: B. 3

For two linear equations to have no solution, the ratios of the coefficients of both variables must be equal, while the ratio of the constant terms must be different. Thus, \(\frac{a}{6}=\frac{4}{8}=\frac{1}{2}\), which gives \(a=3\). Also, \(\frac{10}{25}=\frac{2}{5}\), which is not equal to \(\frac{1}{2}\); therefore, the two lines are parallel and inconsistent. Exam tip: For ‘no solution,’ first check that the variable-coefficient ratios are equal and then verify that the constant-term ratio is different.

Correct answer: C. 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, \(\frac{3}{9}=\frac{15}{45}=\frac{1}{3}\). Therefore, \(\frac{b}{12}=\frac{1}{3}\), which gives \(b=4\). Hence, option C is correct. Exam tip: For infinitely many solutions, check that all three corresponding coefficient and constant ratios are equal.

Correct answer: B. \(p\ne12\)

A pair of linear equations has a unique solution when \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\). Here, \(\frac{8}{2}=4\) and \(\frac{p}{3}\), so the condition is \(4\ne\frac{p}{3}\), which gives \(p\ne12\). Therefore, option B is correct. Notice that when \(p=12\), the coefficient ratios become equal, but the constant ratio \(\frac{24}{7}\) is different; hence the equations have no solution rather than a unique solution. Exam tip: compare the ratios of the coefficients first to test for a unique solution.

Correct answer: D. 32

For infinitely many solutions, both linear equations must represent the same line. The coefficients on the left side of the second equation are twice those in the first equation: 10 = 2×5 and 4 = 2×2. Therefore, the constant on the right side must also be doubled, so n = 2×16 = 32. Hence, option D is correct. Exam tip: For infinitely many solutions, the ratios of the corresponding coefficients and constants must be equal.

Correct answer: B. t ≠ 9

For two linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the pair is inconsistent when a₁/a₂ = b₁/b₂ but c₁/c₂ is different. Here, compare 6x − 2y = 18 with 3x − y = t. The x-coefficient ratio is 6/3 = 2, and the y-coefficient ratio is (−2)/(−1) = 2. Therefore, the equations have the same direction. For them to be inconsistent, they must represent distinct parallel lines, so 18/t must not equal 2. Equality would give t = 9, making the second equation exactly half of the first and producing infinitely many solutions. Hence t ≠ 9, so option B is correct.

Correct answer: C. There are infinitely many solutions

Every term of the second equation is three times the corresponding term of the first: (9,6,24)=3(3,2,8) . Hence both equations represent the same line, so they have infinitely many common points and infinitely many solutions. Therefore, option C is correct. For exams, check the ratios: if a_1/a_2=b_1/b_2=c_1/c_2 , the pair has infinitely many solutions.

Correct answer: B. The first two ratios are equal, but the ratio of the constant terms is different

Here, \(a_1/a_2=4/1=4\) and \(b_1/b_2=8/2=4\), but \(c_1/c_2=12/5\). Thus, \(a_1/a_2=b_1/b_2\ne c_1/c_2\), which is the condition for an inconsistent pair of linear equations. In fact, the first equation simplifies to \(x+2y=3\), while the second is \(x+2y=5\); therefore, the lines are parallel and there is no solution. Exam tip: In the ratio test, always compare the constant-term ratio as well.

Correct answer: C. (7/2 \ne 3/1), so one unique solution

For equations written as \\(a_1x+b_1y+c_1=0\\) and \\(a_2x+b_2y+c_2=0\\), unequal ratios \\(a_1/a_2\\) and \\(b_1/b_2\\) mean that the two lines have different slopes. Lines with different slopes meet at exactly one point, so the pair has a unique solution. Equality of the first two ratios is not required here.

In this pair, \\(7/2=3.5\\), while \\(3/1=3\\). Hence \\(7/2\\ne3/1\\). The lines therefore have different directions and intersect once. The constant ratio is not needed to establish uniqueness after the first two ratios are found unequal. Thus option C correctly states that there is one unique solution.

Correct answer: C. Inconsistent

For the two equations, \(\frac{a_1}{a_2}=\frac{10}{2}=5\) and \(\frac{b_1}{b_2}=\frac{15}{3}=5\), but \(\frac{c_1}{c_2}=\frac{25}{8}\). Since \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\), the two lines are distinct and parallel. Hence, the pair is inconsistent and has no solution. Option B would be correct only if all three ratios were equal. Exam tip: Compare all three coefficient ratios in such questions.

Correct answer: C. Consistent and independent

For the given equations, \\(a_1/a_2=11/5\\) and \\(b_1/b_2=4/2=2\\). Since \\(11/5 \ne 2\\), the two lines intersect at one point and the pair has a unique solution. Therefore, it is a consistent and independent pair. For a consistent dependent pair, all three ratios are equal. Exam tip: Compare \\(a_1/a_2\\) and \\(b_1/b_2\\) first; if they are unequal, there is exactly one solution.

Correct answer: C. Infinitely many solutions

The second equation is (2) times the first. Therefore, both conditions give the same information and have infinitely many solutions.