Conditions for solvability

Compare the ratios of the corresponding coefficients and constants: \(\frac{12}{3}=\frac{4}{1}=4\), but \(\frac{16}{5}\neq 4\). Thus, \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\), so the two lines are distinct and parallel and have no common solution. Therefore, the pair is inconsistent. Exam tip: equal ratios for the first two terms but an unequal ratio for the constant terms indicate an inconsistent pair.

Write the equations in the standard form a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0. Their coefficient ratios are a₁/a₂=13/4 and b₁/b₂=2/5. These values are unequal, so the two lines do not have proportional coefficients and cannot be parallel or coincident. Equivalently, their slopes are -13/2 and -4/5, which are different. Two non-parallel lines in a plane intersect at exactly one point, so the pair is consistent and independent and has one unique solution. Thus option C is correct; options A, B, and D incorrectly describe parallel or coincident cases.

A line is determined by its slope and its y-intercept. If two lines have the same slope and the same y-intercept, they occupy exactly the same position on the coordinate plane. They are not two different lines; they are coincident lines. Consequently, every point on that common line satisfies both equations.

Since every point of the line is a common solution, the pair does not have just one or two solutions. It has infinitely many solutions. In the usual classification, this is called a consistent and dependent pair. “No solution” would describe distinct parallel lines, which have the same slope but different intercepts. Therefore option C, infinitely many solutions, is correct.

The governing graphical idea is that lines with the same slope are parallel, unless they also have the same intercept. Different intercepts mean the lines are distinct: they never meet at a common point. Therefore no ordered pair (x,y) can satisfy both equations at once, so the pair has no solution. A pair with no common solution is called inconsistent. Option A and option D refer to coincident lines, which have infinitely many common points, while option C refers to two non-parallel lines that intersect once. Thus the correct classification is inconsistent, option B.

The slope of a nonvertical line describes how much its y-value changes when its x-value changes. Lines with different slopes have different directions, so they cannot remain parallel or overlap completely. In a plane, two such lines meet at exactly one point. That point gives one ordered pair satisfying both linear equations. Therefore, a pair represented by lines with unequal slopes is called consistent and independent and has one unique solution.

To see the algebraic meaning, different slopes imply that the equations are not scalar multiples with equal coefficient ratios. The two lines therefore cannot describe the same line, and they cannot be distinct parallel lines, because parallel lines have equal slopes. They must intersect once. Hence the number of common solutions is one, which is option C. Infinite solutions correspond to coincident lines, while no solution corresponds to distinct parallel lines.

Compare the coefficients and constants in the two equations. Multiplying 5x + 3y = 2 by 3 gives 15x + 9y = 6, which is exactly the first equation. Thus the two equations are dependent and represent one coincident line rather than two different lines. Every point on this common line is a solution of the pair, so infinitely many solutions exist. A unique solution would require intersecting lines, and distinct parallel lines would have no common point.

For the coefficients, \(\frac{a_1}{a_2}=\frac{14}{2}=7\) and \(\frac{b_1}{b_2}=\frac{-7}{-1}=7\), but \(\frac{c_1}{c_2}=\frac{28}{3}\). Thus, \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\), which represents an inconsistent pair of equations and two distinct parallel lines. Therefore, there is no solution. Option B would be correct only if all three ratios were equal, while option A requires the first two ratios to be unequal. Exam tip: Compare the corresponding ratios in the same order.

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

Here (4/8=3/6) but (10/21) is different so there is no solution. Equal first two ratios and a different third ratio mean parallel lines.

Here (1/3 \ne 5/14) so the lines intersect at one point. Different coefficient ratios give a unique solution.

For a pair of linear equations, if \(a_1/a_2 \ne b_1/b_2\), the two lines have different slopes. Hence, they intersect at exactly one point, and the pair has a unique solution. Parallel lines require \(a_1/a_2 = b_1/b_2 \ne c_1/c_2\). Exam tip: compare the coefficient ratios first to determine the position of the lines and the number of solutions.

For a pair of linear equations, if \(a_1/a_2=b_1/b_2\ne c_1/c_2\), the two lines have the same slope but different y-intercepts. Hence, they are distinct parallel lines and never intersect, so the pair has no solution. Exam tip: remember that the pattern \(a_1/a_2=b_1/b_2\ne c_1/c_2\) represents distinct parallel lines and an inconsistent pair.

When all three ratios are equal both equations make the same line. This situation has infinitely many solutions.

Compare the two equations directly. Multiplying every term of 5x-y=9 by 2 gives 10x-2y=18, which is exactly the second equation. Thus both equations represent the same straight line rather than two different lines. Every point on this common line satisfies both equations, so there are infinitely many common solutions. Such a pair is called consistent and dependent: it is consistent because solutions exist, and dependent because one equation is a scalar multiple of the other. Option A would require one intersection, option B would indicate parallel distinct lines, and option D is not the relevant classification.

Dividing the first equation by 2 gives 3x+y=4, whereas the second equation is 3x+y=5. The ratios of the coefficients of x and y are equal, \(6/3=2/1\), but the ratio of the constant terms is different, \(8/5\ne 2\). Thus, the two lines are parallel and distinct, so the pair has no solution. Infinitely many solutions would require both equations to represent the same line. Exam tip: if \(a_1/a_2=b_1/b_2\ne c_1/c_2\), the pair has no solution.

The ratios of the coefficients of x and y are 2/5 and 7/9, respectively. Since 2/5 ≠ 7/9, the two lines intersect at exactly one point, giving one unique solution. Therefore, option A is correct. Exam tip: For a pair of linear equations, if a₁/a₂ ≠ b₁/b₂, the pair has a unique solution; hence it has neither no solution nor infinitely many solutions.

Distinct parallel lines never intersect, so they have no common point and no common solution. Therefore, the pair is called inconsistent. In contrast, coincident lines have infinitely many solutions. Exam tip: intersecting lines indicate one solution, coincident lines indicate infinitely many solutions, and distinct parallel lines indicate no solution.

In the graphical method, a solution of a pair of linear equations is represented by a common point of the two graphs. If the two lines intersect at exactly one point, only the coordinates of that point satisfy both equations simultaneously. Therefore the pair has exactly one solution, called a unique solution. In standard terminology, the pair is consistent and independent. No solution occurs when distinct lines are parallel, while infinitely many solutions occur when the two equations represent the same coincident line. “Inconsistent” is another name for the no-solution case, so it cannot describe a pair that intersects once. Hence option B is correct.

Every point on the same line satisfies both equations. Therefore such a pair has infinitely many solutions.

Multiply the first equation, 7x+4y=20, by 2. The result is 14x+8y=40, exactly the second equation. Hence both equations have the same set of solutions and their graphs are coincident, not merely parallel. Every point on this common line satisfies both equations, so the pair has infinitely many solutions. In the ratio test, a₁/a₂=7/14=1/2, b₁/b₂=4/8=1/2, and c₁/c₂=20/40=1/2; equality of all three ratios confirms a consistent and dependent pair. Therefore option C is correct; a unique solution would require unequal coefficient ratios.