Graphical method of finding solutions.

Correct answer: C. No solution

(\frac{2}{4}=\frac{-5}{-10}\neq\frac{7}{16}), so the lines are distinct and parallel. Such a pair has no solution.

Correct answer: A. ((72,54))

The direct answer is option A, (72,54), where x is the cost of the first item and y is the cost of the second item. The total cost gives x+y=126. Since the first item is 18 costlier, x-y=18. Add the equations: 2x=144, so x=72. Substitute into x+y=126: 72+y=126, hence y=54. Therefore the two graph lines intersect at (72,54). Option A satisfies both conditions: 72+54=126 and 72-54=18. Option B, (54,72), makes the first item 18 cheaper, not costlier. Option C, (70,56), has total 126 but difference 14. Option D, (74,52), has total 126 but difference 22. The order matters because the question specifically calls the first item more expensive. Memory cue: for “first is more than second,” write x-y=the extra amount, not y-x.

Correct answer: C. (4)

The second equation is (3) times the first, so (3a=12) and (a=4). Include the constant term in ratio checking.

Correct answer: B. (k=40)

The coefficient ratio is (\frac{1}{2}), so coincidence needs (k=42). For (k=40), the lines are distinct and parallel.

Correct answer: B. 2

The intersection point of the two graphs satisfies both equations. Adding the equations gives \(5x=15\), so \(x=p=3\). Substituting this into \(x+y=4\) gives \(y=q=1\). Hence, \(p-q=3-1=2\), so option B is correct. Exam tip: The intersection of the graphs can be found efficiently by solving the two linear equations simultaneously.

Correct answer: B. (5)

For parallel lines, (\frac{7}{14}=\frac{b}{10}), so (b=5). Since (\frac{21}{50}\neq\frac{1}{2}), the lines are not coincident.

Correct answer: A. \(\left(\frac{61}{17},\frac{43}{17}\right)\)

Putting \(x=3y-4\) gives \(5(3y-4)+2y=23\), so \(y=\frac{43}{17}\) and \(x=\frac{61}{17}\). Fractional coordinates can also be correct graphical solutions.

Correct answer: D. ((0,4))

Substituting ((0,4)) gives (4\cdot0-9\cdot4=-36), not (36). Check every point in the equation before drawing the graph.

Correct answer: A. ((14,9))

Solving both equations gives (x=14) and (y=9). On the graph, the intersection of the two lines will be ((14,9)).

Correct answer: A. Distinct parallel

(\frac{4}{8}=\frac{9}{18}\neq\frac{36}{75}), so the lines are distinct and parallel. Distinct parallel lines have no common point.

Correct answer: C. (5)

The second equation is (3) times the first, so (3k=15) and (k=5). In coincident lines, all terms are in the same ratio.

Correct answer: C. (a=4)

For parallel lines, (\frac{3}{9}=\frac{a}{12}), so (a=4). Since (\frac{15}{47}\neq\frac{1}{3}), they will not be coincident.

Correct answer: A. \(2x+y=\frac{7}{2}\), \(x-2y=\frac{11}{2}\)

Substituting \(\left(\frac{5}{2},-\frac{3}{2}\right)\) makes both \(2x+y=\frac{7}{2}\) and \(x-2y=\frac{11}{2}\) true. Check the intersection point in both equations.

Correct answer: A. ((4,0)) and ((0,-9))

Putting (y=0) gives (x=4), and putting (x=0) gives (y=-9). Intercepts give convenient points for graphing.

Correct answer: A. ((-5,12))

Putting (x=-5) gives (-15+2y=9), so (y=12). With a vertical line, the (x)-coordinate is already fixed.

Correct answer: A. ((12,5))

Adding the two equations gives (2p=24), so (p=12) and (q=5). Coordinates of the intersection satisfy both equations together.

Correct answer: C. 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. The lines meet at one point

A consistent and independent pair has one unique solution. On a graph, it appears as one intersection point of two lines.

Correct answer: C. Coincident

If all three ratios are equal, both equations represent the same line. Hence the lines are coincident and give infinitely many solutions.

Correct answer: B. (3)

From the second equation, (y=4x-9). Substituting gives (6x+5(4x-9)=39), so (x=3). The graph intersection gives this (x)-coordinate.

Correct answer: B. \(\left(\frac{36}{11},\frac{32}{11}\right)\)

Putting \(y=7x-20\) in \(x+3y=12\) gives \(22x=72\), so \(x=\frac{36}{11}\) and \(y=\frac{32}{11}\). Fractional coordinates can also be correct graphical solutions.

Correct answer: B. Only ((10,0)) and ((0,12)) lie on the line

((10,0)) and ((0,12)) satisfy the equation, but ((5,6)) does not give (60). Check points before drawing the graph.

Correct answer: B. Inconsistent

(\frac{7}{14}=\frac{-3}{-6}\neq\frac{18}{41}), so the lines are distinct and parallel. An inconsistent pair has no solution.

Correct answer: A. ((4,3))

From the first equation, (y=4x-13). Substituting gives (x+2(4x-13)=14), so (x=4) and (y=3). The intersection point is the graphical solution.

Correct answer: B. (k=50)

The coefficient ratio is (\frac{1}{2}); coincidence needs (k=54). For (k=50), the lines will be distinct and parallel.