Graphical method of finding solutions.

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

From the second equation, (x=3y-11). Substituting gives (9y-33+2y=25), so (y=5). Then (x=5), so the intersection is ((5,5)).

Correct answer: A. (2x+y=4), (x-y=-7)

Substituting ((-1,6)) makes (2x+y=4) and (x-y=-7) both true. The intersection point must lie on both lines.

Correct answer: C. Distinct parallel

(\frac{9}{3}=\frac{12}{4}\neq\frac{45}{20}), so the lines are distinct and parallel. Check the constant term ratio also.

Correct answer: C. (20)

The first equation must be (4) times the second, so (a=20). In coincident lines, all terms change by the same multiplier.

Correct answer: C. (21)

The second equation is (3) times the first, so (m=21). In a coincident line, the coefficient of (y) also changes in the same ratio.

Correct answer: B. (3)

From the first equation, (y=43-8x). Substituting gives (2x-3(43-8x)=-5), so (x=5) and (y=3). Hence the (y)-coordinate is (3).

Correct answer: A. ((45,29))

The direct answer is option A: the graphical solution is (45,29), where the first coordinate represents the first group and the second coordinate represents the second group. Let the numbers of students be x and y. The total gives x+y=74. Since the first group has 16 more students, x-y=16. Adding these equations gives 2x=90, so x=45. Substituting in x+y=74 gives 45+y=74, hence y=29. Therefore the two lines meet at (45,29), which is their common solution. Option A is correct because it satisfies both equations: 45+29=74 and 45-29=16. Option B gives (29,45); its total is 74, but the first group is 16 fewer, not 16 more, so it reverses the groups. Option C gives (44,30); its total is 74, but the difference is only 14, so it fails the second condition. Option D gives (46,28); its difference is 18, not 16, although its total is 74. In a graph, the intersection point represents the ordered pair satisfying both lines. Memory cue: add the equations to find the first variable, and remember that the first coordinate belongs to the first group.

Correct answer: A. (20, 13)

Dividing the first equation by 3 and the second by 4 gives x + y = 33 and x − y = 7. Adding these equations gives 2x = 40, so x = 20 and y = 13. Therefore, the intersection point of the two lines, and hence the graphical solution, is (20, 13). Option B reverses the coordinates; for (13, 20), x − y = −7, so it is incorrect. Exam tip: the point where the two graphs intersect represents the solution of the pair of linear equations.

Correct answer: C. (5)

The second equation is (4) times the first, so (4k=20) and (k=5). For infinite solutions, the lines must be coincident.

Correct answer: A. ((5,8))

From the first equation, (y=38-6x). Substituting gives (3x-2(38-6x)=-1), so (x=5), (y=8). This is the graph intersection.

Correct answer: A. No solution

Two distinct parallel lines never intersect, so they have no common point. Therefore, the pair has no solution and is called an inconsistent pair. Option B is incorrect because an intersection point exists only when the lines meet. Exam tip: distinct parallel lines represent zero solutions.

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

Substituting ((4,3)) does not give \(2x+5y=29\), so it is not correct; the true solution is \(\left(\frac{64}{17},\frac{73}{17}\right)\). Check a point in both equations.

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

Substituting ((4,3)) gives \(2x+5y=31\) but not \(3x-y=7\); the true common point is \(\left(\frac{66}{17},\frac{79}{17}\right)\). Verify in both equations before choosing.

Correct answer: A. (x+y=-3), (2x-y=-9)

Substituting ((-4,1)) makes (x+y=-3) and (2x-y=-9) both true. Substituting the intersection point in both equations is the fastest check.

Correct answer: C. The solution set is the line (7x+8y=56)

The second equation is (2) times the first, so both are the same line. The solution is all points on that line, not the whole plane.

Correct answer: B. -2

From the given equation, \(5y=-10x+25\), so \(y=-2x+5\). Comparing this with \(y=mx+c\), the coefficient of \(x\) gives the slope \(m=-2\); therefore, option B is correct. Option A misses the negative sign. In an exam, isolate \(y\) first to identify the slope directly.

Correct answer: A. ((4,6))

The second equation is (3) times the first, so every point on (3x+4y=36) is a solution. ((4,6)) lies on this line.

Correct answer: C. No solution

(\frac{3}{6}=\frac{-8}{-16}\neq\frac{11}{25}), so the lines are distinct and parallel. Such a pair has no solution.

Correct answer: A. ((88,66))

The direct answer is option A: the graphical solution is (88,66), where x is the price of the first item and y is the price of the second item. Let the prices be x and y. Their total is 154, so x+y=154. The first item is 22 costlier, so x-y=22. Add the equations: 2x=176, giving x=88. Put this into x+y=154: 88+y=154, so y=66. Thus the two graphs intersect at (88,66). Option A is correct because 88+66=154 and 88-66=22. Option B, (66,88), has the correct total but makes the first item 22 cheaper, not costlier. Option C, (87,67), totals 154 but the difference is 20, not 22. Option D, (90,64), totals 154 but the difference is 26. A graphical solution is the common point of the two lines, so both equations must be checked, not just the total. Memory cue: for “total” and “first is more,” use x+y and x-y, then add them.

Correct answer: C. (6)

The second equation is (3) times the first, so (3a=18) and (a=6). Include the constant term in ratio checking.

Correct answer: B. (k=88)

The coefficient ratio is (\frac{1}{2}), so coincidence needs (k=90). For (k=88), the lines are distinct and parallel.

Correct answer: C. 3

In the graphical method, the intersection point of the two lines represents the common solution of both equations. From \(x+y=5\), we get \(y=5-x\). Substituting this in \(5x-y=19\) gives \(5x-(5-x)=19\), so \(6x=24\) and \(x=4\). Consequently, \(y=1\), hence \((p,q)=(4,1)\) and \(p-q=4-1=3\). Therefore, option C is correct. Exam tip: the coordinates of the intersection of two lines give the simultaneous solution of their equations.

Correct answer: C. (6)

For parallel lines, (\frac{9}{27}=\frac{2}{m}), so (m=6). Since (\frac{18}{55}\neq\frac{1}{3}), the lines are not coincident.

Correct answer: A. \(\left(\frac{112}{19},\frac{41}{19}\right)\)

Elimination gives \(19x=112\), so \(x=\frac{112}{19}\) and \(y=\frac{41}{19}\). A graphical solution may also have fractional coordinates.

Correct answer: D. ((2,8))

Substituting ((2,8)) gives \(13\cdot2+4\cdot8=58\), not (52). Check every point in the equation before drawing the graph.