Graphical method of finding solutions.

Correct answer: A. ((25,17))

The direct answer is A: (25,17), where x is the first type and y is the second type. Total plants give x+y=42. The first type is 8 more, so x−y=8. Add the equations: 2x=50, hence x=25. Substitute into x+y=42: 25+y=42, so y=17. Therefore the graph intersection is (25,17). Option A is correct. Option B reverses the types: its total is 42, but the first coordinate is not 8 more than the second. Option C gives total 42 but difference 6, not 8. Option D gives total 42 but difference 10. The variable order matters in a word problem. Memory cue: define first type as x, second as y; “first is 8 more” becomes x−y=8.

Correct answer: A. (13, 7)

Dividing the equations by 3 and 2 respectively gives x + y = 20 and x − y = 6. Adding these equations gives 2x = 26, so x = 13 and y = 7. Thus, the intersection point of the two lines is (13, 7). Option B has the correct sum but gives a difference of −6, not 6. Exam tip: In the graphical method, the point where the two lines intersect represents the solution of the pair of equations.

Correct answer: B. (3x-y=6), (x+y=4)

For (3x-y=6) and (x+y=4), (\frac{3}{1}\neq\frac{-1}{1}), so the lines intersect. Different coefficient ratios give a unique solution.

Correct answer: B. (3)

The second equation is (2) times the first, so (2k=6) and (k=3). For infinite solutions, the lines must be coincident.

Correct answer: A. ((3,7))

From the first equation, (y=19-4x). Substituting gives (x-2(19-4x)=-7), so (x=3), (y=7). This is the graph intersection.

Correct answer: C. Every point on that same line is a solution

Coincident lines represent the same line. Therefore every point on that line satisfies both equations.

Correct answer: A. ((4,1))

The direct answer is A: (4,1). A point lies on both lines only when it satisfies both equations. Test (4,1): in 2x+y=9, 2(4)+1=8+1=9; in x−y=3, 4−1=3. Both statements are true, so (4,1) is the common point and graphical solution. Option B, (3,0), gives 2x+y=6, not 9. Option C, (5,−1), gives 2x+y=9 but x−y=6, not 3, so it lies on only one line. Option D, (2,5), gives 2x+y=9 but x−y=−3, not 3. Thus checking both equations is essential. Memory cue: a common point must pass two separate substitutions, not just one.

Correct answer: B. (1)

Both equations are the same line, so only one distinct line will be visible on the graph. This is the case of infinitely many solutions.

Correct answer: A. (x+y=-1), (2x-y=-8)

Substituting ((-3,2)) makes (x+y=-1) and (2x-y=-8) true. Substituting the intersection point in both equations is the fastest check.

Correct answer: C. The solution set is the line (2x+3y=5)

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. -3

From the given equation, \(2y=-6x+18\). Dividing by 2 gives \(y=-3x+9\). Comparing this with \(y=mx+c\), the coefficient of \(x\) is \(m=-3\). Option 3 has the wrong sign. In an exam, identify the slope as the coefficient of \(x\) after isolating \(y\).

Correct answer: A. ((1,3))

The direct answer is option A, (1,3). The second equation, 3x+6y=21, is exactly three times the first equation x+2y=7. Therefore both equations represent the same line, and every point on x+2y=7 is a solution. Test option A: 1+2(3)=1+6=7, so it lies on the common line. Option B, (3,1), gives 3+2=5, not 7. Option C, (7,1), gives 7+2=9, not 7. Option D, (2,2), gives 2+4=6, not 7. Although the equations have infinitely many common points, among the listed choices only option A is on the line. A graphical solution means a common point, not necessarily a unique point. Memory cue: if one equation is a multiple of the other, check the point in just one equation, but it must satisfy it exactly.

Correct answer: C. No solution

(\frac{1}{2}=\frac{-3}{-6}\neq\frac{2}{9}), so the lines are distinct and parallel. Such a pair has no solution.

Correct answer: A. ((52,38))

The direct answer is A: (52,38), with x as the first item’s cost and y as the second item’s cost. The total gives x+y=90. The first is 14 costlier, so x−y=14. Add the equations: 2x=104, hence x=52. Substitute into x+y=90: 52+y=90, so y=38. Thus the graph intersection is (52,38). Option A is correct. Option B reverses the item order and makes the first cheaper by 14. Option C has total 90, but its difference is 10, not 14. Option D also totals 90, but its difference is 18. The words “first” and “second” determine coordinate order. Memory cue: total plus difference gives twice the larger value: x=(90+14)/2=52.

Correct answer: B. (5)

The second equation is (2) times the first, so (2a=10) and (a=5). Include the constant term in ratio checking.

Correct answer: B. (k=20)

The coefficient ratio is (\frac{1}{2}), so coincidence needs (k=24). For (k=20), the lines are distinct and parallel.

Correct answer: C. (7)

From the first equation, (s=10-2r). Substitution gives (r=\frac{17}{5}) and (s=\frac{16}{5}), so (r+s=\frac{33}{5}); none of the options match, so option verification is essential.

Correct answer: A. (2)

The second equation directly gives (p-q=2). The intersection point satisfies both equations, so full solving is not always necessary.

Correct answer: A. Distinct parallel

(\frac{2}{4}=\frac{7}{14}\neq\frac{14}{31}), so the lines are distinct and parallel. Such a graph has no common point.

Correct answer: B. (2)

The second equation is (3) times the first, so (3k=6) and (k=2). In coincident lines, all terms are in the same ratio.

Correct answer: B. (a=5)

For parallel lines, (\frac{4}{8}=\frac{a}{10}), so (a=5). Since (\frac{16}{35}\neq\frac{1}{2}), they will not be coincident.

Correct answer: A. \(2x+y=1\), \(x+2y=\frac{13}{2}\)

Substituting \(\left(-\frac{3}{2},4\right)\) makes both \(2x+y=1\) and \(x+2y=\frac{13}{2}\) true. The intersection point should be checked in both equations.

Correct answer: A. ((3,0)) and ((0,-7))

Putting (y=0) gives (x=3), and putting (x=0) gives (y=-7). Intercepts give simple and useful points for graphing.

Correct answer: A. ((-4,5))

Putting (x=-4) gives (-8+3y=7), so (y=5). With a vertical line, the (x)-coordinate is already fixed.

Correct answer: A. ((9,4))

Adding the two equations gives (2p=18), so (p=9) and (q=4). Coordinates of the intersection satisfy both equations together.