Algebraic methods: Substitution method and Elimination method.

Correct answer: A. 143/29

Use elimination to solve the simultaneous equations before evaluating the requested expression. Multiply 5x + 2y = 28 by 3 to obtain 15x + 6y = 84. Multiply 7x - 3y = 19 by 2 to obtain 14x - 6y = 38. Adding gives 29x = 122, so x = 122/29. Substitution in the first equation gives y = 101/29. Hence 2x - y = 2(122/29) - 101/29 = 143/29. The originally supplied key and choices were inconsistent with the equations, so option A has been corrected to the verified value. The other numerical choices do not satisfy the calculated expression.

Correct answer: A. 210/23

First clear the denominators correctly. Multiplying x/3 + y/4 = 5 by 12 gives 4x + 3y = 60. Multiplying x/2 - y/5 = 3 by 10 gives 5x - 2y = 30. To eliminate y, multiply the first resulting equation by 2 and the second by 3: 8x + 6y = 120 and 15x - 6y = 90. Adding yields 23x = 210, so x = 210/23. Therefore option A is correct. The other options result from arithmetic or denominator-clearing errors. The originally supplied choices and key stated 270/23, but direct substitution confirms the corrected value: 210/23.

Correct answer: B. 3

The governing idea of the elimination method is to make the coefficients of one variable equal in magnitude and opposite in sign. In the first equation, the coefficient of y is -9, while in the second equation it is +3. Multiplying every term of the second equation by 3 changes it to 15x + 9y = 123. The terms -9y and +9y then cancel when the equations are added, leaving an equation in x alone. Therefore, 3 is the required multiplier, so option B is correct. Multiplying by 2 or 4 would produce 6y or 12y, which would not directly cancel -9y; multiplying by -3 would produce -9y, making the coefficients equal rather than opposite for addition.

Correct answer: D. 41

The governing concept is substitution for a pair of simultaneous linear equations. From x + y = 24, isolate x to obtain x = 24 - y. Substitute this expression into 3x - 2y = 37: 3(24 - y) - 2y = 37. Simplifying gives 72 - 3y - 2y = 37, so 72 - 5y = 37 and 5y = 35. Thus y = 7, and x = 24 - 7 = 17. The requested expression is 2x + y, not either original equation. Substitution of the values gives 2(17) + 7 = 34 + 7 = 41. Therefore option D is correct. The other choices do not result from the required expression and may arise from incomplete substitution or arithmetic mistakes.

Correct answer: C. There is no solution

For a pair of linear equations, if the coefficients of x and y are proportional but the constants are not proportional in the same way, the two lines are parallel and distinct. Multiplying the first equation by 2 gives 6x + 10y = 40. The second equation is 6x + 10y = 43, which has exactly the same left-hand side but a different right-hand side. A pair of numbers cannot make the same expression equal to both 40 and 43. Therefore the equations are inconsistent and have no common solution, so option C is correct. They do not have one or infinitely many solutions; those cases require non-proportional coefficients or complete proportionality, respectively.

Correct answer: C. x = 136/9, y = 100/9

Direct answer: C, x=136/9 and y=100/9. First remove the fractions from x/4+y/5=6 by multiplying every term by 20, the least common multiple of 4 and 5. This gives 5x+4y=120. From x−y=4, write x=y+4. Substitution gives 5(y+4)+4y=120, so 9y+20=120, 9y=100, and y=100/9. Then x=y+4=100/9+36/9=136/9. Check: x−y=36/9=4, and x/4+y/5=34/9+20/9=54/9=6. Thus C is correct. A has a different difference between x and y and fails the second equation. B also gives x−y=36/9 but its first-equation value is wrong. C satisfies both equations exactly. D has x−y=36/9 as well, but its fractional sum is too large. The safest habit is to clear denominators first and then substitute without changing signs.

Correct answer: C. 425/19

Use elimination to solve the simultaneous equations. Multiply 2x + 3y = 41 by 2 to obtain 4x + 6y = 82. Multiply 5x - 2y = 14 by 3 to obtain 15x - 6y = 42. Adding eliminates y and gives 19x = 124, so x = 124/19. Substitute this into the first equation: 2(124/19) + 3y = 41 = 779/19. Hence 3y = 531/19 and y = 177/19. The required expression is therefore 2x + y = 248/19 + 177/19 = 425/19. Option C is correct. The nearby alternatives result from errors in multiplying equations, adding constants, or combining the final fractions, and they do not give the value of the requested expression.

Correct answer: C. 200/7

The price relationships form a pair of simultaneous linear equations, which can be solved by elimination. To eliminate q, multiply 5p + 2q = 300 by 2, obtaining 10p + 4q = 600. Subtract 3p + 4q = 260 from this equation: 7p = 340, so p = 340/7. Substitute into 5p + 2q = 300: 5(340/7) + 2q = 300. Since 300 = 2100/7, we get 1700/7 + 2q = 2100/7, hence 2q = 400/7 and q = 200/7. Thus option C is correct. The corresponding p and q satisfy both original price equations; the other proposed q-values do not.

Correct answer: C. 240/11

Start with x - y = 6, which gives x = y + 6. To clear the denominators 5 and 6 in x/5 + y/6 = 7, multiply the entire equation by their least common multiple, 30. This produces 6x + 5y = 210. Substitution gives 6(y + 6) + 5y = 210, so 11y + 36 = 210 and 11y = 174. Hence y = 174/11. Now calculate x: x = y + 6 = 174/11 + 66/11 = 240/11. Therefore option C is correct. A check confirms x - y = 66/11 = 6, and substituting the values into the fractional equation gives 7. The other choices result from arithmetic or denominator-clearing errors.

Correct answer: B. 63

The key concept is translating place value and digit conditions into a pair of linear equations. Let the tens digit be x and the units digit be y. Then the two-digit number is 10x + y, while the sum of its digits is x + y. The conditions give x = y + 3 and 10x + y = 7(x + y). Expanding the second equation gives 10x + y = 7x + 7y, so 3x = 6y and x = 2y. Substituting into x = y + 3 gives 2y = y + 3, hence y = 3 and x = 6. The number is therefore 10(6) + 3 = 63, so option B is correct. Checking confirms that its digit sum is 9 and 7 × 9 = 63. The other choices fail at least one of the two stated conditions.

Correct answer: B. (x = 2, y = -12/5)

Use elimination by adding the two equations, because +5y and −5y cancel: (8x + 5y) + (4x − 5y) = 4 + 20. This gives 12x = 24, so x = 2. Substitute x = 2 into the first equation: 8(2) + 5y = 4, hence 16 + 5y = 4 and 5y = −12. Therefore y = −12/5, giving the ordered pair (2, −12/5). Option B is correct. Verification in the second equation gives 4(2) − 5(−12/5) = 8 + 12 = 20, so the pair satisfies both equations. The other options fail either the x-value obtained after elimination or direct substitution.

Correct answer: A. There is no solution

Compare the two equations using the elimination principle. Multiplying the first equation by 2 gives 8x + 14y = 62, while the second equation is 8x + 14y = 65. The left-hand sides are identical but the constants are different. Subtracting the two equations would therefore give 0 = 3, which is impossible. Geometrically, the equations represent distinct parallel lines, so they never intersect. Hence the pair is inconsistent and has no solution; option A is correct. Infinitely many solutions would require both equations to be exact multiples, including the constant term, and one or two solutions do not apply to a pair of linear equations in this form.

Correct answer: C. 119/15

This is a pair of simultaneous linear equations, and elimination is efficient because multiplying the second equation by 3 creates opposite y-coefficients. From 8x - 3y = 37, obtain 24x - 9y = 111. Add this to 6x + 9y = 117: the y-terms cancel, so 30x = 228 and x = 38/5. Substitute into the second equation: 8(38/5) - 3y = 37. Since 37 = 185/5, this gives 304/5 - 3y = 185/5, so -3y = -119/5 and y = 119/15. Hence option C is correct. Checking the first equation gives 228/5 + 357/5 = 585/5 = 117. The other fractions fail this simultaneous check.

Correct answer: C. 412/11

Eliminate a by making its coefficients equal. Multiply 6a + 5b = 460 by 2 to obtain 12a + 10b = 920, and multiply 4a + 7b = 444 by 3 to obtain 12a + 21b = 1332. Subtract the first new equation from the second: (12a + 21b) − (12a + 10b) = 1332 − 920. Hence 11b = 412 and b = 412/11. Therefore option C is correct. The coefficient of a cancels exactly, which is the governing idea of the elimination method. The nearby choices differ by 10 in the numerator and would arise from an arithmetic error in the subtraction; substituting b = 412/11 with the corresponding a confirms the system.

Correct answer: D. 32/3

The governing concept is elimination in a simultaneous pair of linear equations. The y-coefficients are -3 and +3, so adding the equations removes y: (7x - 3y) + (14x + 3y) = 20 + 64. This gives 21x = 84 and therefore x = 4. Substitute x=4 in the first equation: 7(4) - 3y = 20, so 28 - 3y = 20 and 3y = 8. Hence y=8/3. The requested expression is not y alone; calculate it carefully: 2x+y = 2(4)+8/3 = 8+8/3 = 24/3+8/3 = 32/3. Thus option D is correct. The integer distractors arise from ignoring the fractional value of y or evaluating the wrong expression.

Correct answer: A. 16 rupees

This word problem is modeled by a pair of linear equations. Let p be the price of one pen and n the price of one notebook. The statements give 3p + 2n = 86 and 2p + 3n = 89. To eliminate n, multiply the first equation by 3 and the second by 2: 9p + 6n = 258 and 4p + 6n = 178. Subtracting the second result from the first gives 5p = 80, so p = 16 rupees. Substituting into 2p + 3n = 89 gives 32 + 3n = 89, hence n = 19. A check gives 3(16)+2(19)=48+38=86. Therefore option A is correct; 19 is the notebook price, not the pen price.

Correct answer: D. 24

The relevant concept is elimination because the y-coefficients, +2 and -2, are opposites. Add the equations: (9x + 2y) + (3x - 2y) = 41 + (-5). The y-terms cancel and 12x = 36, so x = 3. Substitute this into 3x - 2y = -5: 3(3) - 2y = -5, giving 9 - 2y = -5, so -2y = -14 and y = 7. Now evaluate the requested expression, x + 3y = 3 + 3(7) = 3 + 21 = 24. Therefore option D is correct. Options A, B, and C are not the value of the required expression; they can arise from stopping after finding x or from an arithmetic error. Substitution verifies both equations.

Correct answer: A. 1/9

The answer is A, y=1/9. The y terms have opposite coefficients, −9y and +9y, so add the two equations: (4x−9y)+(8x+9y)=31+65. The y terms cancel and 12x=96, giving x=8. Substitute x=8 into the first equation: 4(8)−9y=31, so 32−9y=31. Subtract 32 from both sides: −9y=−1. Dividing by −9 gives y=1/9. Check in the second equation: 8(8)+9(1/9)=64+1=65. Thus A is correct. Values 2, 3 and 4 do not satisfy the original pair with x=8. When opposite terms appear, adding the equations is the quickest elimination step.

Correct answer: B. 18

The governing concept is clearing denominators before applying elimination. The least common multiple of 2 and 3 is 6, so multiply both equations by 6. They become 3x + 2y = 42 and 2x + 3y = 48. Adding these equations gives 5x + 5y = 90. Factor out 5: 5(x+y)=90, and divide by 5 to obtain x+y=18. It is unnecessary to calculate x and y separately because the requested sum appears directly. For verification, subtracting the equations gives x-y=-6; together with x+y=18 this gives x=6 and y=12, which satisfy the original fractional equations: 3+4=7 and 2+6=8. Hence option B is correct.

Correct answer: C. 3

The governing concept is solving a pair of simultaneous linear equations by substitution and using a known component of their common solution. Since x = 4, substitute it into the second equation: 3(4) - 2y = 7, so 12 - 2y = 7 and y = 5/2. Now place x = 4 and y = 5/2 in the first equation: 4a + 2(5/2) = 17. Therefore 4a + 5 = 17, so 4a = 12 and a = 3. A direct check gives 3(4) + 2(5/2) = 12 + 5 = 17 and 3(4) - 2(5/2) = 12 - 5 = 7. Thus option C is correct. Options A, B, and D would make the first equation's left side 9, 13, or 21, respectively, instead of 17.

Correct answer: B. 3

This question applies substitution to simultaneous linear equations containing an unknown coefficient. The common solution has y = 4. Substitute this value into the second equation: 4x - 3(4) = 2, giving 4x - 12 = 2 and therefore x = 7/2. Use x = 7/2 and y = 4 in the first equation: 4(7/2) + 4k = 26. Hence 14 + 4k = 26, so 4k = 12 and k = 3. Verification is important: the second equation gives 14 - 12 = 2, while the first gives 14 + 3(4) = 26. Therefore option B is correct. If k were 2, 4, or 5, the first equation would have left side 22, 30, or 34, respectively, so those distractors cannot satisfy the stated common solution.

Correct answer: C. 19 years

This is a pair of linear equations because the two unknown present ages occur only to the first power. Let Ram’s age be r and Shyam’s age be s. The first statement gives r − s = 4. After five years their ages become r + 5 and s + 5, so their sum is r + s + 10 = 44, or r + s = 34. Adding r − s = 4 and r + s = 34 eliminates s: 2r = 38, hence r = 19. Therefore option C is correct. Checking, Shyam is 15, Ram is 19, and after five years the ages are 24 and 20, whose sum is 44. Options A, B and D do not satisfy both conditions.

Correct answer: C. 15

The governing idea is to model the numbers of correct and wrong answers as two linear equations. Let c be the number of correct answers and w the number of wrong answers. Since all 20 questions are counted, c + w = 20. The marking rule gives 4c − w = 55, because each wrong answer contributes negative one mark. Adding the two equations eliminates w: 5c = 75, so c = 15. Then w = 20 − 15 = 5, and the score check is 15 × 4 − 5 × 1 = 60 − 5 = 55. Thus option C is correct. Options A, B and D would produce scores 47, 52 and 61 respectively when the remaining answers are treated as wrong, so they fail the given total.

Correct answer: A. 7

Use elimination first, then calculate the requested product. Adding the equations removes y because the coefficients are −12 and +12: 15x = −1 + 61 = 60. Hence x = 4. Substitute this into 5x − 12y = −1: 20 − 12y = −1, so −12y = −21 and y = 21/12 = 7/4. Therefore xy = 4 × 7/4 = 7. Option A is correct. A quick check in the second equation gives 10(4) + 12(7/4) = 40 + 21 = 61. The originally supplied options and marked answer did not contain 7, so the options have been corrected; calculating only x or y would not answer the question.

Correct answer: C. 13 cm

Let the rectangle’s length be l cm and breadth be b cm. The two statements translate into the linear equations l + b = 37 and l − b = 11. This is a standard word problem solved by elimination. Subtract the second equation from the first: (l + b) − (l − b) = 37 − 11. The l terms cancel, leaving 2b = 26, so b = 13 cm. Then l = b + 11 = 24 cm. Substitution verifies both conditions: 24 + 13 = 37 and 24 − 13 = 11. Therefore option C is correct. If the breadth were 11, 12, or 14 cm, the corresponding sums would be 33, 35, or 39 cm, not 37 cm.