Algebraic methods: Substitution method and Elimination method.

Correct answer: A. 12

Use elimination to solve the simultaneous equations and then evaluate the requested expression. Add 15x + 2y = 54 to 5x − 2y = 6. The y terms cancel, giving 20x = 60 and therefore x = 3. Substitute x = 3 into the first equation: 15(3) + 2y = 54, so 45 + 2y = 54 and 2y = 9. The question asks for x + 2y, not just y, so calculate 3 + 9 = 12. Hence option A is correct. Verification in the second equation gives 5(3) − 9 = 6. The other options differ from the exact value and cannot be obtained from the solved pair. The revised options are distinct and include the correct result.

Correct answer: A. x=8, y=11/7

The appropriate governing method is elimination because the coefficients of y are -7 and +7, which are opposites. Add the two equations term by term: (2x - 7y) + (4x + 7y) = 5 + 43. The y terms cancel, leaving 6x = 48, so x = 8. Substitute x = 8 into the first equation: 2(8) - 7y = 5. Thus 16 - 7y = 5, so 7y = 11 and y = 11/7. The ordered pair is therefore (8,11/7), which is option A. Checking in the second equation gives 4(8) + 7(11/7) = 32 + 11 = 43. The other listed pairs fail to satisfy both equations simultaneously, even if one equation may appear close for a particular pair.

Correct answer: B. 180/7

The governing idea is to remove denominators before using elimination. The least common multiple of 3 and 4 is 12, so multiply both equations by 12. The first becomes 4x + 3y = 84, and the second becomes 3x + 4y = 96. Add these equations: 7x + 7y = 180. Factor the left side to get 7(x + y) = 180, and divide by 7: x + y = 180/7. Therefore option B is correct. Finding x and y separately is unnecessary because the requested sum appears immediately after addition. The integer choices 24, 26, and 28 are not equal to 180/7 and could arise from careless denominator clearing, incorrect addition, or rounding. Exact fraction arithmetic is required.

Correct answer: C. 68/9

The governing concept is solving a linear system accurately after clearing decimal coefficients. Multiply 0.25x + y = 9 by 4 to obtain x + 4y = 36. Multiply x - 0.5y = 2 by 2 to obtain 2x - y = 4. From the second equation, x = 2 + y/2. Substitute this into the first: 2 + y/2 + 4y = 36. Since y/2 + 4y = 9y/2, we get 9y/2 = 34. Multiplying by 2 gives 9y = 68, hence y = 68/9. Therefore option C is correct. The decimal equations are equivalent to the multiplied equations, so no solution is lost. The values 6, 7, and 9 fail at least one original equation; rounding would also destroy the exact result.

Correct answer: B. 4

Answer: 4, so option B is correct. A common solution means that the same pair of values (x, y) must satisfy both equations. We know x = 5, so first use the second equation to find y: 2x - 3y = 5. Substituting x = 5 gives 2(5) - 3y = 5, or 10 - 3y = 5. Therefore 3y = 5 and y = 5/3. Now substitute x = 5 and y = 5/3 into the first equation: ax + 3y = 25 becomes 5a + 3(5/3) = 25. Since 3(5/3) = 5, we get 5a + 5 = 25, so 5a = 20 and a = 4. Check: 4(5) + 3(5/3) = 20 + 5 = 25, and 2(5) - 3(5/3) = 10 - 5 = 5. If a = 3, the first expression is 20; if a = 5, it is 30; and if a = 6, it is 35. Thus none of A, C, or D works. Memory cue: a common solution must be substituted into both equations, not just one.

Correct answer: C. 5

The key principle is that the stated solution must satisfy both equations. Begin with the first equation and use x = 5: 3(5) + 2y = 28. Thus 15 + 2y = 28, giving 2y = 13 and y = 13/2. Substitute x = 5 and y = 13/2 into the second equation: 5m - 2(13/2) = 12. The product 2(13/2) equals 13, so 5m - 13 = 12. Therefore 5m = 25 and m = 5. Hence option C is correct. A check gives 15 + 13 = 28 in the first equation and 5(5) - 13 = 12 in the second. Values 3, 4, and 6 would make the second left side 2, 7, and 17, respectively, rather than 12.

Correct answer: C. 24

Let c be the number of correct answers and w be the number of wrong answers. Since all 30 questions are counted, c + w = 30. The marking rule gives the second linear equation 5c - 2w = 108, because correct answers add marks and wrong answers subtract marks. From the first equation, w = 30 - c. Substitution gives 5c - 2(30 - c) = 108, so 5c - 60 + 2c = 108, 7c = 168, and c = 24. Thus option C is correct. Verification is immediate: 24 correct answers give 24 × 5 = 120 marks, while 6 wrong answers deduct 6 × 2 = 12 marks, leaving 120 - 12 = 108. The other options do not produce the stated score.

Correct answer: B. 307/35

Use elimination because multiplying the first equation by 2 creates -10y, which cancels the +10y in the second equation. Thus 2(6x - 5y = 8) gives 12x - 10y = 16. Add this to 9x + 10y = 83: 21x = 99, so x = 33/7. Substitute this into the first equation: 6(33/7) - 5y = 8. Since 6(33/7)=198/7 and 8=56/7, we obtain 5y=142/7 and y=142/35. Convert x to denominator 35: x=165/35. Therefore x+y=(165+142)/35=307/35, so option B is correct. The integer distractors 8, 9, and 10 are not the exact sum and do not satisfy the solved system.

Correct answer: C. 41/15

The coefficients of y are +5 and -5, so addition eliminates y immediately. Add the equations: (14x + 5y) + (7x - 5y) = 77 + (-7), giving 21x = 70 and x = 10/3. Substitute this value into 14x + 5y = 77: 14(10/3) + 5y = 77. Since 140/3 is subtracted from 231/3, 5y = 91/3, so y = 91/15. The requested expression is y - x. Convert x to fifteenths: 10/3 = 50/15. Thus y - x = 91/15 - 50/15 = 41/15. Therefore option C is correct. The integer choices 5, 6, and 8 cannot equal this exact fractional value and result from incomplete or incorrect elimination.

Correct answer: B. 15/7

Add the equations because the y coefficients are -7 and +7, so they cancel: (12x - 7y) + (4x + 7y) = 9 + 39. This gives 16x = 48 and therefore x = 3. Substitute x = 3 into the second equation: 4(3) + 7y = 39, so 12 + 7y = 39 and 7y = 27. Hence y = 27/7. Now evaluate the requested expression exactly: 2x - y = 2(3) - 27/7 = 6 - 27/7 = 42/7 - 27/7 = 15/7. Therefore option B is correct. The values 2, 4, and 5 may result from forgetting the coefficient 2, using the wrong sign, or rounding y, but none is the exact value.

Correct answer: C. ₹1400

This is a pair of linear equations solved by elimination. Let c be the price of one chair and t be the price of one table. The statements give 3c + 2t = 4900 and 2c + 3t = 5600. Multiply the first equation by 2 and the second by 3: 6c + 4t = 9800 and 6c + 9t = 16800. Subtracting eliminates c and gives 5t = 7000, so t = 1400. Therefore one table costs ₹1400, making option C correct. Substitution gives c = 700; then 3(700)+2(1400)=4900 and 2(700)+3(1400)=5600, confirming the result. The other options do not satisfy both original totals.