Algebraic methods: Substitution method and Elimination method.

Correct answer: A. (x=4,\ y=3)

From the first equation (y=15-3x); substituting gives (x+45-9x=13), so (x=4) and (y=3). Isolate the simpler variable in substitution.

Correct answer: B. (x=4,\ y=6)

Subtracting the equations gives (x-y=-2), so (x=y-2); substituting gives (y=6) and (x=4). Put the relation obtained by elimination into an original equation.

Correct answer: C. (3, 7)

The answer is C, (3,7). Substitution means replacing a variable in one equation by an expression equal to that variable. Since y=2x+1, put 2x+1 in place of y in x+y=10. This gives x+(2x+1)=10, so 3x+1=10, 3x=9, and x=3. Now substitute x=3 into y=2x+1: y=2(3)+1=7. Therefore the ordered pair is (x,y)=(3,7). Check both original equations: 3+7=10 and 7=2(3)+1, so the pair is correct. A gives 2+8=10 but 8 is not 2(2)+1. B and D also fail at least one equation. Always check both equations because satisfying only one is not enough.

Correct answer: B. 2

The requested expression appears directly when the equations are subtracted. Subtract the first equation, 5x + 6y = 142, from the second equation, 6x + 5y = 144. On the left, 6x - 5x = x and 5y - 6y = -y, so the result is x - y. On the right, 144 - 142 = 2. Therefore x - y = 2, and option B is correct. There is no need to solve separately for x and y because the coefficients are arranged so that subtraction produces exactly the requested expression. Options A, C, and D do not follow from the direct subtraction and would result from an arithmetic or sign error.

Correct answer: D. 79/7

The governing concept is the elimination method for a pair of linear equations. The coefficients of y are -7 and +7, so adding the equations removes y without requiring a preliminary multiplication: (4x - 7y) + (6x + 7y) = 9 + 71. Thus 10x = 80, giving x = 8. Substitute this value into 4x - 7y = 9: 4(8) - 7y = 9, so 32 - 7y = 9, hence 7y = 23 and y = 23/7. Therefore x + y = 8 + 23/7 = 56/7 + 23/7 = 79/7. Option D is correct. Option C may result from an addition error, while A and B do not satisfy the obtained values. Substitution confirms both original equations.

Correct answer: B. 3

The elimination method requires the y-terms to have equal magnitudes and opposite signs before the equations are added. The first equation contains −15y, whereas the second contains +5y. If the second equation is multiplied by k, its y-term becomes 5ky. To cancel −15y, we need 5k = 15, so k = 3. Indeed, multiplying 4x + 5y = 23 by 3 gives 12x + 15y = 69; adding it to 6x − 15y = 9 removes y exactly. Therefore option B is correct. Multipliers 2, 4, and 5 produce 10y, 20y, and 25y, respectively, so none of them directly eliminates the −15y term.

Correct answer: C. 4

The governing idea is elimination using equal coefficients of x. Both equations contain 13x, so subtract the first equation from the second: (13x + 4y) - (13x - 6y) = 48 - 8. The x-terms cancel, and carefully combining the y-terms gives 4y + 6y = 10y. Thus 10y = 40 and y = 4. A consistency check is useful: with y=4, the first equation gives 13x - 24 = 8, so 13x = 32; the second gives 13x + 16 = 48, also yielding 13x = 32. Therefore option C is correct. Values 2, 3, and 5 may result from subtracting the y-coefficients incorrectly or dividing 40 by the wrong number.

Correct answer: C. 5

The elimination method removes one variable by adding or subtracting equations whose coefficients are opposites. Here the y-coefficients are +3 and −3, so add the equations directly: (8x + 3y) + (5x − 3y) = 46 + 19. The y terms cancel, leaving 13x = 65. Dividing both sides by 13 gives x = 5. Substituting x = 5 into the first equation gives 40 + 3y = 46, so y = 2; the second equation checks as 25 − 6 = 19. Hence option C is correct. Options A, B and D do not satisfy the resulting equation 13x = 65, so they cannot be the solution.

Correct answer: C. 4

The governing method is elimination. Both equations have the same coefficient, 13, for x, so subtract the first equation from the second to cancel x: (13x + 9y) − (13x − 6y) = 61 − 1. The x terms disappear and the y terms give 15y = 60. Dividing both sides by 15 yields y = 4, so option C is correct. For verification, place y = 4 in the first equation: 13x − 24 = 1, hence 13x = 25 and x = 25/13. In the second equation, 13(25/13) + 9(4) = 25 + 36 = 61, so the pair is consistent. The other choices would make the difference of the two left sides 30, 45, or 75 instead of 60.

Correct answer: C. 7

Substitution is the most direct method because the first equation already expresses y in terms of x. Replace y in the second equation by 2x + 3: 5x - 2(2x + 3) = 1. Carefully distributing -2 gives 5x - 4x - 6 = 1, so x - 6 = 1 and x = 7. To verify, substitute x = 7 into the first equation: y = 2(7)+3 = 17. The second equation then becomes 5(7)-2(17)=35-34=1, exactly as required. Therefore option C is correct. The other choices can arise from a sign error while expanding -2(2x+3), but they fail the original system.