Algebraic methods: Substitution method and Elimination method.

Correct answer: B. (8,1)

From the second equation, y=x-7. Substituting this into 3x+y=25 gives 3x+(x-7)=25, so 4x=32 and x=8. Then y=8-7=1. Therefore, the solution is (8,1). Although (9,2) satisfies x-y=7, it gives 3x+y=29, so it is not correct. Exam tip: Always verify the ordered pair in both original equations.

Correct answer: A. \((5,2)\)

From the second equation, \(2x-y=8\), we get \(y=2x-8\). Substituting this into \(x+4y=13\) gives \(x+4(2x-8)=13\), so \(9x=45\) and hence \(x=5\). Then \(y=2(5)-8=2\). Therefore, the correct ordered pair is \((5,2)\). Option \((4,3)\) satisfies the first equation but not \(2x-y=8\). Exam tip: always substitute the obtained values into both original equations to verify the answer.

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

The direct answer is option C: (x,y)=(8,2). Add the equations because the y terms have opposite signs: (5x-y)+(x+y)=38+10. The y terms cancel, giving 6x=48, so x=8. Substitute into x+y=10: 8+y=10, hence y=2. Check in the first equation: 5(8)-2=40-2=38; in the second: 8+2=10. Option C is correct. Option A, (6,4), gives 5(6)-4=26, not 38. Option B, (7,3), gives 35-3=32, not 38. Option D, (9,1), gives 45-1=44, not 38. Although these pairs may have sums close to the second equation, they do not satisfy both equations. The useful elimination cue is to add equations when equal variable terms have opposite signs.

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

From the second equation, \(2x-y=6\), we get \(y=2x-6\). Substituting this into \(4x+3y=42\) gives \(4x+3(2x-6)=42\), so \(10x=60\) and hence \(x=6\). Then \(y=2(6)-6=6\). Therefore, the solution is \((6,6)\). For example, \((7,4)\) gives \(40\), not \(42\), in the first equation. Exam tip: always verify the obtained pair in both original equations.

Correct answer: B. (3)

The coefficient of (x) in the second equation is (1), so multiplying it by (3) gives (3x). In elimination, first make coefficients equal.

Correct answer: B. (9, 4)

This pair is solved conveniently by substitution. From x-y=5, we obtain x=y+5. Substituting this value of x into the second equation gives 2(y+5)+3y=30. On simplifying, 2y+10+3y=30, so 5y=20 and y=4. Therefore x=y+5=9. The ordered pair is (9,4), so option B is correct. Option A, (8,3), gives x-y=5 but produces 25 rather than 30 in the second equation. Option C gives x-y=5 but also fails the second equation, while option D does not even satisfy the first equation because 10-5=5 but 20+15=35. Checking both original equations confirms the answer.

Correct answer: C. (7, 1)

Use substitution from x+y=8, which gives x=8-y. Substituting into 3x-4y=17 produces 3(8-y)-4y=17. Thus 24-3y-4y=17, so 24-7y=17, -7y=-7, and y=1. Substitution back into x=8-y gives x=7. Hence the solution is (7,1), making option C correct. A quick verification is useful: 3(7)-4(1)=21-4=17 and 7+1=8, so both equations hold. Option A satisfies the second equation because 6+2=8, but its first-equation value is 18-8=10, not 17. Option B gives 15-12=3, and option D gives 12-16=-4; therefore these are distractors rather than solutions.

Correct answer: C. \((10,2)\)

From the second equation, \(3x-y=28\), we get \(y=3x-28\). Substituting this into the first equation gives \(6x+2(3x-28)=64\), or \(12x-56=64\). Hence, \(x=10\) and \(y=2\). Therefore, the correct solution is \((10,2)\). Exam tip: always verify the ordered pair in both original equations.

Correct answer: D. ( (10,3) )

Subtracting the second from the first gives (4y=12), so (y=3) and (x=10). Removing equal (2x) terms shortens the calculation.

Correct answer: A. (5, 6)

The elimination method is immediate because the y-coefficients are -1 and +1. Adding the two equations, (4x-y)+(2x+y)=14+16, cancels y and gives 6x=30. Therefore x=5. Substituting x=5 into 2x+y=16 gives 10+y=16, so y=6. The solution is (5,6), which is option A. Verification gives 4(5)-6=20-6=14 and 2(5)+6=10+6=16, confirming both equations. Option B reverses the values and gives 24-5=19 in the first equation. Option C gives 16-7=9, and option D gives 28-4=24; neither satisfies the system. The key governing idea is that adding equations with opposite coefficients eliminates one variable without changing the common solution.

Correct answer: B. (2, 6)

The term \(3x\) is the same in both equations. Subtracting the second equation from the first gives \(3y=18\), so \(y=6\). Substituting this into \(3x+2y=18\) gives \(3x+12=18\), hence \(x=2\). Therefore, the solution is \((2, 6)\). The pair \((6, 2)\) does not satisfy both equations. Exam tip: directly subtract equations when a variable has equal coefficients.

Correct answer: C. ( (9,6) )

Putting (x=2y-3) gives (2(2y-3)+y=24), so (y=6) and (x=9). Correct brackets prevent sign errors.

Correct answer: D. ( (7,1) )

Putting (15-2x) in place of (y) gives (x-(15-2x)=6), so (x=7) and (y=1). Always use brackets with a minus sign.

Correct answer: A. ( (8,4) )

Multiplying the first equation by (4) gives (2x+y=20). Solving it with (x-y=4) gives (x=8) and (y=4).

Correct answer: B. \(l=11,\ b=6\)

Divide the first equation \(2l+2b=34\) by 2 to get \(l+b=17\). Adding this to \(l-b=5\) gives \(2l=22\), so \(l=11\). Substituting into \(l-b=5\), we get \(11-b=5\), hence \(b=6\). Option C is not correct because its difference is 7, not 5. Exam tip: Simplify coefficients first, then use addition or subtraction to eliminate one variable.

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

Adding both equations gives (2x=16), so (x=8) and (y=5). In digit problems, do not interchange tens and units.

Correct answer: B. No solution

Multiplying the first equation by (2) should give left side (6x+4y) and right side (24). But the second right side is (25), so there is no solution.

Correct answer: D. ( (3,2) )

The direct answer is option D: \((x,y)=(3,2)\). From \(x+y=5\), write \(x=5-y\). Substitute into \(4x-5y=2\): \(4(5-y)-5y=2\). Expand: \(20-4y-5y=2\), so \(20-9y=2\), \(-9y=-18\), and \(y=2\). Therefore \(x=5-2=3\). Option A, \((4,1)\), gives \(4(4)-5(1)=11\), not 2. Option B, \((2,3)\), gives \(8-15=-7\). Option C, \((5,0)\), gives 20. Option D gives \(4(3)-5(2)=12-10=2\) and also has sum 5, so it is correct. The main danger is forgetting that the negative sign applies to both terms after substitution.

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

Subtracting the second equation from the first gives (5y=25), so (y=5) and (x=8). Eliminating equal (5x) terms is the right step.

Correct answer: C. (3, 9)

From the second equation, y = 4x - 3. Substituting this into the first equation gives 2x + 3(4x - 3) = 33, so 14x = 42 and x = 3. Then y = 4(3) - 3 = 9. Hence, the solution is (3, 9). Option (5, 6) satisfies the second equation, but it gives 2x + 3y = 28, not 33. Exam tip: Always verify the ordered pair in both original equations.

Correct answer: B. \((7,6)\)

From the second equation, \(3x-y=15\), so \(y=3x-15\). Substituting this into \(x+5y=37\) gives \(x+5(3x-15)=37\). Hence, \(16x=112\), so \(x=7\) and \(y=6\). Therefore, \((7,6)\) is the correct solution. For example, \((8,5)\) gives \(33\), not \(37\), in the first equation. Exam tip: always verify the ordered pair in both equations.

Correct answer: C. (4, 7)

Apply elimination by adding the equations. The y terms have coefficients +2 and -2, so (7x+2y)+(7x-2y)=42+14. This simplifies to 14x=56, giving x=4. Substitute x=4 into the first equation: 7(4)+2y=42, so 28+2y=42, 2y=14, and y=7. Therefore the solution is (4,7), option C. Substitution into the second equation also confirms it: 7(4)-2(7)=28-14=14. Option A gives 35+8=43 in the first equation, option B gives 42+6=48, and option D gives 21+16=37, so none of those pairs satisfies the first equation. The cancellation of opposite y terms is the essential elimination step.

Correct answer: D. \((9,10)\)

From the first equation \(2x-y=8\), we get \(y=2x-8\). Substituting this into the second equation gives \(3x+2(2x-8)=47\), so \(7x=63\) and \(x=9\). Then \(y=2(9)-8=10\). Hence, the solution is \((9,10)\). Option \((8,8)\) satisfies the first equation but not the second. Exam tip: always verify the ordered pair in both original equations.

Correct answer: C. ( (9,5) )

Subtracting the second from the first gives (3x=27), so (x=9) and (y=5). Remove equal (3y) terms to solve quickly.

Correct answer: C. \(x=5\)

From the first equation, \(y=4x-17\). Substituting this into the second equation gives \(2x+3(4x-17)=19\). Thus, \(14x-51=19\), so \(14x=70\) and \(x=5\). The nearby option \(x=4\) does not satisfy both equations together. Exam tip: after substitution, multiply 3 by both \(4x\) and \(-17\) while expanding the bracket.