Algebraic methods: Substitution method and Elimination method.

Correct answer: A. 0

From the first equation, y=14-x. Substituting this in 3x-2y=7 gives 3x-2(14-x)=7, so 5x=35 and x=7. Then y=14-7=7. Therefore, x-y=7-7=0. The values 1, 2, and 3 would be possible only if x and y were unequal. Exam tip: after finding both variables, calculate the exact expression asked in the question separately.

Correct answer: C. (3)

From (2x-y=1), put (y=2x-1) in the first equation. In exams, isolate one variable clearly first.

Correct answer: B. (35)

From the second equation, put (y=2x-3), giving (x=5) and (y=7). In exams, recheck both values before finding (xy).

Correct answer: D. (29)

Use (y=17-4x) from the second equation to get (x=3), (y=5). In exams, calculate the asked expression after finding the solution.

Correct answer: B. (3)

Multiply the first equation by (2) and the second by (5) to eliminate (y). In exams, making equal coefficients is an easy method.

Correct answer: B. (15) and (8)

Form (x+y=23) and (x-y=7), then add them. In exams, adding sum-difference equations quickly gives one variable.

Correct answer: A. (74)

Let the tens digit be (x) and the units digit be (y), giving (x+y=11) and (9x-9y=27). In exams, write a two-digit number as (10x+y).

Correct answer: C. (11) cm

The direct answer is option C: the breadth is 11 cm. Let length be l and breadth be b. The perimeter formula is 2(l+b)=58, so l+b=29. The statement that length is 7 cm more gives l-b=7. Add these equations: 2l=36, so l=18. Then b=29-18=11 cm. Option C is correct, and checking gives perimeter 2(18+11)=58 and length difference 18-11=7. Option A, 9 cm, would give length 16 cm and perimeter 50 cm. Option B, 10 cm, would give length 17 cm and perimeter 54 cm. Option D, 12 cm, would give length 19 cm and perimeter 62 cm. The common mistake is to use l+b=58 instead of half the perimeter; remember that a rectangle has two lengths and two breadths.

Correct answer: A. \(83:81\)

Using elimination, multiply the first equation by 5 and the second by 7: \(20x+35y=290\) and \(42x-35y=42\). Adding them gives \(62x=332\), so \(x=166/31\). Substituting this into \(4x+7y=58\) gives \(y=162/31\). Therefore, \(x:y=166:162=83:81\). The option \(3:4\) is incorrect because it does not satisfy the given pair of equations. Exam tip: always reduce a ratio by dividing both terms by their greatest common factor.

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

Multiply the first equation by (2) and the second by (3) to eliminate (y). In exams, eliminating variables with opposite signs is faster.

Correct answer: D. (12)

Make the coefficients equal to eliminate (y) and get (x=5), (y=7). In exams, check the answer in both original equations.

Correct answer: A. (\frac{9}{14})

Let the numerator be (x) and denominator be (y), so (y=x+5) and (\frac{x+1}{y+1}=\frac{2}{3}). In exams, cross multiply when converting a fraction into an equation.

Correct answer: C. \(\frac{9}{2}\)

Substituting \(x=5\) in the second equation gives \(5+y=7\), so \(y=2\). Now substitute \(x=5\) and \(y=2\) in the first equation: \(2(5)+2k=19\). Thus, \(10+2k=19\), so \(2k=9\) and \(k=\frac{9}{2}\). Option \(5\) may result from solving \(2k=9\) incorrectly. In exams, verify the given solution in both equations.

Correct answer: C. (7)

Multiply the second equation by (2), add it to the first, and get (x=4), (y=5). In exams, first find the solution and then evaluate the expression.

Correct answer: B. (2)

Multiply the second equation by (4) to eliminate (x) and get (y=5). In exams, create equal coefficients instead of worrying about large numbers.

Correct answer: C. (10) km/h

Downstream speed is (12) and upstream speed is (8), so (b+s=12), (b-s=8). In exams, first find speeds from distance and time.

Correct answer: C. (17)

Form (x+y=32) and (10x+5y=245), then solve. In exams, money problems usually need separate equations for count and value.

Correct answer: C. 4

Given \(y=3\), substitute it into \(2x-y=5\): \(2x-3=5\), so \(2x=8\) and \(x=4\). Now put \(x=4\) and \(y=3\) in \(ax+3y=25\): \(4a+9=25\). Thus \(4a=16\), giving \(a=4\). Exam tip: First substitute the given variable into the equation that lets you find the other variable most easily.

Correct answer: C. (x=6, y=1)

Multiply the second equation by (2) and add it to the first. In exams, add equations when opposite signs appear.

Correct answer: C. (13)

Multiply the first equation by (2) and subtract from the second to get (y=4). In exams, subtract in the right direction to avoid sign errors.

Correct answer: A. (6) years

Form (f=4s+2) and (f+8=3(s+8)), then solve. In exams, add the same number of years to both ages for future age.

Correct answer: C. Infinitely many solutions

The second equation is (3) times the first, so both lines are the same. In exams, compare ratios to identify dependent equations.

Correct answer: C. Inconsistent

The ratio of variable coefficients is equal, but the ratio of constants is different. In exams, this means parallel lines and no solution.

Correct answer: B. (2)

Putting (x=4) in the second equation gives (y=7). Then the first equation gives (4k+14=16), so (k=\frac{1}{2}); check options carefully in exams.

Correct answer: B. (x=3, y=5)

From (2x+y=11), put (y=11-2x) in the first equation. In exams, combine all terms correctly after substitution.