Algebraic methods: Substitution method and Elimination method.

From the first equation, y = 13 - 2x. Substituting this into the second equation gives x + 3(13 - 2x) = 19, so -5x = -20 and x = 4. Then y = 13 - 2(4) = 5. Hence, the solution is (4, 5). The close distractor (5, 4) gives 2x + y = 14, so it is not correct. Exam tip: Always verify the ordered pair in both original equations.

Putting (x=y+6) gives (\frac{y+6}{3}+y=6), so (y=3) and (x=9). Matching denominators reduces mistakes in fractions.

From the first equation, \(y=2x-7\). Substituting this into \(x+2y=16\) gives \(x+2(2x-7)=16\), so \(5x=30\) and hence \(x=6\). Then \(y=2(6)-7=5\). Therefore, the solution is \((6,5)\). For \((5,6)\), \(2x-y=4\), so it does not satisfy the first equation. Exam tip: always substitute the ordered pair into both original equations to verify the answer.

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.

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.

Adding both equations gives (6x=48), so (x=8) and (y=2). Add opposite (y) terms to eliminate quickly.

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.

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

Putting (x=y+5) gives (2(y+5)+3y=30), so (y=4) and (x=9). Combine like terms after substitution.

Putting (x=8-y) gives (3(8-y)-4y=17), so (y=1) and (x=7). Be careful while combining negative terms.

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.

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

Adding both equations gives (6x=30), so (x=5) and (y=6). Opposite (y) terms cancel when added.

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.

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

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.

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

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.

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

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.